Spektral analyse af en $\delta$-vekselvirkning i en dimension defineret som en kvadratisk form
Translated title
Spectral Analysis of a $\delta$-interaction in One Dimension Defined as a Quadratic Form
Author
Jensen, Peter
Term
4. term
Education
Publication year
2008
Pages
96
Abstract
Specialet udvikler en systematisk behandling af sesquilineare og sektorielle kvadratiske former på Hilbert-rum og anvender denne teori til at definere og analysere Hamilton-operatoren for en δ-vekselvirkning i én dimension. Første del gennemgår lukkede og aflukkelige former, kriterier for aflukkelighed, repræsentation af former med fokus på den første repræsentationssætning, samt Friedrichs-udvidelsen. I anden del konstrueres en selv-adjungeret Hamilton-operator ud fra en kvadratisk form på H1(R), dens domæne bestemmes, og resolventen opbygges ved hjælp af den frie resolvent udtrykt ved den frie Green’s funktion. Det diskrete spektrum lokaliseres som singulariteter i resolventen, og det vises, at Hamilton-operatoren har præcis én egenværdi, verificeret ved en Riesz-projektion af rang én. Appendikserne indeholder resultater om partiel integration i Sobolev-rum og et bevis for eksistensen af ikke-trivielle testfunktioner, som bruges i analysen.
This thesis develops a systematic treatment of sesquilinear and sectorial quadratic forms on Hilbert spaces and applies this framework to define and analyze the one-dimensional delta-interaction Hamiltonian. The first part presents closed and closable forms, criteria for closability, representation of forms culminating in the first representation theorem, and the Friedrichs extension. In the second part, a self-adjoint Hamiltonian is induced from a quadratic form on H1(R); its domain is characterized, and the resolvent is constructed via the free resolvent expressed through the free Green’s function. The discrete spectrum is located as singularities of the resolvent, and it is shown that the Hamiltonian has exactly one eigenvalue, verified by a rank-one Riesz projection at the singularity. Appendices provide results on integration by parts in Sobolev spaces and a proof of the existence of non-trivial test functions used in the analysis.
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