A new representation of solutions to the equation −y ′′ + q(x)y = ω 2 y is obtained. For every x the solution is represented as a Neumann series of Bessel functions depending on the spectral parameter ω. Due to the fact that the representation is obtained using the corresponding transmutation operator, a partial sum of the series approximates the solution uniformly with respect to ω which makes it especially convenient for the approximate solution of spectral problems. The numerical method based on the proposed approach allows one to compute large sets of eigendata with a nondeteriorating accuracy.
A method for approximate solution of spectral problems for Sturm-Liouville equations based on the construction of the Delsarte transmutation operators is presented. In fact the problem of numerical approximation of solutions and eigenvalues is reduced to approximation of a primitive of the potential by a finite linear combination of generalized wave polynomials introduced in [25], [34]. The method allows one to compute both lower and higher eigendata with an extreme accuracy. * Research was supported by CONACYT, Mexico via the project 166141.
We solve the following problem. Given a continuous complex-valued potential q1 defined on a segment [−a, a] and let q2 be the potential of a Darboux transformed Schrödinger operator, that a]. Find an analogous transmutation operator for A2 = d 2 dx 2 − q2(x). It is well known that the transmutation operators can be realized in the form of Volterra integral operators with continuously differentiable kernels. Given a kernel K1 of the transmutation operator T1 we find the kernel K2 of T2 in a closed form in terms of K1. As a corollary interesting commutation relations between T1 and T2 are obtained which then are used in order to construct the transmutation operator for the one-dimensional Dirac system with a scalar potential.
An L-basis associated to a linear second-order ordinary differential operator L is an infinite sequence of functions {ϕ k } ∞ k=0 such that Lϕ k = 0 for k = 0, 1, Lϕ k = k(k−1)ϕ k−2 , for k = 2, 3, . . . and all ϕ k satisfy certain prescribed initial conditions. We study the transmutation operators related to L in terms of the transformation of powers of the independent variable (x − x0) k ∞ k=0 to the elements of the L-basis and establish a precise form of the transmutation operator realizing this transformation. We use this transmutation operator to establish a completeness of an infinite system of solutions of the stationary Schrödinger equation from a certain class. The system of solutions is obtained as an application of the theory of bicomplex pseudoanalytic functions and its completeness was a long sought result. Its use for constructing reproducing kernels and solving boundary and eigenvalue problems has been considered even without the required completeness justification. The obtained result on the completeness opens the way for further development and application of the tools of pseudoanalytic function theory.
Transmutation operators for Sturm-Liouville equationsAccording to the definition given by Levitan [22], let E be a linear topological space, A and B be linear operators: E → E. Let E 1 and E 2 be closed subspaces of E.Definition 1 A linear invertible operator T defined on the whole E and acting from E 1 to E 2 is called a transmutation operator for the pair of operators A and B if it fulfills the following two conditions.1. Both the operator T and its inverse T −1 are continuous in E; 2. The following operator equality is validor which is the same A = T BT −1 .Our main interest concerns the situation when A = − d 2 dx 2 + q(x), B = − d 2 dx 2 , and q is a continuous complex-valued function. Hence for our purposes it will be sufficient to consider the functional space E = C 2 [a, b] with the topology of uniform convergency. For simplicity we will assume that the interval is symmetric with respect to the origin, thus E = C 2 [−a, a].An operator of transmutation for such A and B can be realized in the form (see, e.g., [22] and [23])
A spectral parameter power series (SPPS) representation for regular solutions
of singular Bessel type Sturm-Liouville equations with complex coefficients is
obtained as well as an SPPS representation for the (entire) characteristic
function of the corresponding spectral problem on a finite interval. It is
proved that the set of zeros of the characteristic function coincides with the
set of all eigenvalues of the Sturm-Liouville problem. Based on the SPPS
representation a new mapping property of the transmutation operator for the
considered perturbed Bessel operator is obtained, and a new numerical method
for solving corresponding spectral problems is developed. The range of
applicability of the method includes complex coefficients, complex spectrum and
equations in which the spectral parameter stands at a first order linear
differential operator. On a set of known test problems we show that the
developed numerical method based on the SPPS representation is highly
competitive in comparison to the best available solvers such as SLEIGN2,
MATSLISE and some other codes and give an example of an exactly solvable test
problem admitting complex eigenvalues to which the mentioned solvers are not
applicable meanwhile the SPPS method delivers excellent numerical results.Comment: 25 pages, 5 figures, 7 table
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