An introduction to Krein strings

Abstract: Krein strings appear in the study of the motion of a vibrating string where an irregular density is allowed. This thesis presents the theory from the perspective of integral equations and operator theory. It will be shown that each Krein string gives rise to a unique Stieltjes function, by utilizing the compactness of the resolvent operators for short strings and then approximating any long string with a sequence of short strings. The converse is also true: each Stieltjes function gives rise to a unique Krein string and this bijection is called Krein's correspondence. The existence part is proved by constructing Krein strings for a special class of Stieltjes functions. Then, an arbitrary Stieltjes function can be approximated by this class and the limiting procedure yields a string corresponding to this Stieltjes function. The uniqueness part is not treated in this thesis. Instead, some properties and simple examples of Krein's correspondence will be presented.