This work studies the asymptotic formulas for the solutions of the Sturm-Liouville equation with the polynomial dependence in the spectral parameter. Using these asymptotic formulas it is proved some trace formulas for the eigenvalues of a simple boundary problem generated in a finite interval by the considered Sturm-Liouville equation.
Sturm-Liouville Equation Asymptotic Formulas for Solutions Spectral Parameter Eigenvalue Boundary Value Problem Trace Formula Fractional Integrals and Derivatives1. Introduction
Consider the differential equation
where are complex valued functions and is a complex parameter.
Differential equations of type (1) often appear in connection with some spectral problems and nonlinear evolution equations (see [1] [2] [3] ). In the case the equation is the classical Sturm-Liouville equation and in this case there are a wide class of spectral problems and inverse spectral problems which were investigated by constructing integral representations for the independent solutions of the Sturm-Liouville equation (see [4] ). We studied in [5] , the solutions of the Equation (1) satisfying the initial conditions
and it is proved that in the sectors of complex plane
the solutions have the following integral representations:
where, and,
belong to and respectively. Moreover, if denotes Riemann-Liouville fractional derivative of order (see [6] ) with respect to t, i.e.
then for all the functions and belong to and respectively. Furthermore, the following equalities are valid:
where
In the present paper we use the above facts about special solutions of the Equation (1) to obtain some trace formulas for the boundary value problem generated by the Equation (1) in the segment with simple boundary conditions
2. Asymptotic Formulas and the Trace Formulas
Using (2), (3) and (4) it is easy to prove the following theorem where we seek two solutions which have special representations.
Theorem 1. If and then the Equation (1) has solutions
and
where
Since the solutions and are linearly independent for we have
for the solution of the Equation (1) with initial conditions
Then the Theorem 1 gives
where
Now let us connect the Equation (1) to the boundary conditions
In [2] it is obtained the asymptotic formulas for the eigenvalues of the boundary value problem (1)-(2). Let be a characteristic function of this boundary value problem. Then
where
Let us consider the circles where is sufficiently large integer. On circles the functions and are
bounded by the constants independent of. So we have that the module of the maximum of the function approaches to zero when. Hence, if are the series of eigenvalues of the problem (1), (18) we have
Using (19) and (20) we compute the integrals on the right hand side of the Equation (21) and prove the following theorem.
Theorem 2. If are the series of eigenvalues of the boundary value problem (1), (18) then
where are constants defined by the help of the functions . Here
in which the numbers are defined from the asymptotic equality
From Theorem 2 we have that if the Fourier series are conver-
gent and denoting their sums by we obtain the following regularized trace formulas for the eigenvalues of the boundary value problem (1), (18):
Cite this paper
Adiloglu Nabiev, A. (2016) Asymptotic Formulas of the Solutions and the Trace Formulas for the Polynomial Pencil of the Sturm-Liouville Operators. Applied Mathematics, 7, 2411-2417. http://dx.doi.org/10.4236/am.2016.718189
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