Title: The Liouville transformation in Sturm-Liouville theory Language: English Authors: Kleindeßner, Matthäus Qualification level: Diploma Keywords: Sturm-Liouville Theorie; Liouville Transformation; inverse SpektraltheorieSturm-Liouville theory; Liouville transformation; inverse spectral theory Advisor: Woracek, Harald Issue Date: 2012 Number of Pages: 72 Qualification level: Diploma Abstract: 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:1-60594http://hdl.handle.net/20.500.12708/12928 Library ID: AC07814233 Organisation: E101 - Institut für Analysis und Scientific Computing Publication Type: ThesisHochschulschrift Appears in Collections: Thesis

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