#
Past Seminars Seminários Já Decorridos
2006

## 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] 11:30

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.