Analytic Continuation of Eigenvalues of a Quartic Oscillator

We consider the Schrödinger operator on the real line with even quartic potential x 4  +  α x 2 and study analytic continuation of eigenvalues, as functions of parameter α . We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues ar...

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Bibliographic Details
Published in:Communications in mathematical physics Vol. 287; no. 2; pp. 431 - 457
Main Authors: Eremenko, Alexandre, Gabrielov, Andrei
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.04.2009
Springer
Subjects:
Classical and Quantum Gravitation
Complex Systems
Exact sciences and technology
Functions of a complex variable
Mathematical analysis
Mathematical and Computational Physics
Mathematical methods in physics
Mathematical Physics
Mathematics
Other topics in mathematical methods in physics
Partial differential equations
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Sciences and techniques of general use
Several complex variables and analytic spaces
Theoretical
Cell Decomposition
Meromorphic Function
Riemann Surface
Analytic Continuation
Anharmonic Oscillator
Complex function
Eigenvalues
Eigenfunctions
Analytic functions
Analytic continuation
Mathematical physics
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary:We consider the Schrödinger operator on the real line with even quartic potential x 4  +  α x 2 and study analytic continuation of eigenvalues, as functions of parameter α . We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α -plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α -plane.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-008-0663-6