Exercises in exact quantization

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$onthehalf-line${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 $N\to +\mathrm{\infty }$vs its (solvable) $N=\mathrm{\infty }$ 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 $\det (-{\text{d}}^2\mathrm{/}\text{d}{q}^2+q^4+v{q}^2{)}^{\pm }$ are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic $v\to +\mathrm{\infty }$ properties of Airy type. The third study addresses the potentials $V(q)=q^N+v{q}^{N\mathrm{/}2-1}$ 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.