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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 018 671 631
G\'{o}mez-Ullate, David; Gonz\'{a}lez-L\'{o}pez, Artemio; RTodr\'{\i}guez, Miguel:
Partially solvable problems in quantum mechanics
In: Bajop, Ignacio (ed.) et al. Recent advances in Lie theory.(Vigo, Spain, July 17 - 22, 2000. Lemgo: Heldermann Verlag.. Res. Expo. Math. 25, 211 - 231 (2002)
81Q05Closed and approximate solutions to the Schroedinger, Dirac, Klein-Gordon and other quantum-mechanical equations
Primary Classification:
35C05Solutions in closed form
Secondary Classification:
37J15Symmetries, invariants, invariant manifolds, momentum maps, reduction
47N50Applications in quantum physics
47F05Partial differential operators
Partial differential Schr\"{o}dinger equations; closed solutions; algebraization; partial solvability; Calogero-Sutherland hamiltonians; elliptic model; quasi-exact many-body states; special external field;

Once we find a finite-dimensional invariant space M for the action of
a quantum Hamiltonian H, we may treat the operator H as a
finite-dimensional matrix and construct its eigenstates in M (as well
as the related eigenvalues) by the ``mere" finite-dimensional matrix
diagonalization, i.e., algebraically. One then has to distinguish
between the two cases, viz, the so called exactly solvable (ES) and
quasi-exactly solvable (QES) models (in the latter case, not all of
the existing or ``physical" eigenstates of H are obtainable in this
manner). In both cases, there is a big difference between the
ordinary differential (OD) and partial differential (PD) forms of H
(which is usually chosen to be of the second order only, for physical
reasons). Thus, many OD ES quantum models (not, of course, known as
``quantum" originally) are already available for centuries while the
first ``genuine" PD ES models have only been suggested cca thirty
years ago (by F. Calogero and, independently, B. Sutherland). In
comparison, the popularity of QES Hamiltonians started only recently,
initiated by several independent (and, today, mostly, forgotten)
discoveries of their (really very simple) OD versions in the sixties
and seventies, and re-initiated by several other people who imagined
the importance of QES models (as well as their multiple connections
with other models) cca 15 year ago. In this context, present paper is
a review which maps the fourth and very freshly discovered territory
of PD QES models. The paper offers their systematic review and list
(based mainly on the two recent papers by the same authors) and a
concise discussion of their algebraic background and of some of their
most important characteristic properties. Recommended reading of this
text: Start form the last three pages where an illustrative example
is shown in full detail (and with six nice colored portraits of a
typical PD QES wave-function family). Then read the paper in reverse
order, skipping only ch. 4 on ''elliptic Calogero-Sutherland model"
as the best part which ''coronat opus", the consumption of which is
to be postponed to the very end to be fully appreciated. For some
people it will then be difficult not to return to the original refs.
[11] and [12] and to their Calogero's predecessor [7].
Remarks to the editors:

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