studies concerned with individual eigenvalue or eigenfunction behaviour, we seek the limiting properties of spectral functions, which are symmetric functions of all the eigenvalues at once. We take a semi-rigorous approach, in which we argue a global operational scheme without claiming absolute completeness in every detail.

The paper is organized as follows.

The introductory Sec.1 gives a detailed survey of the exact tools to be used here, while an Appendix collects all the results concerning homogeneous potentials. This is done partly for convenience, given the lack of comprehensive reviews for results that are very scattered in time and place of publication, and also to clarify some parts of our most recent developments where localized inconsistencies went undetected.

Then, Sec.2 (Exercise I) considers the family of homogeneous potentials $V(q)=|q|^N$ as the degree $N$ tends to $+\infty$. This is a most singular problem, for which many explicit results are however available beforehand, and the limiting problem (an infinite square well) is exactly solved by elementary means. We therefore mainly propose and test some general principles of investigation, rather than claim truly new results. In particular, we suggest criteria for sorting out regular vs singular types of limiting behaviour in spectral zeta functions 