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Study on a Quantization Condition and the Solvability of Schrödinger-type Equations

Yuta Nasuda

Abstract

In this thesis, we study a quantization condition in relation to the solvability of Schrödinger equations. This quantization condition is called the SWKB (supersymmetric Wentzel-Kramers-Brillouin) quantization condition and has been known in the context of supersymmetric quantum mechanics for decades. The main contents of this thesis are recapitulated as follows: the foundation and the application of the SWKB quantization condition. The first half of this thesis aims to understand the fundamental implications of this condition based on extensive case studies. It turns out that the exactness of the SWKB quantization condition indicates the exact solvability of a system via the classical orthogonal polynomials. The SWKB quantization condition provides quantizations of energy, which we call the direct problem of the SWKB. We formulate the inverse problem of the SWKB: the problem of determining the superpotential from a given energy spectrum. The formulation successfully reconstructs all conventional shape-invariant potentials from the given energy spectra. We further construct novel solvable potentials, which are classical-orthogonal-polynomially quasi-exactly solvable, by this formulation. We further demonstrate several explicit solutions of the Schrödinger equations with the classical-orthogonal-polynomially quasi-exactly solvable potentials, whose family is referred to as a harmonic oscillator with singularity functions in this thesis. In one case, the energy spectra become isospectral, with several additional eigenstates, to the ordinary harmonic oscillator for special choices of a parameter. By virtue of this, we formulate a systematic way of constructing infinitely many potentials that are strictly isospectral to the ordinary harmonic oscillator.

Study on a Quantization Condition and the Solvability of Schrödinger-type Equations

Abstract

In this thesis, we study a quantization condition in relation to the solvability of Schrödinger equations. This quantization condition is called the SWKB (supersymmetric Wentzel-Kramers-Brillouin) quantization condition and has been known in the context of supersymmetric quantum mechanics for decades. The main contents of this thesis are recapitulated as follows: the foundation and the application of the SWKB quantization condition. The first half of this thesis aims to understand the fundamental implications of this condition based on extensive case studies. It turns out that the exactness of the SWKB quantization condition indicates the exact solvability of a system via the classical orthogonal polynomials. The SWKB quantization condition provides quantizations of energy, which we call the direct problem of the SWKB. We formulate the inverse problem of the SWKB: the problem of determining the superpotential from a given energy spectrum. The formulation successfully reconstructs all conventional shape-invariant potentials from the given energy spectra. We further construct novel solvable potentials, which are classical-orthogonal-polynomially quasi-exactly solvable, by this formulation. We further demonstrate several explicit solutions of the Schrödinger equations with the classical-orthogonal-polynomially quasi-exactly solvable potentials, whose family is referred to as a harmonic oscillator with singularity functions in this thesis. In one case, the energy spectra become isospectral, with several additional eigenstates, to the ordinary harmonic oscillator for special choices of a parameter. By virtue of this, we formulate a systematic way of constructing infinitely many potentials that are strictly isospectral to the ordinary harmonic oscillator.
Paper Structure (171 sections, 3 theorems, 396 equations, 81 figures, 13 tables)

This paper contains 171 sections, 3 theorems, 396 equations, 81 figures, 13 tables.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Backgrounds/Brief History
  4. Motivations
  5. Contributions
  6. Structure of Thesis
  7. Schrödinger Equation
  8. Exactly Solvable Quantum Mechanics
  9. Problem Setting
  10. Classification of Problems
  11. Analyticity of a potential
  12. Solvability of a potential
  13. Formulation
  14. Factorized Hamiltonian
  15. Associated Hamiltonians, Intertwining Relations, Crum's Theorem
  16. ...and 156 more sections

Key Result

Theorem 2.1

Any positive semi-definite hermitian matrix $\mathsf{A}$ can be factorized as a product of a certain matrix, say $\mathsf{P}$ and its hermitian conjugate $\mathsf{P}^{\dag}$, $\mathsf{A} = \mathsf{P}^{\dag}\mathsf{P}$.

Figures (81)

  • Figure 1: Interrelation between the partner Hamiltonians $\mathcal{H}^{[0]}$, $\mathcal{H}^{[1]}$. They are related to each other by Darboux--Crum transformation. The spectral property can be seen as a realization of supersymmetry in quantum mechanics.
  • Figure 2: A visualization of Crum's theorem. It reveals the general structure of the solution space of the one-dimensional Schrödinger equation.
  • Figure 3: Several classes of exactly solvable quantum mechanical systems in connection with the conventional shape-invariant systems.
  • Figure 4: The relation between Sect. \ref{['sec:3-2']} and Sect. \ref{['sec:3-3']}.
  • Figure 5: $g=3$.
  • ...and 76 more figures

Theorems & Definitions (53)

  • Theorem 2.1: Linear algebra
  • Remark 2.1: Multi-degrees of freedom
  • Remark 2.2: Can you always factorize your Hamiltonian?
  • Theorem 2.2: Crum, 1955 10.1093/qmath/6.1.121
  • Remark 2.3: Supersymmetric quantum mechanics
  • Example 2.1: H
  • Example 2.2: L
  • Example 2.3: J
  • Example 2.4: Coulomb potential
  • Remark 3.1
  • ...and 43 more