Exercises in exact quantization

TL;DR

This paper reviews exact 1D quantization formalism and applies it to spectral analysis of specific Schrödinger Hamiltonians, revealing identities, special values, and exact quantization formulas for various potentials.

Contribution

It provides a detailed analytical investigation of spectral determinants and zeta functions for several Schrödinger models, extending known results and introducing new exact quantization formulas.

Findings

Spectral determinants for quartic and higher-degree potentials are explicitly computed.

Identities and functional equations for spectral functions are established.

Exact quantization formulas are derived for a class of potentials, extending harmonic oscillator results.

Abstract

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schr\"odinger Hamiltonians $[-\d^2/\d q^2 + V(q)]^\pm$ on the half-line $\{q>0\}$, with a Dirichlet (-) or Neumann (+) condition at q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential $V(q)=q^N$ as $N \to +\infty$vs its (solvable) $N=\infty$ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-N properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants $\det(-\d^2/\d q^2 + q^4 + v q^2)^\pm$ are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic $v \to +\infty$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v q^{N/2-1}$ of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with v as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2); these results probably reflect the presence of supersymmetric potentials in the family above.