For the example of the infinitely deep well potential, we point out some paradoxes which are solved by a careful analysis of what is a truly self-adjoint operator. We then describe the self-adjoint extensions and their spectra for the momentum and the Hamiltonian operators in different physical situations. Some consequences are worked out, which could lead to experimental checks.

Quantum integrability of classical integrable systems given by quadratic Killing tensors on curved configuration spaces is investigated. It is proven that, using a "minimal" quantization scheme, quantum integrability is insured for a large class of classic examples.

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We consider two indeterminate moment problems: One corresponding to a birth and death process with quartic rates and the other corresponding to the Al-Salam-Carlitz ^-polynomials. Using the Darboux method, we calculate their Nevanlinna matrices and several families of orthogonality measures.