Exercises in exact quantization (Q2711827)

As the author observes, exact quantization, or exact WKB analysis, supplies new tools for the analytical study of the 1D Schrödinger equation, including arbitrary polynomial potentials. This is a clearly written paper where the author reviews in detail the formalism of exact 1D quantization and applies it to the spectral study of three concrete Schrödinger Hamiltonians \([-\partial^2/ \partial q^2+V (q)]^\pm\) on the half-line \(\{q>0\}\), with a Dirichlet \((-)\) or Neumann \((+)\) condition at \(q=0\). Three rather different quantum potentials are chosen to illustrate a variety of situations, which have some basic common features: they are rather simple, with a minimal number of parameters to remain concretely manageable, and one crucial parameter governs the transition to a singular limit, creating an interesting dynamical and analytical situation. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters.NEWLINENEWLINENEWLINEFirst, the homogeneous potential \(V(q)=q^N\) as \(N\to+ \infty\) vs its (solvable) \(N=\infty\) limit (an infinite square well) is discussed and useful distinctions are established between regular and singular behaviours of spectral quantities. Moreover, various identities among the square-well spectral functions are unraveled, as limits of finite-\(N\) properties. In summary, here the author mainly proposes and tests some general principles of investigation, rather than claims truly new results.NEWLINENEWLINENEWLINEThe second model is the quartic anharmonic oscillator family \(V(q)= q^4+vq^2\), which is the most common model for singular perturbation theory (the free harmonic oscillator emerges in the \(v\to +\infty\) limit). Here the zero-energy spectral determinants \(\det(- \partial^2/ \partial q^2+ q^4+vq^2)^\pm\) are explicitly analyzed, 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.NEWLINENEWLINENEWLINEThe third study gives a complete treatment of the similar determinants for a different class of binomial potentials, \(V(q)=q^N +vq^{N/2-1}\) of even degree \(N\). Here the formalism yields fully closed forms for the zero-energy determinants \(\det[-\partial^2/ \partial q^2+V(q)]\) in terms of gamma functions, with exponential prefactors requiring a careful exact calculation. The generalized spectrum (in the \(v\) variable) also becomes completely explicit under an exact quantization condition. The overall analytical structure is a complete generalization of the familiar, exact harmonic-oscillator results, and it seemingly describes a zero-energy cross-section of the formalism for quasi-exactly solvable models. Although all three problems rely on the same background formalism, they can be approached fairly independently from one another. Accordingly, there are no global conclusions and each case carries its own concluding remarks by the author.