Analytically solvable Hamiltonians for quantum systems with a nearest-neighbour interaction
Year: 2009
Type: article
Abstract: We consider quantum systems consisting of a linear chain of n harmonic oscillators coupled by a nearest-neighbour interaction of the form ( 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.
Authors: G Regniers, J Van der Jeugt
Institution: Ghent University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2062333412