Eigenvalues spectrum of quantum integrable systems

The paper considers the eigenvalues spectrum of quantum systems admitting the existence of the first integrals quadratic in momenta within the classical limits. It is stated that a transfer from the classical integrable model to the quantum one is unique, if the integrals of the classical dynamic system depend on momenta quadratically. In this case, both the quantum integrable model and the classical model admit the same three sets of integrable potentials. The sets of multiparameter potentials are studied. The first set is asymptotically isotropic. It represents an infinite-depth well with a finite number of critical points concentrated on the finite area. The second set of multiparameter potentials is a two-dimensional potential barrier. The third set of multiparameter potentials is a two-dimensional potential well with a finite depth. The latter is given as an example.

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