Exercises in exact quantization
A. Voros (CEA/Saclay, SPhT, France)

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 on the half-line , 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 as vs its (solvable) 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 are explicitly…
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