Exercises in exact quantization | Exercices de quantification exacte

URL: http://www-spht.cea.fr/articles/t00/078 Correction et mise à jour du texte (2003)The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians $[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm$ 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 $\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pm$ 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