Periodic Quasi-Exactly Solvable Models

Abstract

Various quasi-exact solvability conditions, involving the parameters of the periodic associated Lamé potential, are shown to emerge naturally in the quantum Hamilton-Jacobi approach. It is found that, the intrinsic nonlinearity of the Riccati type quantum Hamilton-Jacobi equation is primarily responsible for the surprisingly large number of allowed solvability conditions in the associated Lamé case. We also study the singularity structure of the quantum momentum function, which yields the band edge eigenvalues and eigenfunctions. * akksprs@uohyd.ernet.in † akksp@uohyd.ernet.in ‡ prasanta@prl.ernet.in

Supersymmetric partners for the associated Lamé potentials

“…In section 3 we will apply the supersymmetry transformations for generating new periodic and asymptotically periodic potentials which are almost isospectral to the initial associated Lamé potential. In section 4 our general results will be illustrated through the particular case characterized by (m, ℓ) = (3,2). Our conclusions will be finally given at section 5.…”

Solutions of Two-Mode Bosonic and Transformed Hamiltonians

“…x−x k is sum of the moving pole terms [26][27][28][29][30][31][32][33] that can be expressed as P (x)…”

“…On the other hand, quantum Hamilton-Jacobi (QHJ) formalism has generated much interest [22,24,25]. The application of QHJ to eigenvalues has been explored in great detail by Bhalla, Kapoor and collaborators [26][27][28][29][30][31][32][33]. QES systems have been also studied within QHJ approach [31].…”

A quantum Hamilton–Jacobi proof of the nodal structure of the wave functions of supersymmetric partner potentials

“…The use of the singularity structure information has provided interesting insights into the models studied. It has been shown that, the singularity structure of the QMF is markedly different, for the exactly solvable (ES) [6], [7], quasi-exactly solvable (QES) [8], [5], periodic potentials [9], [10] and the new rational potentials, with exceptional polynomials as solutions [11]- [13]. For supersymmetric one dimensional ES potentials [14], [15], the singularity structure of the QMF provided a link to the exactness of the SWKB integral [13], [16].…”