Title: The Liouville transformation in Sturm-Liouville theory
Language: English
Authors: Kleindeßner, Matthäus 
Qualification level: Diploma
Keywords: Sturm-Liouville Theorie; Liouville Transformation; inverse Spektraltheorie
Sturm-Liouville theory; Liouville transformation; inverse spectral theory
Advisor: Woracek, Harald
Issue Date: 2012
Number of Pages: 72
Qualification level: Diploma
This thesis deals with Sturm-Liouville theory, the vast area of mathematical research based on the examination of second-order linear ordinary differential equations of the kind $-(py')'+qy=\lambda r y$ on $(a,b) \subseteq \R$ (SL equation). Herein $\lambda$ is a complex-valued spectral parameter.\\ In 1837 Liouville introduced a transformation (today known as Liouville transformation) which enables reducing the general SL equation to the more special and simpler one $-y''+ ilde{q}y=\lambda y$ on $( ilde{a}, ilde{b}) \subseteq \R$. It is the main task of this thesis to systematically discuss Liouville´s transformation in a rather general way, i.e. we do not only consider the Liouville transformation which transforms $-(py')'+qy=\lambda r y$ into $-y''+ ilde{q}y=\lambda y$, but consider related transformations (which we also call a Liouville transformation) transforming $-(py')'+qy=\lambda r y$ into another equation of this kind (i.e. $p$, $q$, $r$ and $(a,b)$ replaced by $ ilde{p}$, $ ilde{q}$, $ ilde{r}$ and $( ilde{a}, ilde{b})$). Liouville´s original transformation is then obtained as a special case.
In this thesis we examine invariance properties of SL equations under a Liouville transformation and deal with the question to which extent a Liouville transformation can be used to transform a given equation into a simpler one. We see that a Liouville transformation gives rise to a unitary mapping between Hilbert spaces and that certain operators associated with SL equations are unitarily equivalent via this mapping.
A main result of this thesis is an inverse theorem stating sufficient conditions for the existence of a Liouville transformation such that two considered operators are unitarily equivalent via it (considered as a mapping between the associated Hilbert spaces).\\ Clearly, depending on the underlying situation one can expect the functions $p$, $q$ an $r$ to satisfy different conditions. Many attempts have been made to keep necessary conditions for mathematical treatment of SL equations as general as possible - on the other hand, Liouville´s original transformation requires considerable restrictions on the coefficients $p$, $q$ and $r$ for its feasibility. In this thesis we consider the classical right-definite case of Sturm-Liouville Theory and attempt to keep additional restrictions for working with the concept of a Liouville transformation as lean as possible.
URI: https://resolver.obvsg.at/urn:nbn:at:at-ubtuw:1-60594
Library ID: AC07814233
Organisation: E101 - Institut für Analysis und Scientific Computing 
Publication Type: Thesis
Appears in Collections:Thesis

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