Analytically solvable Hamiltonians for quantum systems with a nearest neighbour interaction

Abstract

We consider quantum systems consisting of a linear chain of n harmonic oscillators coupled by a nearest neighbour interaction of the form $-q_r{q}_{r+1}$ ($q_r$ refers to the position of the $r$th oscillator). In principle, such systems are always numerically solvable and involve the eigenvalues of the interaction matrix. In this paper, we investigate when such a system is analytically solvable, i.e. when the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This is the case when the interaction matrix coincides with the Jacobi matrix of a system of discrete orthogonal polynomials. Our study of possible systems leads to three new analytically solvable Hamiltonians: with a Krawtchouk interaction, a Hahn interaction or a q-Krawtchouk interaction. For each of these cases, we give the spectrum of the Hamiltonian (in analytic form) and discuss some typical properties of the spectra.

References in corpus (13)

• Dynamics and manipulation of entanglement in coupled harmonic systems with many degrees of freedom
• Correlations, spectral gap, and entanglement in harmonic quantum systems on generic lattices
• Quantum vs Classical Integrability in Calogero-Moser Systems
• An exactly solvable many-body problem in one dimension
• A class of N-body problems with nearest- and next-to-nearest neighbour interactions
• Many-Body Wigner Quantum Systems
• The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator
• Wigner quantum oscillators. Osp(3/2) oscillators
• A non-commutative n-particle 3D Wigner quantum oscillator
• Harmonic oscillator chains as Wigner Quantum Systems: periodic and fixed wall boundary conditions in gl(1|n) solutions
• Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System
• "A Solvable Hamiltonian System" Integrability and Action-Angle Variables
• Exact solution of a many body problem with nearest and next-nearest neighbour interactions

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• Dynamical analysis of mass-spring models using Lie algebraic methods