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Theoretical and Mathematical Physics

, Volume 197, Issue 2, pp 1626–1634 | Cite as

Asymptotics of Wave Functions of the Stationary Schrödinger Equation in the Weyl Chamber

  • S. Yu. DobrokhotovEmail author
  • D. S. Minenkov
  • S. B. Shlosman
Article
  • 10 Downloads

Abstract

We study stationary solutions of the Schrödinger equation with a monotonic potential U in a polyhedral angle (Weyl chamber) with the Dirichlet boundary condition. The potential has the form \(U\left( x \right) = \sum _{j = 1}^nV\left( {{x_j}} \right),x = \left( {{x_1}, \ldots ,{x_n}} \right) \in {\mathbb{R}^n}\) , with a monotonically increasing function V (y). We construct semiclassical asymptotic formulas for eigenvalues and eigenfunctions in the form of the Slater determinant composed of Airy functions with arguments depending nonlinearly on xj. We propose a method for implementing the Maslov canonical operator in the form of the Airy function based on canonical transformations.

Keywords

stationary Schrödinger equation boundary value problem Weyl-chamber-type polyhedral angle spectrum quantization condition Maslov canonical operator Airy function 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • S. Yu. Dobrokhotov
    • 1
    • 2
    Email author
  • D. S. Minenkov
    • 1
  • S. B. Shlosman
    • 3
    • 4
    • 5
  1. 1.Ishlinsky Institute for Problems of MechanicsMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)DolgoprudnyRussia
  3. 3.Skolkovo Institute for Science and TechnologyMoscowRussia
  4. 4.Aix Marseille Université, Université de Toulon, CNRS, CPTMarseilleFrance
  5. 5.Kharkevich Institute for Information Transmission Problems, RASMoscowRussia

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