Analytic continuation of eigenvalues of a quartic oscillator
We consider the Schrödinger operator on the real line with even quartic potential x4 + α x2 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. [ABSTRACT FROM AUTHOR].
EIGENVALUES, MATRICES, NONLINEAR oscillators
Date of this Version
Eremenko, Alexandre and Gabrielov, Andrei, "Analytic continuation of eigenvalues of a quartic oscillator" (2009). Department of Earth, Atmospheric, and Planetary Sciences Faculty Publications. Paper 41.
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