An exact solution method for 1D polynomial Schr\"odinger equations

TL;DR

This paper introduces an exact solution method for 1D polynomial Schrödinger equations by reducing them to explicit quantization conditions involving spectral determinants, enabling precise eigenvalue computation.

Contribution

It develops a novel reduction of 1D polynomial Schrödinger equations to explicit quantization conditions based on spectral determinants, improving eigenvalue accuracy.

Findings

Numerical schemes show geometric convergence to eigenvalues.

Method applies to symmetric quartic oscillators.

Reduction suggests a fixed-point formulation for solutions.

Abstract

Stationary 1D Schr\"odinger equations with polynomial potentials are reduced to explicit countable closed systems of exact quantization conditions, which are selfconsistent constraints upon the zeros of zeta-regularized spectral determinants, complementing the usual asymptotic (Bohr--Sommerfeld) constraints. (This reduction is currently completed under a certain vanishing condition.) In particular, the symmetric quartic oscillators are admissible systems, and the formalism is tested upon them. Enforcing the exact and asymptotic constraints by suitable iterative schemes, we numerically observe geometric convergence to the correct eigenvalues/functions in some test cases, suggesting that the output of the reduction should define a contractive fixed-point problem (at least in some vicinity of the pure $q^4$ case).