It is well-known that differential equations in physics with discontinuous material properties can be solved by applying boundary conditions on the interface between the different media for the adaption of the primary independent solutions for the media. When the transitions between the different media are sufficiently abrupt, but still smooth, one can often solve such problems approximately by generalized boundary conditions. We will demonstrate this here for one of the simplest second-order differential equations in physics, the Sturm-Liouville equation. This equation is of great importance because it is almost identical to the one-dimensional Schrödinger equation and some special problems of wave propagation in optics can be also reduced to this equation.

Generalized boundary conditions in optics taking into account transition layers between two different media were first derived by Drude [1] [2]. Three additional

parameters (of the kind of moments) l = ∫ 1 2     d z , p = ∫ 1 2     d z   ε , q = ∫ 1 2 d z ε ( ε

permittivity) play a role, where the integration goes over the boundaries of “thin” transition layers with thickness l which, however, is difficult to define exactly. In reflection and refraction formulae they often go at once into numerator and denominator and partially compensate the incorrectness connected with this approach. They were applied, for example, in the monographs about thin films of Mayer [3], Heavens [4], Rozenberg [5], and Vašiček [6].

A relatively new procedure to derive generalized boundary conditions is to involve generalized functions and, in particular, the step function and all its derivatives which are the delta function with its derivatives. This method is appropriate in classical electrodynamics for the derivation of generalized boundary conditions for media with spatial dispersion [7] (including, e.g., optical activity) and for correct derivations in presence of transition layers. A preliminary step to such derivations of boundary conditions was made in [8] [9]. The method was also developed in [10] [11] in application to the optics of media with transition layers, and the reflection was calculated in the first-order approximation of the generalized boundary conditions. In our paper [12] (see also Acknowledgement) we developed this method for light propagation in media with spatial dispersion (e.g., optical activity or gyrotropy) and transition layers and tried to represent the general procedure of continuing it to higher-order approximations. The main role played expansions of rapidly in infinity decreasing functions into series of delta functions and their derivatives which we call moment series because their coefficients are moments of the expanded function. The further development in the field of reflection and refraction not considered here can be seen, in particular, in the monograph of Lekner [13]. It is fairly comprehensive and represents many different aspects of reflection and refraction in optics and acoustics. In chap. 3 it uses integral invariants which are partially analogous to the parameters which we call moments of a function and in chap. 18 it considers chiral media including boundary conditions (a special case of spatial dispersion [7] which in the new edition of vol. 8 of Landau and Lifshits [14], chap. XII and also in our paper [12] appear under this last name with one of its special cases as natural optical activity).

In the present article we try to demonstrate the method of derivation of generalized boundary conditions in detail at a simple one-dimensional equation and as such we have chosen the Sturm-Liouville equation (e.g., special monographs [15] [16] [17] and many books of functional analysis) which is nearly the one-dimensional Schrödinger equation and to which many other equations can be reduced (e.g., article [18] in [19] ). We need for this purpose exact solutions for asymptotic potentials of the one-dimensional Schrödinger equation (e.g., [20] [21] [22] [23] and others). As said it is easier to make and to explain for this equation the derivations of generalized boundary conditions between two asymptotic potentials up to high-order approximations than for the full Maxwell equations with transition layers between two media and with spatial dispersion in the constitutive equations.

In Section 2 we discuss the restrictions for the applicability of the approach to potential problems for the one-dimensional Schrödinger equation. From Section 3 on we develop systematically the approach of the derivation of boundary conditions for the Sturm-Liouville equation between two asymptotic potentials with solutions supposed as known, in principle. In Section 4 we show how we can obtain finite approximations of the infinite system of boundary conditions and explain the special new situation when we go to the third-order approximation. From Section 5 on we apply this to the special case of two different but asymptotically constant potentials and calculate reflection and refraction problems in Section 6 and discuss in Sections 7 and 8 the appearance of discrete eigenstates which are in certain sense equivalent to total reflection and to surface waves for the three-dimensional Maxwell equations. In Section 9 we give a matrix form of the boundary condition which is well appropriate for a layered system where the output of one layer may serve as input of the next layer. The Appendix A gives a detailed derivation of the moment series in delta functions and their derivatives with moments of the expanded functions as coefficients and some related formulae for Generalized functions. Appendix B brings special examples for such expansions which are interesting for potential barriers and wells and for transition between potentials.

The one-dimensional time-dependent Schrödinger equation for a particle with mass m in a state | Ψ ( t ) 〉 in abstract representation possesses the form

i ℏ ∂ ∂ t | Ψ ( t ) 〉 = H | Ψ ( t ) 〉 ,   ( H ≡ T ( P ) + U ( Q ) ) , (2.1)

where H is the Hamilton operator which is the sum of the operator of kinetic energy T ( P )

T ( P ) = P 2 2 m , (2.2)

and of potential energy U ( Q ) . With P and Q we denote a pair of canonical coordinates which in quantum mechanics obey the commutation relations

[ Q , P ] ≡ Q P − P Q = i ℏ I , (2.3)

where I is the identity operator in representation (Hilbert) space. In representation of the (non-normalizable) eigenstates | q 〉 of the operator Q according to

Q | q 〉 = q | q 〉 ,   〈 q | Q = q 〈 q | ,   P | q 〉 = i ℏ ∂ ∂ q | q 〉 , 〈 q | P = − i ℏ ∂ ∂ q 〈 q | ,   〈 q ′ | q 〉 = δ ( q − q ′ ) , (2.4)

we find from (2.1) the time-dependent Schrödinger equation in q-representation¹

i ℏ ∂ ∂ t Ψ ( q , t ) = { − ℏ 2 2 m ∂ 2 ∂ q 2 + U ( q ) } Ψ ( q , t ) ,   Ψ ( q , t ) ≡ 〈 q | Ψ ( t ) 〉 . (2.5)

With the supposition

Ψ ( q , t ) = exp ( − i E t ℏ ) ψ 0 ( q , E ) , (2.6)

where E is the energy of the particle one obtains from (2.1) the time-independent Schrödinger equation

{ − ℏ 2 2 m ∂ 2 ∂ q 2 + U ( q ) } ψ 0 ( q , E ) = E ψ 0 ( q , E ) . (2.7)

It will be taken as our starting equation for the derivation of boundary conditions for the approximate solution of (one-dimensional) potential problems.

First, we make an estimation of the order of magnitudes with which we have to do in calculations with the Schrödinger Equation (2.7) if q is a spatial coordinate. Starting from Einstein’s equivalence of mass m and energy E = m c 2 de Broglie [24] conjectured that particles due to the dualism of waves and particles should possess also a wavelength which in analogy to Planck’s and Einstein’s relations for the energy E of particles and light is given by the momentum   p together with the dispersion equation and that according to hypothesis it is determined by (c light velocity, m 0 rest mass, E 0 rest energy)

E = ℏ ω ,     p = ℏ k ,   E 2 − c 2 p 2 = ( m 0 c 2 ) 2 ,   E 0 = m 0 c 2 . (2.8)

Herein ω is the (circular) frequency, k the wave vector and ℏ Planck’s action quantum (divided by 2 π )². From classical theory of a free particle it is

known and from relativistic theory for small velocities v 2 c 2 = c 2 p 2 E 0 2 ≤ 1 (or E ≈ E 0 ) it follows that its kinetic energy is

E kin ≡ E − E 0 ≈ p 2 2 m 0 = ℏ 2 k 2 2 m 0 , (2.9)

and from this dispersion equation resolved to | p | = ℏ | k | = ℏ 2 π λ = 2 m 0 E kin one obtains for the wavelength λ of the moving free particles with velocity v ≡ | v | and momentum p = m 0 v and for frequency ν

λ ≈ 2 π ℏ 2 m 0 E kin = ( h m 0 c ) m 0 c 2 2 E kin = h m 0 v ,   ν ≈ m 0 c 2 2 π ℏ . (2.10)

From this dispersion relations follows a phase velocity ω | k | larger than c (is not a signal velocity)

ω | k | ≡ λ ν ≈ c   E 0 2 E kin ≫ c , (2.11)

and from (2.8) the group velocity v ≡ ∂ ω ∂ k for the free particle waves equal to particle velocity v

∂ ω ∂ k = ∂ E ∂ p = c 2 p E = c c k ω ≈ c ( h m 0 c ) k 2 π = c v c k | k | ≪ c k | k |   . (2.12)

The quantity h m 0 c is the Compton length of the particle and m 0 c 2 the rest

energy of the particle. For an electron with mass m e the following values are well known from experiment

h m e c ≈ 2.43 × 10 − 10 cm ,   m e c 2 ≈ 0.511   MeV = 0.511 × 10 6 eV . (2.13)

Therefore, the de-Broglie wavelength for a free electron with kinetic energy E kin is approximately

λ e = 1 2 ( h m e c ) m e c 2 E kin ≈ 1.23 × 10 − 7 1   eV E kin   cm . (2.14)

For comparison, the first Bohr radius is a 0 = ℏ 2 m e e 2 ≈ 5.29 × 10 − 9 cm (e elementary charge), and the Rydberg constant R e (ionization potential) for infinitely high mass of the proton is R e = 1 2 α 2 m e c 2 ≈ 13.6   eV ; ( α ≈ 1 / 137 ) . For energies of the electron up to the order of a few eV the wavelengths are large in comparison to the first Bohr radius. Furthermore, 1 eV is equivalent to energy κ B T corresponding to temperature T of 11,606 K ( κ B Boltzmann constant).

The approximate boundary conditions which we derive in the following for the Schrödinger Equation (2.7) are applicable if the wavelengths λ e (more generally, λ ) are large in comparison to the order of lengths where the potential function U ( q ) changes essentially. Oppositely, if the wave length of the particle is small in comparison to the order of lengths where the changes of the potential function play a role then another well-known approximation, the WKB-method³ or quasi-classical approximation, can be applied (e.g., [20] ). For high kinetic energies E kin of the order of the rest energy of the electron of about 0.5 MeV and higher the (non-relativistic) Schrödinger equation is no more relevant.

We usually consider in this paper Equation (2.7) in an infinite interval. If U ( q ) in (2.7) is larger than U ( ± ∞ ) then one speaks about a potential barrier and if it is smaller than U ( ± ∞ ) then about a potential well. Both and more general cases which are not to classify in such simple way are included into the following calculations.

With the substitutions which preserve the coordinate as a quantity with dimension of a length

q ≡ z ,   U ( q ) ≡ ℏ 2 2 m u ( z ) ,   E ≡ ℏ 2 2 m λ ,   ψ 0 ( q , E ) ≡ ψ ( z , λ ) , (2.15)

Equation (2.7) can be transformed to the more neutrally looking form ( λ now spectral parameter)

{ − ∂ 2 ∂ z 2 + u ( z ) − λ } ψ ( z , λ ) = 0. (2.16)

This is the standard form of the Sturm-Liouville equation to which also some other differential equations of physics can be reduced as mentioned in the Introduction. An important example is the wave propagation in one-dimensional inhomogeneous media which is described by Landau and Lifshits [14] (chap. X, section 88). The case of polarization of the electric field E ( z ; k ⊥ , ω ) perpendicular to the incidence plane is more directly connected with Equation (2.16)

(substitutions: u ( z ) → − ω 2 c 2 ε ( z ; ω ) ; λ → − k ⊥ 2 ) than the case of polarization in

the incidence plane which requires more subtile substitutions. One has to remove then the arising first-order derivatives in the equation [14] [15] [16]. Central-symmetric potentials in two and three dimensions with transition regions can be dealt with as quasi-one-dimensional cases of the Schrödinger equation and can be transformed to a Sturm-Liouville equation with additional terms to the potential function (effective potentials) and possible applications to scattering problems. We cannot deal with this here.

If ψ 1 ( z , λ ) and ψ 2 ( z , λ ) are two solutions of the Sturm-Liouville equation to the same potential u ( z ) and the same parameter λ then in known way one derives the conservation theorem

∂ ∂ z ( ψ 1 ( z , λ ) ∂ ∂ z ψ 2 ( z , λ ) − ψ 2 ( z , λ ) ∂ ∂ z ψ 1 ( z , λ ) ) = 0. (2.17)

The same is true if the second solution is the complex-conjugate solution of the first solution for real parameter λ that means (we add then a factor “I” to get a real-valued conservation quantity)

i ∂ ∂ z ( ψ ( z , λ ) ∂ ∂ z ψ * ( z , λ ) − ψ * ( z , λ ) ∂ ∂ z ψ ( z , λ ) ) = 0,   ( λ = λ * ) . (2.18)

Other general properties we find, for example, in [15] [16] [17] [18].

We derive in this Section at an example generalized boundary conditions using the theory of Generalized functions (e.g., [25] [26] [27] [28], see also Appendix A). As starting point of the further treatment we take the following ordinary second-order differential Equation (2.16)

( L − λ ) ψ ( z , λ ) = 0,   L ≡ − ∂ 2 ∂ z 2 + u ( z ) , (3.1)

which in this form is commonly called the Sturm-Liouville equation. The operator L is the Sturm-Liouville operator and λ its eigenvalues. The eigensolutions ψ = ψ ( z , λ ) depend on λ as parameter.

We assume that the real-valued potential u ( z ) is defined on the entire real z-axis ( − ∞ < z < + ∞ ). The complex-valued solutions ψ ( z , λ ) are considered in dependence on the spectral parameter λ which can take on arbitrary complex values. As a linear second-order differential equation, the Sturm-Liouville equation possesses two linearly independent solutions for each complex value of the spectral parameter λ . As mentioned the Sturm-Liouville Equation (3.1) can also be seen as the eigenvalue problem of the Sturm-Liouville operator L. Then the real eigenvalues λ of the discrete and of the continuous spectrum may be separated from the totality of complex values of the spectral parameter λ by appropriate normalization conditions of the solutions. The one-dimensional Schrödinger equation in energy representation is identical with (3.1) after simple linear substitutions of the variables (2.15), where the energy becomes proportional to λ . It is by no means a disadvantage to assume that u ( z ) and ψ ( z , λ ) are Generalized functions and that the derivatives in (2.1) are also Generalized functions because this includes or generalizes all classical cases and the point-wise definition of functions is perhaps an unnecessary (and may be unnatural) concept in physics.

For the purpose of the illustration only the asymptotic potentials u ∓ ( z ) for z → ± ∞ are chosen as constant potentials u − ( z ) = u ( − ∞ ) , u + ( z ) = u ( + ∞ ) . The first partial figure shows the genuine potential and below this its approximation by a step function. The upper partial figure on the right-hand side shows in hatched way the error which we make by the approximation and below this the difference of potential and its approximation. The derivations show that the parts to the left and to the right of the chosen boundary are, in general, separately to deal with.

We assume now that the potential u ( z ) can be approximated for z → − ∞ by u − ( z ) and for z → + ∞ by u + ( z ) and that the totality of solutions of the following two Sturm-Liouville equations

0 = { − ∂ 2 ∂ z 2 + u ∓ ( z ) − λ } ψ ∓ ( z , λ ) , (3.2)

is known. The asymptotic potentials u − ( z ) and u + ( z ) have to be of such kind that the differences u ( z ) − u − ( z ) and u ( z ) − u + ( z ) become rapidly decreasing functions for z → + ∞ or z → − ∞ , respectively. The rapid decrease of functions in infinity means that they decrease in infinity more rapidly than any power function | z | n with arbitrary negative numbers n or, more generally expressed, that they can be defined as linear continuous functionals over the space of smooth and moderately increasing functions, especially, the polynomials. The effective lengths of the transition regions should be small compared with the characteristic lengths of changes in the solutions for the asymptotic potentials u + ( z ) and u − ( z ) (e.g., wavelengths). Usually, u + ( z ) and u − ( z ) are chosen as such potentials for which we know the totality of solutions in form of elementary or well-known higher transcendental functions as, for instance, for constant potential (Exponential or Trigonometric functions), linear potential (Airy functions) and quadratic potential (Hermite functions or more generally Parabolic-cylinder functions).

An exact boundary between the two regions of z → − ∞ and z → + ∞ is not defined and for approximations of the derived boundary conditions a “good” choice of the reference point z₀ is necessary. A “good” choice in our case is z 0 = 0 and is drawn in Figure 1. The partial Figures illustrate the influence of the reference point z₀ onto the hatched correction functions.

There are also some other well-known special potentials with known exact solutions of the Sturm-Liouville equation as, for instance, the reflection-less Bargmann potentials with their importance in soliton theory of the Korteweg-de Vries equation (the famous Hyperbolic Secans square as the simplest nontrivial one). The solutions for the most common potentials can be found, e.g., in Vol. III (Quantum mechanics) of Landau and Lifshitz [20] (chap. 3) and furthermore in [21] (chap. 12.3) and in [22]. The derivation of relations between the asymptotic solutions ψ + ( z , λ ) and ψ − ( z , λ ) belongs to reflection and refraction or scattering problems of the Sturm-Liouville equation. By means of the generalized boundary conditions which we will derive now, one can obtain approximate solutions of the direct scattering problem of Equation (3.1).

The almost smooth transition of the potential u ( z ) from the asymptotic potential u − ( z ) for z → − ∞ to the asymptotic potential u + ( z ) for z → + ∞ has to be concentrated mainly on the neighborhood of a value z₀ of the variablez (Figure 1) which is not exactly determined and can be varied in certain way. We have then to choose a suitable point

z = z 0 , (3.3)

as the boundary or the reference point of the expansions in the proposed method.

A small variation of the reference point z₀ should not destroy the whole approximation (Figure 2). The special case of a potential barrier (Figure 3) or a potential well (by taking instead negative values of that for the barrier) (Figure 4) we consider more detailed in Sections 6 and 7.

Next we represent the solution ψ ( z , λ ) of Equation (2.16) after separation of

the asymptotic solutions ψ ∓ ( z , λ ) to the asymptotic potentials u ∓ ( z ) by means of Heaviside’s step function θ ( z ) in the form ( θ ( z ) = 0 , ( z < 0 ) , θ ( z ) = 1 , ( z > 0 ) )

ψ ( z , λ ) = ψ − ( z , λ ) θ ( z 0 − z ) + ψ + ( z , λ ) θ ( z − z 0 ) + ψ ′ ( z 0 ; z , λ ) , (3.4)

where ψ ′ ( z 0 ; z , λ ) depending in addition on the reference point z₀ is a small deviation from the sum of the two asymptotic solutions in the transition region and turns out to be a function of z rapidly decreasing for z → ± ∞ and mainly concentrated on the neighborhood of the point z₀. Below we make a moment-series expansion of ψ ′ ( z 0 ; z , λ ) into a delta and its derivatives (see Appendix A).

Inserting (3.4) into the Sturm-Liouville Equation (2.16) the following equation results

0 = { − ∂ 2 ∂ z 2 + u ( z ) − λ } ( ψ − ( z , λ ) θ ( z 0 − z ) + ψ + ( z , λ ) θ ( z − z 0 ) + ψ ′ ( z 0 ; z , λ ) ) , (3.5)

which we have to solve. For this purpose we apply Generalized functions and an expansion of rapidly in infinity decreasing functions into delta function and its derivatives called moment series which we explain in Appendix A.

Using the Leibniz rule for the n-fold differentiation of a product of functions f ( z ) g ( z )

∂ n ∂ z n f ( z ) g ( z ) = ∑ k = 0 n n ! ( n − k ) ! k ! f ( n − k ) ( z ) g ( k ) ( z ) , (3.6)

we can make the following identical transformation of a part of the right-hand side of (3.5) which brings the Heaviside step functions in front of the derivatives

− ∂ 2 ∂ z 2 ( ψ − ( z , λ ) θ ( z 0 − z ) + ψ + ( z , λ ) θ ( z − z 0 ) ) = − θ ( z 0 − z ) ∂ 2 ∂ z 2 ψ − ( z , λ ) − θ ( z − z 0 ) ∂ 2 ∂ z 2 ψ + ( z , λ )     + 2 δ ( z − z 0 ) ( ∂ ∂ z ψ − ( z , λ ) − ∂ ∂ z ψ + ( z , λ ) )     + δ ( 1 ) ( z − z 0 ) ( ψ − ( z , λ ) − ψ + ( z , λ ) )

= − θ ( z 0 − z ) ∂ 2 ∂ z 2 ψ − ( z , λ ) − θ ( z − z 0 ) ∂ 2 ∂ z 2 ψ + ( z , λ )     + δ ( z − z 0 ) ( ψ − ( 1 ) ( z 0 , λ ) − ψ + ( 1 ) ( z 0 , λ ) )     + δ ( 1 ) ( z − z 0 ) ( ψ − ( z 0 , λ ) − ψ + ( z 0 , λ ) ) . (3.7)

We have applied the identity (A.5) to make the coefficients of the delta function and its first derivative independent on variable z. Equation (3.5) using this identity can be written

0 = θ ( z 0 − z ) { − ∂ 2 ∂ z 2 + u ( z ) − λ } ψ − ( z , λ )   + θ ( z − z 0 ) { − ∂ 2 ∂ z 2 + u ( z ) − λ } ψ + ( z , λ )

  + δ ( z − z 0 ) ( ψ − ( 1 ) ( z 0 , λ ) − ψ + ( 1 ) ( z 0 , λ ) )   + δ ( 1 ) ( z − z 0 ) ( ψ − ( z 0 , λ ) − ψ + ( z 0 , λ ) )   + { − ∂ 2 ∂ z 2 + u ( z ) − λ } ψ ′ ( z 0 ; z , λ ) . (3.8)

With the asymptotic Equations (3.2) we may transform this equation to

0 = + δ ( z − z 0 ) ( ψ − ( 1 ) ( z 0 , λ ) − ψ + ( 1 ) ( z 0 , λ ) )   + δ ( 1 ) ( z − z 0 ) ( ψ − ( z 0 , λ ) − ψ + ( z 0 , λ ) )   + θ ( z 0 − z ) ( u ( z ) − u − ( z ) ) ψ − ( z , λ )   + θ ( z − z 0 ) ( u ( z ) − u + ( z ) ) ψ + ( z , λ )   + { − ∂ 2 ∂ z 2 + u ( z ) − λ } ψ ′ ( z 0 ; z , λ ) . (3.9)

Our aim is to represent this equation as a superposition of the delta function and of its derivatives δ ( n ) ( z − z 0 ) , ( n = 0 , 1 , 2 , ⋯ ) with respect to variable z where the coefficients should not depend on variable z. For the sum term with the function ψ ′ ( z 0 ; z , λ ) we make this below and consider now the two sum terms in the second line of (3.9).

The deviation Δ u ( z 0 ; z ) of the potential u ( z ) in the transition region from the two asymptotic potentials u − ( z ) and u + ( z ) depends on the chosen boundary z₀ and will be defined by

Δ u ( z 0 ; z ) ≡ u ( z ) − u − ( z ) θ ( z 0 − z ) − u + ( z ) θ ( z − z 0 ) = ( u ( z ) − u − ( z ) ) θ ( z 0 − z ) + ( u ( z ) − u + ( z ) ) θ ( z − z 0 ) . (3.10)

However, since it does not find entrance in Equation (3.9) as a whole but is combined with the solutions ψ − ( z , λ ) and ψ + ( z , λ ) we have to deal with the two parts u ( z ) − u − ( z ) for z ≤ z 0 and u ( z ) − u + ( z ) for z ≥ z 0 separately.

Since u ( z ) − u − ( z ) and u ( z ) − u + ( z ) by presupposition are concentrated in the neighborhood of the chosen boundary z = z 0 and are rapidly vanishing for z → ∓ ∞ one may make moment-series expansions of the form (see Equations (A.23) and (A.24) in Appendix A)

( u ( z ) − u − ( z ) ) θ ( z 0 − z ) = ∑ m = 0 ∞     u − , m ( z 0 ) δ ( m ) ( z − z 0 ) , ( u ( z ) − u + ( z ) ) θ ( z − z 0 ) = ∑ m = 0 ∞     u + , m ( z 0 ) δ ( m ) ( z − z 0 ) , (3.11)

with coefficients u − , m ( z 0 ) and u + , m ( z 0 ) which are left-hand and right-hand moments of the hatched error parts of the potential which do not depend on variable z (but on reference point z₀) and are given by

u − , m ( z 0 ) ≡ ( − 1 ) m m ! ∫ − ∞ z 0     d z ( u ( z ) − u − ( z ) ) ( z − z 0 ) m                           = 1 m ! ∫ − ∞ z 0     d z ( u ( z ) − u − ( z ) ) ( z 0 − z ) m ︸ ≥   0 ,

u + , m ( z 0 ) ≡ ( − 1 ) m m ! ∫ z 0 + ∞     d z ( u ( z ) − u + ( z ) ) ( z − z 0 ) m ︸ ≥   0 . (3.12)

These are integrals over the hatched parts in last partial figure in Figure 2 multiplied by ( z − z 0 ) m .

We have now to multiply the expressions in (3.11) with ψ − ( z , λ ) and ψ + ( z , λ ) , respectively, and using the multiplication formula (A.5) after reordering of the arising double sums we obtain

( u ( z ) − u − ( z ) ) θ ( z 0 − z ) ψ − ( z , λ ) + ( u ( z ) − u + ( z ) ) θ ( z − z 0 ) ψ + ( z , λ ) = ∑ m = 0 ∞ ( u − , m ( z 0 ) δ ( m ) ( z − z 0 ) ψ − ( z , λ ) + u + , m ( z 0 ) δ ( m ) ( z − z 0 ) ψ + ( z , λ ) ) = ∑ m = 0 ∞ ∑ k = 0 m ( − 1 ) k m ! k ! ( m − k ) ! ( u − , m ( z 0 , λ ) ψ − ( k ) ( z 0 , λ ) + u + , m ( z 0 , λ ) ψ + ( k ) ( z 0 , λ ) ) δ ( m − k ) ( z − z 0 ) = ∑ n = 0 ∞ ∑ k = 0 ∞ ( − 1 ) k ( n + k ) ! n ! k ! ( u − , n + k ( z 0 ) ψ − ( k ) ( z 0 , λ ) + u + , n + k ( z 0 ) ψ + ( k ) ( z 0 , λ ) ) δ ( n ) ( z − z 0 ) , (3.13)

The hatched error function to the approximation does not depend on the reference point z₀ but the moments depend on it. However, the first moment ∫ − ∞ + ∞     d z ( u ( z ) − u ( ∞ ) ) is independent of the chosen reference point z₀. Similar is the case of a potential well between two asymptotic equal potentials by changing the sign of u ( z ) .

The hatched error function to the approximation does not depend on the reference point z₀ but the moments depend on it. However, the first moment ∫ − ∞ + ∞     d z ( u ( z ) − u ( ∞ ) ) is independent of the chosen reference point z₀.

There remains to represent the additional term ψ ′ ( z 0 ; z , λ ) to the asymptotic solution in (3.5) which is concentrated onto the neighborhood of z = z 0 as a superposition of the delta function and of their derivatives that means the expansion into a moment series according to (see Appendix A)

ψ ′ ( z 0 ; z , λ ) = ∑ m = 0 ∞     ψ ′ m ( z 0 , λ ) δ ( m ) ( z − z 0 ) , (3.14)

The hatched error function to the approximation does not depend on the reference point z₀ but the moments dependon it. However, the first moment ∫ − ∞ + ∞     d z ( u ( z ) − u ( ∞ ) ) is independent of the chosen reference point z₀.

With the coefficients ψ ′ m ( z 0 , λ ) which do not depend on variable z but on the reference point z₀

ψ ′ m ( z 0 , λ ) ≡ ( − 1 ) m m ! ∫ − ∞ + ∞     d z   ψ ′ ( z 0 ; z , λ ) ( z − z 0 ) m . (3.15)

Now we have to apply the operator L − λ of the Sturm-Liouville equation to the moment-series expansion of the “small” additional part ψ ′ ( z 0 ; z , λ ) of the solution. Using the multiplication theorem (A.5) and the expansion (3.14) we find

− ∂ 2 ∂ z 2 ψ ′ ( z 0 ; z , λ ) = − ∑ n = 2 ∞     ψ ′ n − 2 ( z 0 , λ ) δ ( n ) ( z − z 0 ) ,

u ( z ) ψ ′ ( z 0 ; z , λ ) = ∑ n = 0 ∞ ( ∑ k = 0 ∞ ( − 1 ) k ( n + k ) ! n ! k ! u ( k ) ( z 0 ) ψ ′ n + k ( z 0 , λ ) ) δ ( n ) ( z − z 0 ) ,

− λ ψ ′ ( z 0 ; z , λ ) = − λ ∑ n = 0 ∞     ψ ′ n ( z 0 , λ ) δ ( n ) ( z − z 0 ) . (3.16)

Thus we have prepared the representation of all parts for insertion into Equation (3.9).

Inserting now (3.13) and (3.16) into Equation (3.9) and collecting all terms proportional to δ ( z − z 0 ) and to its derivatives δ ( n ) ( z − z 0 ) , ( n = 1 , 2 , ⋯ ) we find

0 = { ψ − ( 1 ) ( z 0 , λ ) − ψ + ( 1 ) ( z 0 , λ ) + ∑ k = 0 ∞ ( − 1 ) k ( u − , k ( z 0 ) ψ − ( k ) ( z 0 , λ )         + u + , k ( z 0 ) ψ + ( k ) ( z 0 , λ ) + u ( k ) ( z 0 ) ψ ′ k ( z 0 , λ ) ) − λ ψ ′ 0 ( z 0 , λ ) } δ ( z − z 0 )           + { ψ − ( z 0 , λ ) − ψ + ( z 0 , λ ) + ∑ k = 0 ∞ ( − 1 ) k ( 1 + k ) ! 1 ! k ! ( u − , 1 + k ( z 0 ) ψ − ( k ) ( z 0 , λ )         + u + , 1 + k ( z 0 ) ψ + ( k ) ( z 0 , λ ) + u ( k ) ( z 0 , λ ) ψ ′ 1 + k ( z 0 , λ ) ) − λ ψ ′ 1 ( z 0 , λ ) } δ ( 1 ) ( z − z 0 )           + ∑ n = 2 ∞ { ∑ k = 0 ∞ ( − 1 ) k ( n + k ) ! n ! k ! ( u − , n + k ( z 0 ) ψ − ( k ) ( z 0 , λ ) + u + , n + k ( z 0 ) ψ + ( k ) ( z 0 , λ )         + u ( k ) ( z 0 ) ψ ′ n + k ( z 0 , λ ) ) − ( ψ ′ n − 2 ( z 0 , λ ) + λ ψ ′ n ( z 0 , λ ) ) } δ ( n ) ( z − z 0 ) . (3.17)

All coefficients in front of the derivatives of the delta functions in this equation are now independent on the variable z and according to (3.12) and (3.13) they have to vanish since the left-hand side of (3.17) vanishes. This leads to the intended boundary conditions.

Summarizing, the whole solution of the Sturm-Liouville equation was taken in the general form

ψ ( z , λ ) = ψ − ( z , λ ) θ ( z 0 − z ) + ψ + ( z , λ ) θ ( z − z 0 ) + ∑ n = 0 ∞     ψ ′ n ( z 0 , λ ) δ ( n ) ( z − z 0 ) , (3.18)

and after application of the Sturm-Liouville operator to (3.18) it has to vanish and by separation of the asymptotic solutions (3.2) there remains a part which could be represented as a superposition of the delta function and its derivatives with constant coefficients (3.17) which do not depend on the variable z but only on the chosen boundary at z = z 0 and the parameter λ and all the coefficients have to vanish. In this way we arrived at the intended boundary conditions. Since this system of boundary conditions is infinite we have to look for approximations. This is done in next Section.

The boundary conditions which are as explained the vanishing of the (z-independent) coefficients in front of the delta function and of its derivatives in (3.17) are the following:

1) Vanishing of coefficient in front of δ ( z − z 0 )

0 = ψ − ( 1 ) ( z 0 , λ ) − ψ + ( 1 ) ( z 0 , λ ) + ∑ k = 0 ∞ ( − 1 ) k ( u − , k ( z 0 ) ψ − ( k ) ( z 0 , λ )           + u + , k ( z 0 ) ψ + ( k ) ( z 0 , λ ) + u ( k ) ( z 0 ) ψ ′ k ( z 0 , λ ) ) − λ ψ ′ 0 ( z 0 , λ ) , (4.1)

2) Vanishing of coefficient in front of δ ( 1 ) ( z − z 0 )

0 = ψ − ( z 0 , λ ) − ψ + ( z 0 , λ ) + ∑ k = 0 ∞ ( − 1 ) k ( 1 + k ) ( u − , 1 + k ( z 0 ) ψ − ( k ) ( z 0 , λ )           + u + , 1 + k ( z 0 ) ψ + ( k ) ( z 0 , λ ) + u ( k ) ( z 0 ) ψ ′ 1 + k ( z 0 , λ ) ) − λ ψ ′ 1 ( z 0 , λ ) , (4.2)

3) Vanishing of coefficients in front of δ ( n ) ( z − z 0 ) ,   ( n = 2,3,4, ⋯ )

0 = ∑ k = 0 ∞ ( − 1 ) k ( n + k ) ! n ! k ! ( u − , n + k ( z 0 ) ψ − ( k ) ( z 0 , λ ) + u + , n + k ( z 0 ) ψ + ( k ) ( z 0 , λ )           + u ( k ) ( z 0 ) ψ ′ n + k ( z 0 , λ ) ) − ( ψ ′ n − 2 ( z 0 , λ ) + λ ψ ′ n ( z 0 , λ ) ) . (4.3)

Apart from the given asymptotic potentials u ∓ ( z ) the properties of the complete potential function u ( z ) are contained in the moments u ∓ , m ( z 0 ) (lower indices m = 0 , 1 , 2 , ⋯ ) defined in (3.12) and in the k-th derivatives u ( k ) ( z 0 ) of u ( z ) at reference point z = z 0 (upper indices ( k = 0 , 1 , 2 , ⋯ ) ) and their definitions are given in last Section, where functions with lower indices mean moments and with upper indices derivatives of the corresponding functions.

All approximations of the boundary condition should be contained in (4.1), (4.2) and (4.3) but truncations for approximations have to be made in non-contradictory way. A difficulty is that in each of the conditions an infinite number of potential properties (summations over k) is involved and therefore that the truncation is not a simple problem. The properties of the transition region between the two asymptotic parts is described after the choice of a reference point z₀ by the moments u ∓ , n ( z 0 ) defined by (3.12) and in connection with (3.11). As N-th approximation we will denote the case where all moments n ≥ N are neglected that means

u ∓ , n ( z 0 ) = 0 ,   ( n ≥ N ) . (4.4)

It seems that an appropriate strategy to obtain finite approximation of boundary condition is to truncate the expansion of the solution (3.18) from a finite n ≥ N on as follows

ψ n ( z 0 , λ ) = 0 ,   ( n ≥ N ) . (4.5)

Then the approximate solution possesses the form

ψ ( z , λ ) = ψ − ( z , λ ) θ ( z 0 − z ) + ψ + ( z , λ ) θ ( z − z 0 ) + ∑ n = 0 N − 1     ψ ′ n ( z 0 , λ ) δ ( n ) ( z − z 0 ) . (4.6)

We call both conditions (4.4) and (4.5) taken together the N-th approximation of the boundary conditions. The boundary conditions (4.3) are then identically satisfied in N-th approximation for n ≥ N + 2 and have to be not taken into account for n ≥ N . Thus we have N + 2 boundary conditions for N + 4 quantities ψ ∓ ( 1 ) ( z 0 , λ ) and ψ ∓ ( z 0 , λ ) (4 quantities) plus ψ ′ 0 ( z 0 , λ ) , ⋯ , ψ ′ N − 1 ( z 0 , λ ) (N quantities) in the N-th approximation. For N → ∞ all terms in the boundary conditions are taken into account. In a sufficiently high approximation one may neglect additional terms if they prove to be negligible. We now consider explicitly the three lowest approximations to the basic boundary conditions (or zeroth approximation).

For the illustration of the derived boundary conditions and as important application we consider now problems of reflection and refraction for incident waves at a transition layer. Our notations for the amplitudes of the corresponding waves are schematically shown in Figure 5.

One of the simplest asymptotic potentials are potentials which go rapidly to constant values u − ( z ) → u − as z → − ∞ and to u + ( z ) → u + as z → + ∞ and, therefore, where the deviations Δ u ( z 0 ; z ) from these potentials are rapidly vanishing for z → ± ∞ in the inner region. The lengths in the inner region where the potential changes essentially must be small in comparison to characteristic lengths of the problem (e.g., wavelengths). Similar problems are sometimes calculated for concrete transition potentials, in case of potential barriers or potential wells for rectangular potentials. The WKB-approximation where the lengths of changes of the potentials are large in comparison to the characteristic lengths of the problem is the opposite case to our calculations (e.g., [20] ).

Thus we consider now the following asymptotic potentials u − ( z ) and u + ( z )

u − ( z → − ∞ ) ≡ u − ,   u + ( z → − ∞ ) ≡ u + . (5.1)

Only solutions with the same value of the parameter λ are coupled at the boundary. The general asymptotic solutions of Equation (3.2) for the constant

asymptotic potentials u ∓ for each value of the parameter λ are superpositions of two independent basic solutions and we set

ψ − ( z , λ ) = A − i e i k − z + A − r e − i k − z , ψ + ( z , λ ) = A + i e − i k + z + A + r e i k + z , (5.2)

with the dispersion relations (in principle, also true for complex values λ )

k − = λ − u − ,   k + = λ − u + , ⇔   λ = k − 2 + u − = k + 2 + u + ,   u − − u + = k + 2 − k − 2 . (5.3)

The complex amplitudes of the two possible waves on each side of the boundary z = z 0 are denoted by A − i , r and A + i , r (upper index “i” stands for “incident” and upper index “r” for “reflected” or “refracted” in dependence on the side from which the incident and reflected or refracted waves are considered but only for real λ and then chosen k − > 0 and k + > 0 these notations are uniquely justified). From (5.2) follows for the derivatives of the solutions

ψ − ( 1 ) ( z , λ ) = + i k − ( A − i e i k − z − A − r e − i k − z ) , ψ + ( 1 ) ( z , λ ) = − i k + ( A + i e − i k + z − A + r e i k + z ) . (5.4)

For λ ≥ u ∓ both solutions k ∓ possess real values and denote then wavelengths and for λ ≤ u ∓ both solutions k ∓ possess imaginary values and denote decay constants. For u − < λ < u + or u − > λ > u + only one of the parameters k ∓ is real and the other k ± is imaginary, respectively.

The derived boundary conditions at z = z 0 in arbitrary approximation can be written in form of a matrix equation, for example, in the following way

S i ( A − i e + i k − z 0 A + i e − i k + z 0 ) = S r ( A − r e − i k − z 0 A + r e + i k + z 0 ) , (5.5)

where S i , r are matrices depending on the considered approximations. These matrices S i , r (or S ∓ ) become more complicated with each higher approximation of the boundary conditions and the jump from the second approximation (4.17) to the third approximation (4.22) is already very large and leads to complicated expression.

In the second-order approximation (4.17) we find for the considered asymptotically constant potentials the following matrices in case of (5.9)

S i = ( ( 1 − u − ,1 ( z 0 ) ) k − − i u − ,0 ( z 0 ) ( 1 + u + ,1 ( z 0 ) ) k + − i u + ,0 ( z 0 ) 1 + u − ,1 ( z 0 ) − 1 + u + ,1 ( z 0 ) ) , S r = ( ( 1 − u − ,1 ( z 0 ) ) k − + i u − ,0 ( z 0 ) ( 1 + u + ,1 ( z 0 ) ) k + + i u + ,0 ( z 0 ) − 1 − u − ,1 ( z 0 ) 1 − u + ,1 ( z 0 ) ) . (5.6)

Due to two-dimensional Hamilton-Cayley identity of an arbitrary two-dimensional matrix S

S 2 − 〈 S 〉 S + [ S ] I = 0,   [ S ] ≡ 1 2 ( 〈 S 〉 2 − 〈 S 2 〉 ) , (5.7)

S ≡ ( S 11 , S 12 S 21 , S 22 ) ,   S ¯ ≡ ( S 22 , − S 12 − S 21 , S 11 ) ,   〈 S 〉 ≡ S 11 + S 22 ,   [ S ] ≡ S 11 S 22 − S 12 S 21 .

However, already in 3D-case the advantage is mostly important and we prefer them everywhere.

where 〈 S 〉 is the trace and [ S ] the determinant of S and I denotes the identity matrix one finds the inverse matrix S − 1 in coordinate-invariant way by⁴

S ¯ ≡ 〈 S 〉 I − S ,   ⇒   〈 S ¯ 〉 = 〈 S 〉 ,   [ S ¯ ] = [ S ] ,   S ¯ S = [ S ] I ,   S − 1 = S ¯ [ S ] , (5.8)

where S ¯ by definition denotes the associated matrix to the matrix S .

By multiplication of (5.6) with the inverse matrix ( S r ) − 1 to S r one may resolve the equations to the reflected or refracted wave amplitudes in dependence on the amplitudes of the incident waves.

( A − r e − i k − z 0 A + r e + i k + z 0 ) = S r ¯ S i [ S r ] ( A − i e + i k − z 0 A + i e − i k + z 0 ) ≡ R ( A − i e + i k − z 0 A + i e − i k + z 0 ) = ( R − − R − + R + − R + + ) ( A − i e + i k − z 0 A + i e − i k + z 0 ) . (5.9)

We find for the numerators of the matrix elements of the reflection-refraction matrix R

[ S r ] R − − = ( 1 + u − ,1 ( z 0 ) u + ,1 ( z 0 ) ) ( k − − k + ) − ( u − ,1 ( z 0 ) + u + ,1 ( z 0 ) ) ( k − + k + )                             − i ( u − ,0 ( z 0 ) + u + ,0 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) ) ,

[ S r ] R − + = 2 ( 1 − u + ,1 2 ( z 0 ) ) k + ,

[ S r ] R + − = 2 ( 1 − u − ,1 2 ( z 0 ) ) k − ,

[ S r ] R + + = − ( 1 + u − ,1 ( z 0 ) u + ,1 ( z 0 ) ) ( k − − k + ) + ( u − ,1 ( z 0 ) + u + ,1 ( z 0 ) ) ( k − + k + )                               − i ( u − ,0 ( z 0 ) + u + ,0 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) ) , (5.10)

and for the determinant of the matrix S r

[ S r ] = ( 1 + u − ,1 ( z 0 ) u + ,1 ( z 0 ) ) ( k − + k + ) − ( u − ,1 ( z 0 ) + u + ,1 ( z 0 ) ) ( k − − k + )   + i ( u − ,0 ( z 0 ) + u + ,0 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) ) . (5.11)

Already into this second-order approximation the moments u − , ν ( z 0 ) and u + , ν ( z 0 ) are separately involved and not only as their sums u − , ν ( z 0 ) + u + , ν ( z 0 ) , ( ν = 0,1 ) . The formulae (5.9) together with (5.10) and (5.11) describe the already complicated second-order approximation of reflection and refraction at a transition layer between two asymptotically constant potentials u − and u + which are contained in the expressions (5.3) for k − and k + .

In first-order approximation we neglect the moments u ∓ ,1 ( z 0 ) and obtain from (5.10) and (5.11) the following formulae for the amplitudes of the reflected and refracted waves in dependence on the amplitudes of the incident waves

( A − r e − i k − z 0 A + r e + i k + z 0 ) = ( k − − k + − i   u 0 ( z 0 ) k − + k + + i   u 0 ( z 0 ) 2 k + k − + k + + i   u 0 ( z 0 ) 2 k − k − + k + + i   u 0 ( z 0 ) k + − k − − i   u 0 ( z 0 ) k − + k + + i   u 0 ( z 0 ) ) ( A − i e + i k − z 0 A + i e − i k + z 0 ) , (6.1)

with the inversion

( A − i e + i k − z 0 A + i e − i k + z 0 ) = ( k − − k + + i   u 0 ( z 0 ) k − + k + − i   u 0 ( z 0 ) 2 k + k − + k + − i   u 0 ( z 0 ) 2 k − k − + k + − i   u 0 ( z 0 ) k + − k − + i   u 0 ( z 0 ) k − + k + − i   u 0 ( z 0 ) ) ( A − r e − i k − z 0 A + r e + i k + z 0 ) . (6.2)

The moment u 0 ( z 0 ) is the sum of the moments u − ,0 ( z 0 ) and u + ,0 ( z 0 ) and corresponds to the integral over the deviation of the genuine potential from the asymptotic potentials (see Figure 1 and Figure 2)

u 0 ( z 0 ) ≡ u − ,0 ( z 0 ) + u + ,0 ( z 0 ) = ∫ − ∞ + ∞     d z ( u ( z ) − u − θ ( z 0 − z ) − u + θ ( z − z 0 ) ) . (6.3)

Due to constant asymptotic potentials its Taylor series in powers of z₀ possesses only two terms (see also (4.13))

u 0 ( z 0 ) = u 0 ( 0 ) − ( u − − u + ) z 0 = u 0 ( 0 ) − ( k + 2 − k − 2 ) z 0 , (6.4)

where (5.3) is taken into account. For u − − u + ≠ 0 (genuine transition between asymptotically different potentials) the reference point z₀ of the expansions (the “boundary”) can be uniquely chosen in such way that the moment u 0 ( z 0 ) vanishes leading to

u 0 ( z 0 ) = 0,   ⇒   z 0 = u 0 ( 0 ) u − − u + = u 0 ( 0 ) k + 2 − k − 2 . (6.5)

This is likely the “best” choice of the point z₀ since in this case the matrix of reflection and refraction takes on the usual form without a transition layer and the presence of the last is then involved in the additional phase factors at the amplitudes in (6.1) in comparison to the usual formulae in case of step-wise (abrupt) transition at z 0 = 0 . Formally, with substitutions they are even identical with the relations between the amplitudes of the electric field of incident, reflected and refracted waves for oblique incidence when the electric field is polarized perpendicular to the incidence plane (s-waves; see [14], chap. X, section 86, Eqs. (86.4)).

The matrix Equation (6.1) can be also represented in the form

( A − r A + r ) = ( k − − k + − i   u 0 ( z 0 ) k − + k + + i   u 0 ( z 0 ) e + i 2 k − z 0 2 k + k − + k + + i   u 0 ( z 0 ) e + i ( k − − k + ) z 0 2 k − k − + k + + i   u 0 ( z 0 ) e + i ( k − − k + ) z 0 k + − k − − i   u 0 ( z 0 ) k − + k + + i   u 0 ( z 0 ) e − i 2 k + z 0 ) ( A − i A + i ) . (6.6)

We show now that the matrix in (6.6) which determines the relation between the amplitudes of the incident and reflected waves is nearly unsensible against small variations of the boundary z₀. For example, for the first matrix element in (6.6) we have

k − − k + − i   u 0 ( z 0 ) k − + k + + i   u 0 ( z 0 ) e + i   2 k − z 0 = { k − − k + − i   u 0 ( z 0 ) } e + i ( k − + k + ) z 0 { k − + k + + i   u 0 ( z 0 ) } e − i ( k − − k + z 0 ) ≈ { k − − k + − i ( u 0 ( 0 ) − ( u − − u + ) z 0 ) } ( 1 + i ( k − + k + ) z 0 ) { k − + k + + i ( u 0 ( 0 ) − ( u − − u + ) z 0 ) } ( 1 − i ( k − − k + ) z 0 ) ≈ k − − k + − i   u 0 ( 0 ) + i ( u − − u + + ( k − 2 − k + 2 ) ) z 0 k − + k + + i   u 0 ( 0 ) − i ( u − − u + + ( k − 2 − k + 2 ) ) z 0 = k − − k + − i   u 0 ( 0 ) k − + k + + i   u 0 ( 0 ) , (6.7)

due to (5.3) and where we neglected sum terms proportional to z 0 2 in numerator and denominator as “small” quantities. In full analogy we may treat the other matrix elements where in the non-diagonal elements we have only to transform the denominators. Thus if z = z 0 is already a “good” approximation for the boundary we may simplify (6.6) to

( A − r A + r ) = ( k − − k + − i   u 0 ( 0 ) k − + k + + i   u 0 ( 0 ) 2 k + k − + k + + i   u 0 ( 0 ) 2 k − k − + k + + i   u 0 ( 0 ) k + − k − − i   u 0 ( 0 ) k − + k + + i   u 0 ( 0 ) ) ( A − i A + i ) . (6.8)

In view of the complexity of the second-order approximation of the reflection and refraction formulae (5.9), (5.10) and (5.11) we did not check them in analogous way with respect to their approximate invariance under small variations of the reference point z₀.

The formulae of the preceding Sections 5 and 6 are true, in principle, for complex values λ in connection with (5.2) and (5.3).

For real values λ in cases of λ < u − or λ < u + we find according to (5.3) imaginary values of k − or k + and we set in these special cases, respectively

k − → i κ − ,   k + → i κ + , (7.1)

corresponding to exponentially decreasing or increasing factors for z → ∓ ∞ . From these factors only e + κ − z for z → − ∞ and e − κ + z for z → + ∞ may correspond to finite sources in “infinity”. This implies that in addition to the considered general case we have to distinguish interesting special cases with exponential vanishing of the solutions in infinity. If we concretely assume now u − ≤ u + , we have first the case of semi-bound states

u − ≤ λ ≤ u + ,

k − ≡ λ − u − ,   κ + ≡ − i k + = u + − λ ≥ 0, (7.2)

with the form of the solutions (we set A + i = 0, A + r ≡ A + )

ψ − ( λ , z ) = A − i e + i k − z + A − r e − i k − z ,   ψ + ( z , λ ) = A + e − κ + z . (7.3)

The second interesting case concerns bound states

λ ≤ u − ≤ u + ,

κ − ≡ − i k − = u − − λ ≥ 0,   κ + ≡ − i k + = u + − λ ≥ 0, (7.4)

with the form of the solutions (we set A − i = A + i = 0 , A − r ≡ A − , A + r ≡ A + )

ψ − ( λ , z ) = A − e + κ − z ,   ψ + ( λ , z ) = A + e − κ + z . (7.5)

In the semi-bound case in the first-order approximation of the boundary conditions we find from (6.1) for the amplitudes of the reflected wave A − r on the negative side and the exponentially decreasing wave A + on the positive side in dependence on the amplitude of the incident wave A − i on the negative side and in dependence on the reference point z₀

A − r e − i k − z 0 = k − − i κ + − i u 0 ( z 0 ) k − + i κ + + i u 0 ( z 0 ) A − i e i k − z 0 , A + e − κ + z 0 = 2 k − k − + i κ + + i u 0 ( z 0 ) A − i e i k − z 0 , (7.6)

with

k − ≡ λ − u − ≥ 0,   κ + ≡ u + − λ ≥ 0. (7.7)

Semi-bound states as seen from (7.2) exist as solutions of the Sturm-Liouville equation for asymptotically constant potentials for a whole band of values λ in the range u − ≤ λ ≤ u + and therefore are not discrete states. They are non-degenerate that means for each possible value λ in the given range exists one state. In some aspects they are analogous to total reflection in optics.

We come now to bound states which are characterized by the condition (7.4). The dispersion equation can be obtained from vanishing of the denominators in the matrix elements of in (6.1) but the amplitude relation are difficult to find directly from them. Therefore we go back for the case λ ≤ u − ≤ u + directly to the boundary relations in first-order approximation with the proposition for the solution

ψ − ( z , λ ) = A − e + κ − z ,   ψ − ( 1 ) ( z , λ ) = + κ − A − e + κ − z , ψ + ( z , λ ) = A + e − κ + z ,   ψ + ( 1 ) ( z , λ ) = − κ − A + e − κ + z , (7.8)

with the relations (7.4)

κ + = u + − λ ≥ κ − = u − − λ ≥ 0. (7.9)

Inserting (7.8) into the boundary conditions (4.11) we find written in matrix form

0 = ( κ − + u 0 ( z 0 ) 2 κ + + u 0 ( z 0 ) 2 − 1 1 ) ( A − e + κ − z 0 A + e − κ + z 0 ) . (7.10)

Condition for the existence of non-trivial solutions is the vanishing of the determinant of the matrix in (7.10), i.e.

0 = κ − + κ + + u 0 ( z 0 ) ,   ⇒   u 0 ( z 0 ) = − κ − − κ + ≤ 0, (7.11)

and for the amplitude relations using (7.11) we obtain

A − e + κ − z 0 = A + e − κ + z 0 . (7.12)

We now resolve the dispersion Equation (7.11) with respect to λ using (7.9). Squaring κ + + κ − from (7.11) follows

u 0 2 ( z 0 ) − ( κ + 2 + κ − 2 ) = 2 κ + κ − . (7.13)

Squaring both sides of this equation again we arrive at an equation which using (7.9) is already rational in λ and can be written

0 = u 0 4 ( z 0 ) − 2 ( κ + 2 + κ − 2 ) u 0 2 ( z 0 ) + ( κ + 2 − κ − 2 ) 2 = u 0 4 ( z 0 ) − 2 ( u + + u − − 2 λ ) u 0 2 ( z 0 ) + ( u + − u − ) 2 . (7.14)

From this we find in first-order approximation of the boundary conditions only one solution for λ

λ = −   u 0 4 ( z 0 ) − 2 ( u + + u − ) u 0 2 ( z 0 ) + ( u + − u − ) 2 4 u 0 2 ( z 0 ) . (7.15)

This solution is below the lower of the values of u − and u + , in considered case (7.4) lower than u − that is also seen from the following representation of (7.15)

λ − u − = − ( u 0 2 ( z 0 ) − ( u + − u − ) 2 u 0 ( z 0 ) ) 2 ≤ 0. (7.16)

According to (7.9) inserting (7.15) for λ and choosing the signs of the arising square roots in the way that corresponding to (7.11) the moment u 0 ( z 0 ) defined in (6.3) is non-positive this leads to

κ + = −   u 0 2 ( z 0 ) + ( u + − u − ) 2 u 0 ( z 0 ) ≥ κ − = −   u 0 2 ( z 0 ) − ( u + − u − ) 2 u 0 ( z 0 ) ≥ 0, (7.17)

and this unambiguously is nearer to a potential well than to a potential barrier (next Section).

The bound state (excluding here its limiting case κ + ≥ κ − = 0 )

ψ ( z , λ ) = A − e κ − z θ ( z 0 − z ) + A + e − κ + z θ ( z − z 0 ) , (7.18)

is normalizable. As necessary and sufficient conditions for its presence follows from (7.11) and (7.17)

u 0 ( z 0 ) < 0 ,   u 0 2 ( z 0 ) > u + − u − > 0. (7.19)

The state (7.18) is concentrated to the neighborhood of the considered boundary z₀. In this sense it is analogous to surface waves at a boundary between two media in electrodynamics only with the difference that in last case there is also a real propagation constant parallel to the surface.

We consider now the special case of equal asymptotically constant potentials u − = u +

u − = u + ≡ u ∞ , (8.1)

from which follows that the first moment u 0 ( z 0 ) ≡ u − ( z 0 ) + u + ( z 0 ) of the deviation of the real potential from the asymptotic potentials becomes independent on the reference point z₀, therefore

u 0 ( z 0 ) ≡ ∫ − ∞ + ∞     d z ( u ( z ) − u ∞ ) = u 0 ( 0 ) . (8.2)

For u 0 ( 0 ) > 0 one may speak about a potential barrier and for u 0 ( 0 ) < 0 about a potential well but one may use these notions also in a stronger sense of u ( z ) − u ∞ > 0 and of u ( z ) − u ∞ < 0 only for all z.

All formulae derived in the last two Sections for the general case u − ≠ u + remain true with the specialization (8.1) and only some formulae where u − − u + stands in the denominator (such as (6.5) for a good choice of the reference point z₀) are no more applicable. For a “good” choice of the reference point z₀ one may choose instead that the first moment u 1 ( z 0 ) defined now

u 1 ( z 0 ) ≡ u − , 1 ( z 0 ) + u + , 1 ( z 0 ) = − ∫ − ∞ + ∞     d z ( u ( z ) − u ∞ ) ( z − z 0 ) = u 1 ( 0 ) + z 0 u 0 ( 0 ) , (8.3)

vanishes, from which follows for such choice

0 = u 1 ( z 0 ) = u 1 ( 0 ) + z 0 u 0 ( 0 ) ,   ⇒   z 0 = − u 1 ( 0 ) u 0 ( 0 ) . (8.4)

The normalizable bound states which vanish in infinity are now represented by (see (7.8))

ψ − ( z , λ ) = A − e + κ z ,   ψ + ( z , λ ) = A + e − κ z , ψ − ( 1 ) ( z , λ ) = κ A − e + κ z ,   ψ + ( 1 ) ( z , λ ) = − κ A + e − κ z . (8.5)

For the only discrete state in considered first-order approximation of the boundary conditions follows then from (7.15)

λ − u ∞ = − u 0 2 ( 0 ) 4 < 0 ,   κ = u ∞ − λ = − u 0 ( 0 ) 2 > 0. (8.6)

and the amplitude relations are

A − e + κ z 0 = A + e − κ z 0 . (8.7)

The parameter λ of the discrete state is in every case below the asymptotically constant potential u ∞ . All potential wells with the same area u 0 ( 0 ) below the asymptotic value u ∞ (see also FigureA1 in Appendix B) are in first approximation equivalent

We now calculate the equation for normalizable discrete eigenstates in the third-order approximation of the boundary conditions. With the form of the solutions (8.5) follows from the boundary conditions in third-order approximation (4.22) two homogeneous equations which can be written in the following matrix form

S ( 2 ) ( A − e + κ z 0 A + e − κ z 0 ) = ( 0 0 ) , (8.8)

with the matrix ( λ = − κ 2 )

S ( 3 ) ≡ ( ( 1 − u − ,1 ( z 0 ) ) κ + u − ,0 ( z 0 ) + ( u ( z 0 ) − 2 λ ) u − ,2 ( z 0 ) 1 + u − ,1 ( z 0 ) − 2 u − ,2 ( z 0 ) κ             ( 1 + u + ,1 ( z 0 ) ) κ + u + ,0 ( z 0 ) + ( u ( z 0 ) − 2 λ ) u + ,2 ( z 0 ) − 1 + u + ,1 ( z 0 ) + 2 u + ,2 ( z 0 ) κ ) . (8.9)

The condition of solvability of (8.8) is the vanishing of the determinant of S ( 3 ) which leads to the following equation of third degree for κ

0 = 8   u − ,2 ( z 0 ) u + ,2 ( z 0 ) κ 3 − 4 { u − ,1 ( z 0 ) u + ,2 ( z 0 ) − u + ,1 ( z 0 ) u − ,2 ( z 0 ) } κ 2           − 2 { 1 + u − ,1 ( z 0 ) u + ,1 ( z 0 ) − u − ,0 ( z 0 ) u + ,2 ( z 0 ) − u + ,0 ( z 0 ) u − ,2 ( z 0 )           − 2 u ( z 0 ) u − ,2 ( z 0 ) u + ,2 ( z 0 ) } κ − { u 0 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 )           + u + ,0 ( z 0 ) u − ,1 ( z 0 ) + u ( z 0 ) ( u 2 ( z 0 ) + u − ,1 ( z 0 ) u + ,2 ( z 0 ) − u + ,1 ( z 0 ) u − ,2 ( z 0 ) ) } , (8.10)

and for the amplitude relations we find from (8.8)

( 1 + u − ,1 ( z 0 ) − 2 u − ,2 ( z 0 ) κ ) A − e κ z 0 = ( 1 − u + ,1 ( z 0 ) − 2 u + ,2 ( z 0 ) κ ) A + e − κ z 0 . (8.11)

Our notations specialized to the case (8.1) are ( m = 0 , 1 , 2 )

u m ( z 0 ) ≡ u − , m ( z 0 ) + u + , m ( z 0 ) = ( − 1 ) m m ! ∫ − ∞ + ∞     d z ( u ( z ) − u ∞ ) ( z − z 0 ) m . (8.12)

As a third-degree equation with real coefficients it may possess three real solutions or one real plus two complex-conjugate solutions where due to the smallness of the coefficients in front of κ 3 and of κ 2 the real solution (8.6) is only slightly modified and the new solutions are small from their modulus in comparison to (8.6).

In case of a layer system with each single layer of the considered form it is favorable to represent the boundary conditions in a form where the amplitudes to each side of the layer stand on different sides that means

S − ( A − i e + i k − z 0 A − r e − i k − z 0 ) = S + ( A + r e + i k + z 0 A + i e − i k + z 0 ) . (9.1)

Then the output of one side of the transition layer may serve as input for the next transition layer which have to be in the asymptotically range because in other case we must add new boundary relations for the patch position of the two transition layers.

The resolution of (9.1) to the right-hand side of the amplitudes is

( A + r e + i k + z 0 A + i e − i k + z 0 ) = S + ¯ S − [ S + ] ( A − i e + i k − z 0 A − r e − i k − z 0 ) ≡ T ( A − i e + i k − z 0 A − r e − i k − z 0 ) ≡ ( T ri T rr T ii T ir ) ( A − i e + i k − z 0 A − r e − i k − z 0 ) . (9.2)

In addition, it would be favorable to substitute in case of many layers when their connection is in the asymptotic ranges the lower indices “−” and “+” by specific notations of the involved layers to both sides of the transitions. The

matrices T ≡ S + ¯ S − [ S + ] we may call transmission matrices.

In second-order approximation we obtain for the matrices S + and S − under the same propositions (5.2) as before

S − = ( k − ( 1 − u − ,1 ( z 0 ) ) − i u − ,0 ( z 0 ) − k − ( 1 − u − ,1 ( z 0 ) ) − i u − ,0 ( z 0 ) 1 + u − ,1 ( z 0 ) 1 + u − ,1 ( z 0 ) ) , S + = ( k + ( 1 + u + ,1 ( z 0 ) ) + i u + ,0 ( z 0 ) − k + ( 1 + u + ,1 ( z 0 ) ) + i u + ,0 ( z 0 ) 1 − u + ,1 ( z 0 ) 1 − u + ,1 ( z 0 ) ) . (9.3)

The determinant [ S + ] of the matrix S + is

[ S + ] = 2 ( 1 − u + ,1 ( z 0 ) ) ( 1 + u + ,1 ( z 0 ) ) k + . (9.4)

Except trivial cases this determinant and the determinant [ S − ] of the matrix S − too cannot vanish and solutions which vanish only on one side of the transition layer do not exist. For the components of the transmission matrices T we calculate from (9.2)

[ S + ] T ri = ( 1 + u + ,1 ( z 0 ) u − ,1 ( z 0 ) ) ( k + + k − ) + u 1 ( z 0 ) ( k + − k − )                           − i ( u 0 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) ) ,

[ S + ] T rr = ( 1 + u + ,1 ( z 0 ) u − ,1 ( z 0 ) ) ( k + − k − ) + u 1 ( z 0 ) ( k + + k − )                           − i ( u 0 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) ) ,

[ S + ] T ii = ( 1 + u + ,1 ( z 0 ) u − ,1 ( z 0 ) ) ( k + − k − ) + u 1 ( z 0 ) ( k + + k − )                           + i ( u 0 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) ) ,

[ S + ] T ir = ( 1 + u + ,1 ( z 0 ) u − ,1 ( z 0 ) ) ( k + + k − ) + u 1 ( z 0 ) ( k + − k − )                           + i ( u 0 ( z 0 ) + u + ,0 ( z 0 ) u − ,1 ( z 0 ) − u − ,0 ( z 0 ) u + ,1 ( z 0 ) ) . (9.5)

In the first-order approximation we neglect the moments u + ,1 ( z 0 ) and u − ,1 ( z 0 ) that we do not write down separately.

In this article, we have chosen the important Sturm-Liouville equation as one of the simplest ordinary second-order differential equations to demonstrate important aspects of the derivation of generalized boundary conditions by means of Generalized functions or, more especially, of step functions, delta functions and their derivatives. A countable infinite system of boundary conditions has been obtained and its first few low-order approximations are derived for this equation in case of sufficiently smooth potentials. It was necessary to choose two asymptotic “volume” potentials with known solutions of the corresponding Sturm-Liouville equation and an imaginary’ boundary surface in the transition region between these asymptotic potentials. The information about the deviations of the real potential from the chosen asymptotic potentials on both sides of the “imaginary” boundary surface is involved in the boundary conditions in form of the moments of these deviations. The condition for the applicability of low-order approximations of the boundary conditions requires that the “effective” thickness of the transition region must be small in comparison to the characteristic lengths in alterations of the “volume” solutions in the transition region (wavelengths in the case of asymptotic constant potentials, for instance). Although this is intuitively clear, it is convenient to discuss this more precisely not in general abstract form but in connection with concrete potentials and problems. We applied it to reflection and refraction problems for asymptotically constant potentials.

The general problems of the derivation of boundary conditions by means of the theory of Generalized functions could be solved for the Sturm-Liouville equation. It seems that there are no principal difficulties to derive boundary conditions in an analogous way for other linear scalar or vectorial differential or integro-differential equations as was already done for electrodynamics. However, we do not want to conceal that in the case of nonlinear equations there are some new principal difficulties with the presented method.

Lately, I am able to express my great gratitude to Prof. G. Vojta (from Editorial Board of Ann. d. Physik in former GDR) who advocated the publication of my paper [12] against the rejection without a written report by the Chief Editor of Ann. d. Physik Prof. G. Richter from GDR who at that same time was also a Member of our Central Institute of Optics and Spectroscopy (ZOS). The arguments of G. Richter were unjustified and unacceptable (see my home page)⁵. Furthermore, in particular, I thank Dr. Th. Jung from “Zentralinstitut für Elektronenphysik” der “Akademie der Wissenschaften der DDR” who managed to give a colloquium lecture in mentioned Institute and also Prof. V.M. Agranovich from Moscow where during a one-month visit I could give a seminar lecture in his “Institute of Spectroscopy” nearby Moscow to the problems dealt with in [12]. In my former “Zentralinstitut für Optik und Spektroskopy” I never gave a lecture about these problems to avoid further trouble with some persons.

The author declares no conflicts of interest regarding the publication of this paper.

Wünsche, A. (2021) Boundary Conditions for Sturm-Liouville E- quation with Transition Regions and Barriers or Wells. Advances in Pure Mathematics, 11, 254-295. https://doi.org/10.4236/apm.2021.114018

We derive in this Appendix some identities for generalized functions which we use in the main text. Generalized functions f ( x ) of a function space K' are defined as linear continuous functionals over basis functions φ ( x ) of a space K of (sufficiently well-behaved, e.g., smooth, i.e., continuous and infinitely continuously differentiable) functions which we may write as generalization of the integral on the right-hand side in the relation [25] [26] [27] [28]

( f , φ ) ≡ ∫ − ∞ + ∞     d x   f ( x ) φ ( x ) ,   l i m k → ∞ ( f k , φ ) = ( f , φ ) . (A.1)

The delta function with real displacement parameter x 0 and their n-th derivatives δ ( n ) ( x − x 0 ) according to the rule of partial integration are defined by

∫ − ∞ + ∞     d x   δ ( n ) ( x − x 0 ) φ ( x ) = ( − 1 ) n ∫ − ∞ + ∞     d x   δ ( x − x 0 ) φ ( n ) ( x ) = ( − 1 ) n φ ( n ) ( x 0 ) , (A.2)

with the important special cases for φ ( x ) = 1

∫ − ∞ + ∞     d x   δ ( n ) ( x − x 0 ) = δ n ,0 = { 1, n = 0 0, n ≠ 0 , (A.3)

showing that all “integrals” over genuine (i.e., n ≥ 1 ) derivatives of the delta function taken over the real axis vanish. With possible (e.g., smooth) multiplication functions α ( x ) using the Leibniz rule for differentiation of a product one finds

∫ − ∞ + ∞     d x   α ( x ) δ ( n ) ( x − x 0 ) φ ( x ) = ( − 1 ) n ∫ − ∞ + ∞     d x   δ ( x − x 0 ) ( α ( x ) φ ( x ) ) ( n ) = ( − 1 ) n ∑ k = 0 n n ! k ! ( n − k ) ! α ( k ) ( x 0 ) φ ( n − k ) ( x 0 ) = ∑ k = 0 n ( − 1 ) k n ! k ! ( n − k ) ! α ( k ) ( x 0 ) ∫ − ∞ + ∞     d x   δ ( n − k ) ( x − x 0 ) φ ( x ) , (A.4)

and since this is correct for arbitrary basis functions φ ( x )

α ( x ) δ ( n ) ( x − x 0 ) = ∑ k = 0 n ( − 1 ) k n ! k ! ( n − k ) ! α ( k ) ( x 0 ) δ ( n − k ) ( x − x 0 ) . (A.5)

On the right-hand side of this identity we have only constant coefficients in front of the derivatives of the delta function δ ( x − x 0 ) .

We give two examples for (A.5). The first example with α ( x ) = x m is

x m δ ( n ) ( x − x 0 ) = ∑ k = 0 { m , n } ( − 1 ) k m ! n ! k ! ( m − k ) ! ( n − k ) ! x 0 m − k δ ( n − k ) ( x − x 0 ) = L m , n ( x 0 , ∂ ∂ x ) δ ( x − x 0 ) , (A.6)

with the special case x 0 = 0

x m δ ( n ) ( x ) = ( − 1 ) m n ! ( n − m ) ! δ ( n − m ) ( x ) = L m , n ( 0, ∂ ∂ x ) δ ( x ) . (A.7)

In (A.6) L m , n ( x , y ) denotes the Laguerre 2D polynomials defined by

L m , n ( x , y ) ≡ ∑ k = 0 { m , n } ( − 1 ) k m ! n ! k ! ( m − k ) ! ( n − k ) ! x m − k y n − k = ( − 1 ) n n ! x m − n L n m − n ( x y ) = ( − 1 ) m m ! y n − m   L m n − m ( x y ) , (A.8)

where L n ν ( z ) denotes the Generalized Laguerre polynomials or Laguerre-Sonin polynomials in standard notation.

A second example for illustration with α ( x ) = exp ( − x 2 a 2 ) is

exp ( − x 2 a 2 ) δ ( n ) ( x − x 0 ) = exp ( − x 0 2 a 2 ) ∑ k = 0 n n ! k ! ( n − k ) ! 1 a k H k ( x 0 a ) δ ( n − k ) ( x − x 0 ) = exp ( − x 0 2 a 2 ) 1 a n H n ( x 0 a + a 2 ∂ ∂ x ) δ ( x − x 0 ) , (A.9)

with the special case x 0 = 0

exp ( − x 2 a 2 ) δ ( n ) ( x ) = ∑ l = 0 [ n 2 ] ( − 1 ) l n ! l ! ( n − 2 l ) ! 1 a 2 l δ ( n − 2 l ) ( x ) = 1 a n H n ( a 2 ∂ ∂ x ) δ ( x ) , (A.10)

where H n ( z ) denotes the Hermite polynomials

H n ( z ) ≡ ( − 1 ) n exp ( z 2 ) ∂ n ∂ z n exp ( − z 2 ) = exp ( − 1 4 ∂ 2 ∂ z 2 ) ( 2 z ) n = ∑ k = 0 [ n 2 ] ( − 1 ) k n ! k ! ( n − 2 k ) ! ( 2 z ) n − 2 k , (A.11)

in different representations which can be used as alternative definitions.

An equation of the form

0 = ∑ n = 0 ∞     c n ( x 0 ) δ ( n ) ( x − x 0 ) = c 0 ( x 0 ) δ ( x − x 0 ) + c 1 ( x 0 ) δ ( 1 ) ( x − x 0 ) + ⋯ , (A.12)

with coefficients c n ( x 0 ) which are independent of the variable x is only satisfied if all coefficients c n ( x 0 ) vanish. This follows by multiplication of this equation with the functions ( x − x 0 ) m , ( m = 0,1,2, ⋯ ) using (A.7) and succeeding integration over the whole x-axis using (A.3) according to

0 = ∫ − ∞ + ∞     d x ∑ n = 0 ∞     c n ( x 0 ) ( x − x 0 ) m δ ( n ) ( x − x 0 ) = ∑ n = 0 ∞     c n ( x 0 ) ( − 1 ) m n ! ( n − m ) ! ∫ − ∞ + ∞     d x   δ ( n − m ) ( x − x 0 ) = ( − 1 ) m m ! c m ( x 0 ) ,   ⇒   c m ( x 0 ) = 0 ,   ( m = 0 , 1 , 2 , ⋯ ) . (A.13)

It means that the systems of functions ( x − x 0 ) m and generalized functions δ n ( x − x 0 ) with arbitrary parameter x₀ are bi-orthogonal. For the derivation of boundary conditions for an equation one has to split the asymptotic parts and has to transform the remaining parts of the equation to the principal form (A.12) and find then the boundary condition from the vanishing of the coefficients c m , ( m = 0,1,2, ⋯ ) . From (A.13) follows with variation of the parameter x₀

c n ( x 0 ) = ∑ k = 0 n x 0 k k ! c n − k ( 0 ) . (A.14)

This means that with variation of the parameter x₀ the whole system of boundary conditions transforms into each other. However, by truncation of the boundary conditions with finite numbers n = N this full equivalence becomes an approximate equivalence and a “best” choice of the parameter x₀ becomes important.

We now generalize the preceding considerations to expansions of generalized functions which we call moment series because the coefficients of these expansions are determined by the moments of the function. For this purpose we first define the Fourier transform f ˜ ( k ) of f ( x ) which together with its inversion is given by

f ˜ ( u ) = ∫ − ∞ + ∞     d y   f ( y ) exp ( − i y u ) ,   ⇔   f ( x ) = 1 2 π ∫ − ∞ + ∞     d u   f ˜ ( u ) exp ( i u x ) . (A.15)

Then we represent the function f ( x ) as convolution (notation “*”) of this function with the delta function and apply the displacement operator of the argument of a function as follows

f ( x ) = f ( x ) ∗ δ ( x ) = ∫ − ∞ + ∞     d y   f ( y ) δ ( x − y ) = ∫ − ∞ + ∞     d y   f ( y ) exp ( − y ∂ ∂ x ) δ ( x ) = f ˜ ( − i ∂ ∂ x ) δ ( x ) , (A.16)

where we applied in last step the definition (A.15) of the Fourier transform. Making now expansions of the function δ ( x − y ) in a Taylor series with respect to powers of y and f ˜ ( k ) in a Taylor series at k = 0 we find from (A.16)

f ( x ) = ∑ n = 0 ∞     f n ( 0 ) δ ( n ) ( x ) = ∑ n = 0 ∞ f ˜ ( n ) ( 0 ) n ! ( − i ∂ ∂ x ) n δ ( x ) , (A.17)

with the moments f n ( 0 ) of the function f ( x ) defined by

f n ( 0 ) ≡ ( − 1 ) n n ! ∫ − ∞ + ∞     d y   f ( y ) y n = ( − i ) n n ! f ˜ ( n ) ( 0 ) . (A.18)

This shows that the moments of f ( x ) are related to the derivatives of the Fourier transforms f ˜ ( u ) at x = 0 . Conversely, the derivatives of the function f ( x ) at x 0 = 0 are related to the moments of its Fourier transform f ˜ ( u ) by

f ( n ) ( 0 ) = ( − i ) n 2 π n ! f ˜ n ( 0 ) , (A.19)

with f ˜ n ( 0 ) defined by

f ˜ n ( 0 ) ≡ ( − 1 ) n n ! ∫ − ∞ + ∞     d u   f ˜ ( u ) u n = 2 π i n n ! f ( n ) ( 0 ) . (A.20)

The relations (A.18) and (A.19) express a certain duality of the moments of a function and of the derivatives of its Fourier transforms at the coordinate origin. Upper indices in the form f ( n ) ( x ) mean in every case the n-th derivative of f ( x ) and lower indices in the form f n ( x 0 ) moments of a function to reference point x₀⁷.

The convergence of moment series is weak convergence in the sense of the convergence of the sequence of partial sums as linear functionals over the space of basis functions such as written in (A.1). The linear continuous functionals over basis functions φ ( x ) can be transformed by Fourier transformation (A.15) in linear continuous functionals over the space of Fourier transforms to the basis functions of the space D usually denoted by Z as follows

( f , φ ) ≡ ∫ − ∞ + ∞     d x   f ( x ) φ ( x ) = ∫ − ∞ + ∞     d x 1 2 π ∫ − ∞ + ∞     f ˜ ( u ) e i u x φ ( x ) = ∫ − ∞ + ∞     d u   f ˜ ( u ) 1 2 π ∫ − ∞ + ∞     d x   φ ( x ) e i x u = 1 2 π ∫ − ∞ + ∞     d u   f ˜ ( u ) φ ˜ ( − u ) ≡ 1 2 π ( f ˜ , φ ˜ * ) , (A.21)

where we changed the order of integrations. The obtained relation is the Parseval theorem.

The expansion (A.17) can be generalized to an arbitrary reference point x₀ instead of x 0 = 0 and instead of (A.16) we have

f ( x ) = ∫ − ∞ + ∞     d y   f ( y ) δ ( x − x 0 + x 0 − y ) = ∫ − ∞ + ∞     d y   f ( y ) exp ( ( x 0 − y ) ∂ ∂ x ) δ ( x − x 0 ) = exp ( x 0 ∂ ∂ x ) f ˜ ( − i ∂ ∂ x ) δ ( x − x 0 ) . (A.22)

The operators exp ( x 0 ∂ ∂ x ) and f ˜ ( − i ∂ ∂ x ) commute and using this it is easy to return to the form (A.16) which also follows as special case setting x 0 = 0 . From (A.22) follows

f ( x ) = ∑ n = 0 ∞     f n ( x 0 ) δ ( n ) ( x − x 0 ) , (A.23)

with the moments f n ( x 0 ) determined by

f n ( x 0 ) ≡ ( − 1 ) n n ! ∫ − ∞ + ∞     d y   f ( y ) ( y − x 0 ) n = ∑ k = 0 n x 0 k k ! f n − k ( 0 ) = ∑ k = 0 n ( − i ) n − k k ! ( n − k ) ! x 0 k f ˜ ( n − k ) ( 0 ) = 1 n ! { ( x 0 − i ∂ ∂ u ) n f ˜ ( u ) } u = 0 , (A.24)

where the relation of the moments at reference point x = x 0 to the moments at reference point x = 0 is given. The zeroth moment f 0 ( x 0 ) does not depend on the reference point x₀

f 0 ( x 0 ) = f 0 ( 0 ) = f ˜ ( 0 ) = ∫ − ∞ + ∞     d y   f ( y ) , (A.25)

and, therefore, is “absolute”. Furthermore, this formula shows how the moments to the reference points x₀ of a function can be expressed in simple way by the Fourier transform of this function.

For example, for a “normalized” Gaussian function together with its Fourier transform given by

f ( x ) = 1 π a 2 exp ( − x 2 a 2 ) ,   ⇔   f ˜ ( u ) = exp ( − a 2 u 2 4 ) , (A.26)

we find the following moment series

1 π a 2 exp ( − x 2 a 2 ) = exp ( a 2 4 ∂ 2 ∂ x 2 ) δ ( x ) = ∑ m = 0 ∞ a 2 m m ! 2 2 m δ ( 2 m ) ( x ) , (A.27)

with the moments f n ( 0 ) determined by

f 2 m ( 0 ) = a 2 m m ! 2 2 m ,   f 2 m + 1 ( 0 ) = 0 ,   ( m = 0 , 1 , 2 , ⋯ ) . (A.28)

Due to symmetry of the Gaussian function the odd moments are vanishing with respect to the reference point x 0 = 0 but not with respect to an arbitrary reference point x₀. Further examples for moment series of symmetric functions one may find in Appendix B.

As special cases we consider now the moment series for functions which are non-vanishing only to the left or to right of a reference point x = x 0 .

For functions f − ( x ) which are non-vanishing only for variable x < x 0

f − ( x ) ≡ f ( x ) θ ( x 0 − x ) = { f ( x ) , x < x 0 0 , x > x 0 , (A.29)

we find the moment series to reference point x = x 0

f − ( x ) = ∑ n = 0 ∞     f − , n ( x 0 ) δ ( n ) ( x − x 0 ) ,   f − ( x 0 + y ) = ∑ n = 0 ∞     f − , n ( x 0 ) δ ( n ) ( y ) , (A.30)

with the moments

f − , n ( x 0 ) ≡ ( − 1 ) n n ! ∫ − ∞ 0     d y   f ( x 0 + y ) y n = 1 n ! ∫ − ∞ x 0     d x   f ( x ) ( x 0 − x ) n ︸ ≥ 0 = ∫ − ∞ x 0     d x n ∫ − ∞ x n     d x n − 1 ⋯ ∫ − ∞ x 2     d x 1 ∫ − ∞ x 1     d x   f ( x ) , (A.31)

given together with an equivalent multiple integral.

For functions f + ( x ) which are non-vanishing only for variable x > x 0

f + ( x ) ≡ f ( x ) θ ( x − x 0 ) = { f ( x ) , x > x 0 0 , x < x 0 , (A.32)

we find the moment series to reference point x = x 0

f + ( x ) = ∑ n = 0 ∞     f + , n ( x 0 ) δ ( n ) ( x − x 0 ) ,   f + ( x 0 + y ) = ∑ n = 0 ∞     f + , n ( x 0 ) δ ( n ) ( y ) , (A.33)

with the moments

f + , n ( x 0 ) ≡ ( − 1 ) n n ! ∫ 0 + ∞     d y   f ( x 0 + y ) y n = ( − 1 ) n n ! ∫ x 0 + ∞     d x   f ( x ) ( x − x 0 ) n ︸ ≥ 0 = ( − 1 ) n ∫ x 0 + ∞     d x n ∫ x n + ∞     d x n − 1 ⋯ ∫ x 2 + ∞     d x 1 ∫ x 1 + ∞     d x   f ( x ) , (A.34)

given again together with an equivalent multiple integral.

The sum of the moments f − , n ( x 0 ) and f + , n ( x 0 ) is

f ( x ) = f − ( x ) + f + ( x ) ,   f − , n ( x 0 ) + f + , n ( x 0 ) = f n ( x 0 ) , (A.35)

to the moment series (A.23) with the moments determined by (A.24).

It should be mentioned here that the identities in the formulae (A.31) and (A.34) are special cases of the more general identity

f ( − ( n + 1 ) ) ( a ; y ) ≡ ∫ a y     d x n ∫ a x n     d x n − 1 ⋯ ∫ a x 2     d x 1 ∫ a x 1     d x   f ( x ) = 1 n ! ∫ a y     d x   f ( x ) ( y − x ) n , (A.36)

with arbitrary parameter a that can be proved by complete induction with reordering of the arising double integral and accomplishing then the inner integral. With the choice a = 0 and n → μ − 1 as an arbitrary real number it is often taken as starting point for the introduction of “ μ -fold” fractional integration by the convolution integral on the right-hand side which in our written case corresponds to possible ( n + 1 ) -fold integration of the function f ( x ) (e.g., [29], Chap. XIII).

The moment series (A.23) can be made also for functions f ( x ) which are not smooth and even not continuous and, roughly speaking, they are well defined if only all moments exist. The approximate reconstruction of the function f ( x ) from the moment series can be obtained from the limits of the linear functionals ( f k , φ ) when f k denotes the truncated moment series up to the term with the k-th moment (see (A.1)). As basis functions φ ( x ) it seems to be sufficient to take, for example, all displaced normalized Gaussian functions

1 π ε 2 exp ( − ( x − x ′ ) 2 ε 2 ) with the limit ε → 0 . However also the Fourier transforms f ˜ ( u ) are appropriate if their Taylor series may be summed up and if then the inverse Fourier transformation f ( x ) to f ˜ ( u ) is calculable.

In this Appendix we consider for illustration some examples of functions f ( x ) which are appropriate as potential barriers or potential wells together with their Fourier transforms f ˜ ( u ) (notations as in (A.15)) and their moment series and, furthermore, functions which are appropriate to simulate transitions between two potentials for which the Sturm-Liouville equation can be exactly solved (e.g., constant potentials).

In FigureA1 we present six from seven explicitly given symmetric functions f ν ( x ) , ( ν = 1, ⋯ ,7 ) which are appropriate as potential barriers and, negatively taken, as potential wells

(1): f 1 ( x ) = θ ( 1 − 2 | x | a ) ,   f ˜ 1 ( u ) = 2 u sin ( a 2 u ) ,

(2): f 2 ( x ) = 1 − ( π x 2 a ) 2   θ ( 1 − π | x | 2 a ) ,   f ˜ 2 ( u ) = π u J 1 ( 2 a π u ) ,

(3): f 3 ( x ) = ( 1 − | x | a ) θ ( 1 − | x | a ) ,   f ˜ 3 ( u ) = a ( sin ( a 2 u ) a 2 u ) 2 .

(4): f 4 ( x ) = exp ( − 2 | x | a ) ,   f ˜ 4 ( u ) = a 1 + ( a 2 u ) 2 ,

(5): f 5 ( x ) = exp ( − π x 2 a 2 ) ,   f ˜ 5 ( u ) = a exp ( − a 2 4 π u 2 ) ,

(6): f 6 ( x ) = 1 ch 2 ( 2 x a ) ,   f ˜ 6 ( u ) = a π a u 4 sh ( π a 4 u ) ,

(7): f 7 ( x ) = 1 ch ( π x a ) ,   f ˜ 7 ( u ) = a ch ( a 2 u ) . (B.1)

Their first three initial terms of the moment series are

(1): f 1 ( x ) = a { δ ( x ) + a 2 24 δ ( 2 ) ( x ) + a 4 1920 δ ( 4 ) ( x ) + ⋯ } ,

(2): f 2 ( x ) = a { δ ( x ) + a 2 2 π 2 δ ( 2 ) ( x ) + a 4 12 π 4 δ ( 4 ) ( x ) + ⋯ } ,

(3): f 3 ( x ) = a { δ ( x ) + a 2 12 δ ( 2 ) ( x ) + a 4 360 δ ( 4 ) ( x ) + ⋯ } ,

(4): f 4 ( x ) = a { δ ( x ) + a 2 4 δ ( 2 ) ( x ) + a 4 16 δ ( 4 ) ( x ) + a 6 64 δ ( 6 ) ( x ) + ⋯ } ,

(5): f 5 ( x ) = a { δ ( x ) + a 2 4 π δ ( 2 ) ( x ) + a 4 32 π 2 δ ( 4 ) ( x ) + ⋯ } ,

(6): f 6 ( x ) = a { δ ( x ) + π 2 a 2 96 δ ( 2 ) ( x ) + 7 π 4 a 4 92160 δ ( 4 ) ( x ) + ⋯ } ,

(7): f 7 ( x ) = a { δ ( x ) + a 2 8 δ ( 2 ) ( x ) + 5 a 4 384 δ ( 4 ) ( x ) + ⋯ } . (B.2)

All considered examples of functions f ν ( x ) possess analytic Fourier transforms f ˜ ν ( u ) and, therefore, well-defined moment series.

We now consider transition functions. Examples for transition functions can be constructed from the potential functions f ν ( x ) in (B.2) by integration in the following way

F ν ( x ) = 1 a ∫ 0 x     d y   f ν ( y ) ,   F ν ( 1 ) ( x ) = f ν ( x ) a ,   F ν ( ∓ ∞ ) = ∓ 1 2 . (B.3)

Since we have chosen ∫ − ∞ + ∞     d x   f ( x ) = a as the zeroth moment of the function f ( x ) we obtain by (B.3) a transition step of the amount F ( + ∞ ) − F ( − ∞ ) = 1 that means a unit step and an increase in the center x = 0 of F ( 1 ) ( 0 ) = 1 a . For the functions in (B.2) these are the functions

(1): F 1 ( x ) = { − 1 2 , x ≤ − a 2 x a , − a 2 < x < + a 2 + 1 2 , + a 2 < x ,

(2): F 2 ( x ) = { − 1 2 , x ≤ − 2 a π x 2 a 1 − ( π x 2 a ) 2 + 1 π arcsin ( π x 2 a ) , − 2 a π < x < + 2 a π + 1 2 , + 2 a π < x ,

(3): F 3 ( x ) = { − 1 2 , x ≤ − a x a ( 1 − | x | 2 a ) , − a < x < + a + 1 2 , + a < x ,

(4): F 4 ( x ) = 1 2 ( 1 − exp ( − 2 | x | a ) ) x | x | ,

(5): F 5 ( x ) = 1 2 erf ( π x 2 a ) ,

(6): F 6 ( x ) = 1 2 th ( 2 x a ) ,

(7): F 7 ( x ) = 2 π arctg ( th ( π x 2 a ) ) , (B.4)

shown in FigureA2. In the first approximation of the boundary conditions all these transition functions are equivalent and only in higher approximations they provide differences in the results.