C. A. coulson and the surface energy of metals: The distribution of eigenvalues as a difficult problem in number theory

An article by Coulson on the relationhship of physics and mathematics has attracted our attention to problems already discussed by Titchmarsh, namely the distribution of quantum energy levels of a particle confined in a square or a cube with infinite potential‐energy walls. These problems are identical to some classical problems in the theory of lattice points, which belongs to number theory. They are seemingly simple, but in fact they are very difficult and not yet completely solved. We summarize the most relevant results from number theory. Whereas the beautiful theorems bearing directly on the error with respect to elementary estimates only provide indications concerning the order of that error; theorems concerning mean values can be used as a basis for a physicist's guess on its asymptotic mean value. This paper has also been intended to contribute to bridging the gap between quantum mechanics and number theory.