Toward a general theory of transmutation

Abstract

AbstractA general construction of transmutation operators is developed for selfad-jointoperatorsinGelfandtriples. Theoremsregardinganalyticity ofgeneralizedeigenfunctions and Paley-Wiener properties are proved. 1 Introduction The idea of transmutation operator (or transformation operator) B such that BP= QB for P and Q ordinary differential operators goes back to Gelfand, Levitan,Marchenko, Naimark, et. al. in the early 1950’s (cf. [23;25;28]). It was picked upagain by Delsarte and Lions, who established some fundamental ideas (cf. [26;31]),and subsequently it was developed in many directions (see e.g. [2;10-14;17;22;31]). Inthis article we indicate some constructions of a general nature which will be furtherenhanced in subsequent papers. We develop the theory via selfadjoint operators inGelfand triples and give some constructions of transmutation operators with variousdomains. Then various properties such as analyticity of generalized eigenfunctionsand Paley-Wiener properties are discussed, with results of various kinds.