Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case

Abstract

By Lomov’s S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a ”simple” turning point was considered, i.e., one eigenvalue vanishes for t = 0 and has the form t m / n a ( t ) (limit operator is discretely irreversible). The regularization method allows us to construct an asymptotic solution that is uniform over the entire segment [ 0 , T ] , and under additional conditions on the parameters of the singularly perturbed problem and its right-hand side, the exact solution.

Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2

“…Moreover, there is still no complete mathematical theory for singularly perturbed problems with an unstable spectrum, although they began to be studied from a general mathematical standpoint about 50 years ago. Of particular interest among such problems are those in which the spectral features are expressed in the form of point instability (see, for example, [9][10][11][12]). In works devoted to singularly perturbed problems, some of the features of this type are called turning points, and their classification is as follows:…”

“…Here, we give links to several recent studies in the framework of the method of regularization of singularly perturbed problems with singularities in the spectrum of the limit operator of the indicated form: for a simple turning point, see papers [9,10], for a weak turning point, see [11], and for a strong turning point, see [12,13].…”