Doctoral theses
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| Identifier |
uch.physics.phd//2000pliakis |
| Title |
Asymptotic behaniour of Schrodinger and Euler Lagrange equations near singular points |
| Alternative Title |
Ασυμπτωτική συμπεριφορά εξισώσεων Schrodinger και Euler-Lagrange κοντά σε ιδιόμορφα σημεία |
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Author
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Πλιάκης, Δημήτρης
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Thesis advisor
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Φλωράτος, Ε.
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| Abstract |
In this thesis there are treated problems of singularities that are encountered in the Schrodinger aand Euler-Lagrange equations: there are provided asympotic expansions of functions that are interesting in Physics. Specifically in the first part it is studied the small time asymptotic expansion of the heat trace of the general Schrodinger with regular singularities. This asymptotic expansin allows us to examine the conditions for the spectral determination of the potential appearing in such an operator. This answer is affirmative provided the potential is an analytical function in an interval (0,R) with converging expansion in powers and logarithms as χ Υ 0+. In the sequel we prove a generalization of Hardy' s inequality to the following direction: let P be a generic real homogeneous polynomial of degree m in n-variables and with roots on the algebraic set V(p), there exists C>0 such that for any f Ξ C0 ₯ (Rn V(P)) there holds ς Rn P-2/m f 2 dx £ C ς Rn (grad f)2 dx. This inequality as well as certain related inequalities require the use of techniques of algebraic geometry in order to study singular algebraic varieties and is the cornerstone for the treatment of inverse spectral problems in higher dimension. In the second part there are treated singular problems of the calculus of variations. Starting out with the motion of light particle coupled to a strong Yang-Mills field, in which case the Lagrangian function is linear with respect to the velocities, we proceed to study the generic Lagrangian function that leads to singular Legendre transformation and hence to singular Euler-Lagrange equations. The study focuses on the conditions under which the singularities are removed and therefore we have the introduction of an algebra of admissible functions. We specify the form of these functions near the singular locus of the Euler-Lagrange equations. The study requires the proof of certain division theorems in the category smooth functions
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| Language |
English |
| Issue date |
2000-03-01 |
| Date available |
2000-03-31 |
| Collection
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School/Department--School of Sciences and Engineering--Department of Physics--Doctoral theses
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Type of Work--Doctoral theses
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| Permanent Link |
https://elocus.lib.uoc.gr//dlib/c/4/d/metadata-dlib-2000pliakis.tkl
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| Views |
765 |
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