Quasi-exactly solvable models and their physical realization
By: Oleg B. Zaslavskii
From: Faculty of Mechanics and Mathematics, Kharkov National University,
Svobody sq., 4, Kharkov, Ukraine
At: Complexo Interdisciplinar, Anfiteatro
[2006-12-06]
We discuss exact correspondence that exists between energy spectra of some spin (pseudospin) systems and low-lying states of a particle moving in potentials of a certain form. For the coordinate system this gives rise to partial algebraization of the spectrum (so-called quasi-exactly solvable models - QES). In this sense, QES occupies an intermediate position between non-solvable and exactly solvable models of quantum mechanics. For spin systems this leads to the exact effective potentials that enables to develop well-elaborated technique for description of their properties (in particular, the phenomenon of spin tunneling). The similar partial algebraization of the spectrum occurs also in spin-boson and pure boson systems. In the the two- and many-dimensional case the Schrodinger equation corresponding to QES models is in general defined on curved Riemann manifolds. Thus, QES represents a rare case when the underlying algebraic structure of the Schrodinger equation that ensures the existence of exact solutions has direct physical meaning by itself. QES unifie such so different objects as quantum spin systems, quasiparticles, curved manifolds typical of general relativity, etc.