Equivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun differential equations, eigenstates degeneration, and the rabi model

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Abstract

The initial aim of the present paper is to provide a complete description of the eigenvalue problem for the non-commutative harmonic oscillator (NcHO), which is defined by a (two-by-two) matrix-valued self-adjoint parity-preserving ordinary differential operator [28], in terms of Heun's ordinary differential equations, the second-order Fuchsian differential equations with four regular singularities in a complex domain. This description has been achieved for odd eigenfunctions in Ochiai [25] nicely but missing up to now for the even parity which is more important from the viewpoint of determination of the ground state of the NcHO. As a by-product of this study examining the monodromy data (characteristic exponents, etc.) of the Heun equation, we prove that the multiplicity of the eigenvalue of the NcHO is at most two. Moreover, we give a condition for the existence of a finite-type eigenfunction (i.e., given by essentially a finite sum of Hermite functions) for the eigenvalue problem and an explicit example of such eigenvalues, from which one finds that doubly degenerate eigenstates of the NcHO actually exist even in the same parity Also, we determine the possible shape of (so-called) Heun polynomial solutions of the Heun equations, which are obtained by the eigenvalue problem.

Original languageEnglish
Pages (from-to)759-794
Number of pages36
JournalInternational Mathematics Research Notices
Volume2016
Issue number3
DOIs
Publication statusPublished - Jan 1 2016

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Heun Equation
Degeneration
Harmonic Oscillator
Eigenvalue Problem
Equivalence
Parity
Differential equation
Eigenfunctions
Eigenvalue
Hermite Functions
Characteristic Exponents
Polynomial Solutions
Monodromy
Finite Type
Model
Ground State
Differential operator
Multiplicity
Ordinary differential equation
Odd

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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title = "Equivalence between the eigenvalue problem of non-commutative harmonic oscillators and existence of holomorphic solutions of Heun differential equations, eigenstates degeneration, and the rabi model",
abstract = "The initial aim of the present paper is to provide a complete description of the eigenvalue problem for the non-commutative harmonic oscillator (NcHO), which is defined by a (two-by-two) matrix-valued self-adjoint parity-preserving ordinary differential operator [28], in terms of Heun's ordinary differential equations, the second-order Fuchsian differential equations with four regular singularities in a complex domain. This description has been achieved for odd eigenfunctions in Ochiai [25] nicely but missing up to now for the even parity which is more important from the viewpoint of determination of the ground state of the NcHO. As a by-product of this study examining the monodromy data (characteristic exponents, etc.) of the Heun equation, we prove that the multiplicity of the eigenvalue of the NcHO is at most two. Moreover, we give a condition for the existence of a finite-type eigenfunction (i.e., given by essentially a finite sum of Hermite functions) for the eigenvalue problem and an explicit example of such eigenvalues, from which one finds that doubly degenerate eigenstates of the NcHO actually exist even in the same parity Also, we determine the possible shape of (so-called) Heun polynomial solutions of the Heun equations, which are obtained by the eigenvalue problem.",
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