Regularized asymptotics of the solution of a singularly perturbed Cauchy problem for an equation of Schrodinger with potential $Q(x)=x^2$

A. G. Eliseev ; T. A. Ratnikova ; D. A. Shaposhnikova

Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 31-48.

Résumé

In the proposed work, we construct a regularized asymptotics for the solution of a singularly perturbed inhomogeneous Cauchy problem for the Schrodinger equation. The potential $q(x)=x^2$ chosen in the paper leads to a singularity in the spectrum of the limit operator in the form of a strong turning point. The main problem that the researcher faces when applying the regularization method is related to the search and description of regularizing functions that contain a non-uniform singular dependence of the solution of the desired problem, highlighting which, you can search for the rest of the solution in the form of power series in a small parameter. The development of the regularization method led to the understanding that this search is closely related to the spectral characteristics of the limit operator. In particular, it is established how the singular dependence of the asymptotic solution on a small parameter should be described under the condition that the spectrum is stable. When stability conditions are violated, things are much more complicated. 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. In papers devoted to singularly perturbed problems, some of the singularities of this type are called turning points. Based on the ideas of asymptotic integration of problems with an unstable spectrum by S.A. Lomov and A.G. Eliseev, it is indicated how and from what considerations regularizing functions and additional regularizing operators should be introduced, the formalism of the regularization method for the problem posed is described in detail, and justification of this algorithm and an asymptotic solution of any order with respect to a small parameter is constructed.

Détail

Keywords: singularly perturbed problem, asymptotic solution, regularization method, turning point.

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