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Difference Schrödinger operators. (English. Russian original) Zbl 0970.34074
Proc. Steklov Inst. Math. 224, 250-265 (1999); translation from Tr. Mat. Inst. Steklova 224, 275-290 (1999).
This paper is mainly a survey on difference and differential second-order Schrödinger operators $$L$$ on simplexes: $$L:C^k\to C^k$$. The operator $$L$$ is called factorizable if it is of the form $$L=Q^+Q+W_0$$ with a potential $$W_0$$, where $$Q:C^k\to C^{k+q}$$, $$Q^+:C^{k+q}\to C^k$$ are first-order operators. Together with this operator, its Euler-Darboux transform $$\widetilde L=QQ^++W_0$$ is considered. Most of the results stated in this paper relate to the survey article of the author and I. A. Dynnikov [Russ. Math. Surv. 52, No. 5, 1057-1116 (1997); translation from Usp. Mat. Nauk 52, No. 5, 175-234 (1997; Zbl 0928.35107)] and the two papers of A. P. Veselov and the author [Russ. Math. Surv. 50, No. 6, 1316-1317 (1995); translation from Usp. Mat. Nauk 50, No. 6, 171-172 (1995; Zbl 0855.35031)] and of the author and A. P. Veselov [in: Buchstaber, V. M. (ed.) et al., Solitons, geometry, and topology: on the crossroad. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 179(33), 109-132 (1997; Zbl 0922.35141)].
After a short review on differential operators, the author focusses on difference operators. Special attention is paid to the one-dimensional case and lattices on two-dimensional manifolds. Finally, exactly solvable operators in the sense that $$L=Q^+Q$$ are considered.
For the entire collection see [Zbl 0942.00074].
##### MSC:
 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 39A10 Additive difference equations 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations