# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### An introduction to Krein strings

(2017) In Master's Theses in Mathematical Sciences MATM01 20162
Mathematics (Faculty of Sciences)
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... (More)
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. (Less)
Popular Abstract (Swedish)
I den klassiska modellen för en endimensionell vibrerande sträng antas massan vara likformigt fördelad, vilket leder till den ordinära differentialekvationen f''= zg f, där densiteten g är konstant. Kreins strängteori handlar om samma ekvation, men massfördelningen tillåts variera. Denna teori används även för att lösa problemet att förutsäga framtiden med hjälp av information från en ändlig del -2T < t < 0 av dåtiden för endimensionella stokastiska normalprocesser med väntevärde 0. Detta arbete ger en behandling av Kreins strängteori, med fokus på Kreins korrespondens -- problemet där spektraldata är givna i form av en så kallad Stieltjesfunktion och vi vill veta så mycket som möjligt om strängen som funktionen kommer från.
author
supervisor
organization
course
MATM01 20162
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Spectral theory, Krein strings, vibrating strings, non-constant density
publication/series
Master's Theses in Mathematical Sciences
report number
LUNFMA-3090-2017
ISSN
1404-6342
other publication id
2017:E11
language
English
id
8905739
2017-04-18 17:13:56
date last changed
2017-04-18 17:13:56
```@misc{8905739,
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.},
author       = {Truong, Tien},
issn         = {1404-6342},
keyword      = {Spectral theory,Krein strings,vibrating strings,non-constant density},
language     = {eng},
note         = {Student Paper},
series       = {Master's Theses in Mathematical Sciences},
title        = {An introduction to Krein strings},
year         = {2017},
}

```