The Transmutation Method and Boundary-Value Problems for Singular Elliptic Equations

Abstract

The main content of this book is composed from two doctoral theses: by V. V. Katrakhov (1989) and by S. M. Sitnik (2016). In our work, for the first time in the format of a monograph, we systematically expound the theory of transmutation operators and their applications to differential equations with singularities in coefficients, in particular, with Bessel operators. Along with detailed survey and bibliography on this theory, the book contains original results of the authors. Significant part of these results is published with detailed proofs for the first time. In the first chapter, we give historical background, necessary notation, definitions, and auxiliary facts. In the second chapter, we give the detailed theory of Sonin and Poisson transmutations. In the third chapter, we describe an important special class of the Buschman-Erde´lyi transmutations and their applications. In the fourth chapter, we consider new weighted boundary-value problems with Sonin and Poisson transmutations. In the fifth chapter, we consider applications of the Buschman-Erde´lyi transmutations of special form to new boundary-value problems for elliptic equations with significant singularities of solutions. In the sixth chapter, we describe a universal compositional method for construction of transmutations and its applications. In the concluding seventh chapter, we consider applications of the theory of transmutations to differential equations with variable coefficients: namely, to the problem of construction of a new class of transmutations with sharp estimates of kernels for perturbed differential equations with the Bessel operator, and to special cases of the well-known Landis problem on exponential estimates of the rate of growth for solutions of the stationary Schro¨dinger equation. The book is concluded with a brief biographic essay about Valeriy V. Katrakhov, as well as detailed bibliography containing 648 references.

Solving the Schrödinger equation by reduction to a first-order differential operator through a coherent states transform

“…For those systems explicit solutions can be obtained through standard procedures. Mathematically our work is close to the framework of transmutations of PDE [13,26,31,42,50].…”

A representation of the transmutation kernels for the Schrödinger operator in terms of eigenfunctions and applications

“…G(x, s)u(s) ds (7) relating the operator L := d 2 dx 2 − q(x) with the operator B := d 2 dx 2 as follows. Let u ∈ C 2 [0, π] and u ′ (0) = 0.…”

On One Integral Equation in the Theory of Transform Operators

“…The theory of transform operators is an important branch of modern mathematics, which has numerous applications (see [1][2][3][4][5][6]). The possibility of representation (3) with a sufficiently "good" kernel for a wide class of potentials underlies the classical methods for solving inverse problems in the quantum scattering theory [7,8].…”

“…For Sturm-Liouville equations, transform operators of form (3) were first constructed by B.Ya. Levin (see [1][2][3][4][5][6]).…”