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On the singularities of potentials of exactly soluble Schrödinger equations and on Hadamard's problem

The theorem is deduced from the lemma by simple local analysis in the neighbourhood of a non-singular point of a singular hypersurface of the potential: it is proved that in any such neighbourhood the singularity must be a hyperplane. In the known examples of multidimensional exactly soluble Schrödinger equations—the quantized Calogero–Moser problems and their generalizations (see [1], [2]), the singularities form special configurations of hyperplanes. Our result clarifies this circumstance. The problem of describing all these configurations has been solved only in dimension 1 [3] and also in dimension 2 under the assumption that all lines go through one point [4]. The special interest in this problem is explained in particular by its connection with the famous Hadamard problem of describing all second-order hyperbolic equations satisfying the Huygens principle (see, for example, [5]).