Analytic continuation of eigenvalues of a quartic oscillator

TL;DR

This paper investigates the analytic continuation of eigenvalues of a quartic oscillator's Schrödinger operator, revealing their multi-valued nature and finite algebraic singularities, confirming several conjectures.

Contribution

It proves that eigenvalues form two multi-valued analytic functions with finitely many algebraic ramification points, confirming conjectures by Bender, Wu, Loeffel, and Martin.

Findings

Eigenvalues are branches of two multi-valued analytic functions.

Singularities are only algebraic ramification points.

Finitely many singularities over each compact subset.

Abstract

We consider the Schrodinger operator on the real line with even quartic potential and study analytic continuation of eigenvalues, as functions of the coefficient of the potential. 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 plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the plane.