Shooting function for 1D Schrodinger operators

TL;DR

This paper introduces a shooting function approach for 1D Schrödinger operators with potentials tending to infinity, proving the function's entire nature and its zeros correspond to eigenvalues.

Contribution

The paper develops a novel shooting function method for analyzing eigenvalues of 1D Schrödinger operators with unbounded potentials, establishing key mathematical properties.

Findings

Shooting function is entire for the considered operators.

Zeros of the shooting function correspond to eigenvalues.

Provides a new analytical tool for spectral analysis.

Abstract

For Schrodinger operators with suitable 1D potentials, focussing particularly on those that go to infinity at infinity, a characteristic function is constructed, via shooting functions. It is proved to be entire and its zeroes to be the eigenvalues.