Multiparameter spectral theory

Abstract

Introduction. On an occasion calling for a topic of general mathematical interest, I owe my audience some explanation for having chosen one of the less known areas of analysis as my subject today. The idea of considering eigenvalue problems with several parameters occurs quite naturally in certain boundary-value problems, in particular those which lead to the Mathieu functions; the area in question is thus not unknown to the applied mathematician or the specialist in the "special functions of analysis." However, the exploitation of this area in the spirit of classical analysis, in particular for ordinary differential equations, has reached no more than a preliminary stage. The same is true when we examine this area in the spirit of linear algebra or again of functional analysis. It is when we look at this area in a more abstract way that another justification appears for recommending it as worthy of general interest* The formal theory of eigenvalue problems with several parameters makes extensive use of algebraic ideas which, though not new, have gained great vogue in recent years.