Primary classification:
81Q05Closed and approximate solutions to the Schroedinger, Dirac, Klein-Gordon and other quantum-mechanical equations
Secondary classification:
39B22Equations for real functions

A. G. Ushveridze, in his monograph ``Quasi-Exactly Solvable Models in Quantum Mechanics" (IOP Publishing, Bristol, 1994, ISBN: 0750302666), reviews many models where a ``sufficiently elementary" wave function (or a finite multiplet of such wave functions) satisfies Schroedinger equation for a suitably (or ``selfconsistently") determined interaction. The subject proved interesting in various applications and one is quite surprised to read that the Lokshin's note in question (giving precisely a new simple result of this type) cites no (!) references. Not that this would be perceived as unusual in this apparently ``almost trivial" context where one picks up ANY wave function and differentiates it twice. In fact, the Ushveridze's list of references is also extremely scarce. Still, each of the results of this type (citing the other authors or not) deserves attention, especially when it has some non-trivial component in it. This criterion seems satisfied by the short note in question, where the ``necessary and sufficient" functional-equation relation (1) between the ansatz and potentials is solved, in Lemma 2, completely. One feels sorry that due to the above-mentioned connection, the absence of citations transferred this text from the category of very interesting contributions to the field to the category of texts which remain unnoticed by the vast majority of the eligible interested readers. Especially because the latter argument becomes significantly strengthened by the parallel closeness of the subject to another broad area of the so called Calogero-Moser-Sutherland exactly solvable models and of their Inozemtsev-type partially solvable generalizations. Let me skip the details here: Reviewers are not expected to write the extended abstracts which would be much longer than the papers they re-comment.

title < "On a family of exactly solvable multiparticle Schr"odinger equations with pair potential.">
keywords <"N particles on a line; Schroedinger equation; pair potential; ground state; elementary solution; ">
author <"Lokshin, A.A.">
journal <"Math. Notes 70, No.1, 137-142; translation from Mat. Zametki 70, No.1, 150-154 (2001). [ISSN 0001-4346]">