There are many physical examples of quantum mechanical problems in which Position-Dependent Mass (PDM) makes its appearance and leads to effective interactions giving interesting results. Application of PDM Schrö- dinger equations is found in different branches of Physics like Condensed Matter Physics, Material Science, Nuclear Physics etc. Special application of principal concept of PDM is found in the investigation of electronic properties of semiconductors [1], quantum dots and quantum wells [2, 3], quantum liquids [4], Nuclear many body problems[5] etc. In these cases, Exact Analytic Solution (EAS) of PDM Schrödinger equation for a physical potential provides maximum information of the system. Therefore, there has been a sustained effort in calculating EAS of PDM Schrödinger equation providing us with energy eigenvalues and corresponding eigenfunctions for physical potentials. Such studies use different methods known for solving constant mass Schrödinger equations or an extension of them. Point Canonical Transformation (PCT) [6-10], Nikiforov-Uvarov (NU) method [11-13], Supersymmetry (SUSY) quantum mechanics approach [14,15], Quadratic Algebra [16], Darboux Transformation (DT) [17,18] etc. are different approaches used in the study of PDM Schrödinger equations.

In this paper, we are trying to generate Exactly Solvable Potentials (ESP)s of PDM Schrödinger equation by using Extended Transformation (ET) method [19]. Ethics of the method is not to solve PDM Schrödinger equation to obtain wave functions, energy eigenvalues and mass function for a particular physical potential, but to generate exactly solvable potentials of PDM Schrödinger equation from an already solved potential. The method provides us with direct way of calculating the wave functions, energy eigenvalues and mass function for the generated system from the already solved potential. The spirit of ET method is similar to that of SUSY quantum mechanics approach, but in ET method, one needs no concern about the concept of supersymmetry and shape invariance which is essential in SUSY quantum mechanics approach. We start with a known exactly solved Quantum System (QS). Then we invoke a Co-ordinate Transformation (CT) followed by Functional Transformation (FT) of wave function and a set of plausible ansätze to mould the transformed equation to standard form of PDM Schrödinger equation. In addition, we introduce transformation of mass function which transforms the mass function of known system to that of generated system. Most of the potentials generated by this method are Sturmian, and some of them can be made normal as well as physical by system specific regrouping method. In transformation method, the normalization of the wave functions of the generated QS can easily be verified in most cases.

The paper is organized as follows: the method is discussed in Section 2, implementation of the method in practical QS with Hulthen potential is shown in Section 3 and finally, conclusions and findings of results are discussed in Section 4.

We start with Hermitian effective Hamiltonian for a Position-Dependent Mass (PDM) system in one dimension (2m₀ = ħ = 1) [20]

where V^eff(x) is the effective potential given by

Here V(x) is the real potential profile and is the modification emerging from position-dependent mass given by

where α, β, γ are ambiguity parameters satisfying the constraint relation, and m(x) is the dimensionless mass function related to constant mass m₀ and PDM M(x) by [21]. The prime stands for derivative of the function with respect to its argument.

Thus one dimensional position-dependent mass (1DPDM) Schrödinger equation is written as

The above equation finally takes the form

We have considered a real quantum mechanical potential V_A(x), which is termed as A-quantum system (A-QS). 1D-PDM Schrodinger equation for A-QS is

Equation (5) is exactly solvable for V_A(x), i.e. the normalized eigenfunctions ψ_A(x), energy eigenvalues and mass function m_A(x) are known for given V_A(x).

We now invoke a Co-ordinate Transformation (CT)

followed by a Functional Transformation (FT) of the wave function

where ψ_B(x) stands for wave function of transformed quantum system, hereafter called the B-QS. The transformation function g_B(x) must be a differentiable function of at least class C³. The g_B(x) and the modulated amplitude have to be specified within the framework of ET.

Application of ET to Equation (5) of the A-QS, gives

Since mass depends on position, we introduce transformation of mass i.e. m_A(x) of A-QS transforms to new mass function m_B(x) of B-QS, as given below

Therefore Equation (8) for B-QS, takes the intermediate form

Consistency demands that the co-efficient of in Equation (10) must be identical to which gives

and changes Equation (10) to

where

is the Schwartzian derivative symbol and N_B is the integration constant which plays the role of normalization constant of the energy eigenfunctions.

In order to mould Equation (12) to the standard form of 1D-PDM Schrödinger equation, the following plausible ansätze have to be made, which are an integral part of the ET method.

Among these ansätze Equation (14) specifies the functional form of g_B(x). Equations (15) to (18) give the BQS potential V_B(x) as

and effective potential as

Finally 1D-PDM Schrödinger equation for B-QS takes the form

We have considered Hulthen potential as a typical representative of an exactly solved quantum system whose exact solution of 1D-PDM Schrödinger equation is available/possible [12].

The deformed Hulthen potential, denoted as V_A(x) is given by

whereV₀ is the compact form of the three parameters which are atomic number (in atomic system), deformed parameter q and screening parameter. The Hulthen potential is a short range potential which behaves like a Coulomb potential for small values of x and decreases exponentially for large values of x. The deformed Hulthen potential reduces to the Hulthen form for q = 1, to standard Wood-Saxon potential for q = −1 and to the exponential potential for q = 0. It is a special case of the Eckart potential which has been used in many different areas like Nuclear, Atomic, Condensed matter and Chemical Physics.

For

the discrete energy eigenvalues are

and the corresponding eigenfunctions are

where

and

are well known Jacobi Polynomials and N_A is normalization constant.

The parameters and  satisfy the relation

We have presented a method of generating ESPs of PDM Schrödinger equation from a known potential. Two ESPs are generated by successive application of ET on Hulthen potential. The generated potentials are non-power law potential. New mass functions for newly generated systems are also obtained. In constant, mass problems, mass of the generated system and that of parent system remain same, but in case of PDM problems, mass function of parent system gets transformed to new mass function of the generated system. This is a unique feature of this method. Wave functions of generated systems are analytically verified. In case of non-power law potentials, the generated potentials are found to be generally Sturmian. We have converted the Sturmian form of the B-QS potential to normal form by system specific regrouping method. It is evident that the ET may be applied successively any number of times to generate new QSs when we are considering a non power law potential. The present formalism can be generalized to N-dimensional PDM Schrödinger equation.

We are thankful to (Late) Prof. S. A. S. Ahmed, Gauhati University, Guwahati, India, who had initiated the problem and also extended valuable help in the initial stage of the work.