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Name:  
Miloslav Znojil  
Reviewer number:  
9689  
Email:  
znojil@ujf.cas.cz  
Item's zblNumber:  
DE 018 671 631  
Author(s):  
G\'{o}mezUllate, David; Gonz\'{a}lezL\'{o}pez, Artemio; RTodr\'{\i}guez, Miguel:  
Shorttitle:  
Partially solvable problems in quantum mechanics  
Source:  
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)  
Classification:  
 
Primary Classification:  
 
Secondary Classification:  
 
Keywords:  
Partial differential Schr\"{o}dinger equations; closed solutions; algebraization; partial solvability; CalogeroSutherland hamiltonians; elliptic model; quasiexact manybody states; special external field;  
Review:  
Once we find a finitedimensional invariant space M for the action of a quantum Hamiltonian H, we may treat the operator H as a finitedimensional matrix and construct its eigenstates in M (as well as the related eigenvalues) by the ``mere" finitedimensional matrix diagonalization, i.e., algebraically. One then has to distinguish between the two cases, viz, the so called exactly solvable (ES) and quasiexactly 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 reinitiated 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 wavefunction family). Then read the paper in reverse order, skipping only ch. 4 on ''elliptic CalogeroSutherland 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:  