---
author:
- |
    [**André Voros**]{}\
    \
    CEA, Service de Physique Théorique de Saclay\
    F-91191 Gif-sur-Yvette CEDEX (France)\
    [ E-mail : [voros@spht.saclay.cea.fr]{}]{}\
    \
    and\
    \
    Institut de Mathématiques de Jussieu–Chevaleret\
    CNRS UMR 7586\
    Université Paris 7\
    2 place Jussieu, F-75251 Paris CEDEX 05 (France)
title: Exercises in exact quantization$^1$
---

The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger 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 