Exercises in exact quantization

Details Exercises in exact quantization Exercises in exact quantization

2000-05-30

arxiv.org/abs/math-ph/0005029v2

J.Phys.A33:7423-7450,2000

Physics

Mathematical Physics

latex txt.tex, 2 files, 34 pages [SPhT-T00/078]; v2: corrections and updates as indicated by footnotes

Scientific paper

10.1088/0305-4470/33/41/311

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.

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