Sergii M. Torba

Parabolic Type Equations and Markov Stochastic Processes on Adeles

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.

Transmutations, L-bases and complete families of solutions of the stationary Schrödinger equation in the plane☆

An L-basis associated to a linear second-order ordinary differential operator L is an infinite sequence of functions {φk}k=0∞ such that Lφk=0 for k=0,1, Lφk=k(k−1)φk−2, for k=2,3,… and all φk 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 Schrodinger 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.

Transmutations and spectral parameter power series in eigenvalue problems

We give an overview of recent developments in Sturm-Liouville theory concerning operators of transmutation (transformation) and spectral parameter power series (SPPS). The possibility to write down the dispersion (characteristic) equations corresponding to a variety of spectral problems related to Sturm-Liouville equations in an analytic form is an attractive feature of the SPPS method. It is based on a computation of certain systems of recursive integrals. Considered as families of functions these systems are complete in the L22-space and result to be the images of the nonnegative integer powers of the independent variable under the action of a corresponding transmutation operator. This recently revealed property of the Delsarte transmutations opens the way to apply the transmutation operator even when its integral kernel is unknown and gives the possibility to obtain further interesting properties concerning the Darboux transformed Schrodinger operators.

Spectral parameter power series for perturbed Bessel equations

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.

Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems

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 Khmelnytskaya et?al. (2013), and Kravchenko and Torba (2014). The method allows one to compute both lower and higher eigendata with an extreme accuracy.

Wave polynomials, transmutations and Cauchy’s problem for the Klein–Gordon equation

Abstract We prove a completeness result for a class of polynomial solutions of the wave equation called wave polynomials and construct generalized wave polynomials, solutions of the Klein–Gordon equation with a variable coefficient. Using the transmutation (transformation) operators and their recently discovered mapping properties we prove the completeness of the generalized wave polynomials and use them for an explicit construction of the solution of the Cauchy problem for the Klein–Gordon equation. Based on this result we develop a numerical method for solving the Cauchy problem and test its performance.

Transmutations for Darboux transformed operators with applications

We solve the following problem. Let q1 be a continuous complex-valued potential of a stationary Schr?dinger operator defined on a segment [ ? a, a] and q2 be the potential of a Darboux transformed Schr?dinger operator, that is , where f is a nonvanishing solution of the equation . Suppose a transmutation operator T1 is known such that for any u ? C2[ ? a, a]. Find an analogous transmutation operator for . 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.

Modified spectral parameter power series representations for solutions of Sturm–Liouville equations and their applications☆

Abstract Spectral parameter power series (SPPS) representations for solutions of Sturm–Liouville equations proved to be an efficient practical tool for solving corresponding spectral and scattering problems. They are based on a computation of recursive integrals, sometimes called formal powers. In this paper new relations between the formal powers are presented which considerably improve and extend the application of the SPPS method. For example, originally the SPPS method at a first step required to construct a nonvanishing (in general, a complex-valued) particular solution corresponding to the zero-value of the spectral parameter. The obtained relations remove this limitation. Additionally, equations with “nasty” Sturm–Liouville coefficients 1 / p or r can be solved by the SPPS method. We develop the SPPS representations for solutions of Sturm–Liouville equations of the form p ( x ) u ′ ′ + q ( x ) u = ∑ k = 1 N λ k R k u , x ∈ ( a , b ) where R k u ≔ r k ( x ) u + s k ( x ) u ′ , k = 1 , … N , the complex-valued functions p , q , r k , s k are continuous on the finite segment a , b . Several numerical examples illustrate the efficiency of the method and its wide applicability.

Construction of Transmutation Operators and Hyperbolic Pseudoanalytic Functions

A representation for integral kernels of Delsarte transmutation operators is obtained in the form of a functional series with exact formulae for the terms of the series. It is based on the application of hyperbolic pseudoanalytic function theory and recent results on mapping properties of the transmutation operators. The kernel