Spectral Theory for One-Dimensional Three-Body Quantum Systems
Translated title
Spektralteori for en-dimensionale tre-legeme kvantesystemer
Author
Have, Jonas
Term
4. term
Education
Publication year
2016
Submitted on
2016-06-10
Pages
76
Abstract
Denne afhandling behandler spektalteori for ét-dimensionelle tre-legeme-kvantesystemer. Først defineres en selvadjungeret Schrödinger-operator med Dirac-delta-interaktioner (punktinteraktioner) ved hjælp af en seskvilineær form. Vi angiver operatorens domæne og bestemmer det essentielle spektrum, den del af spektret der ikke består af isolerede egenværdier. For at identificere det essentielle spektrum beviser vi en særlig udgave af HVZ-sætningen tilpasset denne model. Vi etablerer også resultater om resolventen (operatoren (H − z)^{-1}), som bruges til at analysere spektret. I en afsluttende behandling af et beslægtet tre-legeme-system bruger vi perturbationsteori til at påvise en diskret egenværdi (bundet tilstand) og beskriver, hvordan den afhænger af koblingskonstanten κ. For små κ har egenværdien størrelsesordenen O(κ^4).
This thesis studies spectral theory for one-dimensional three-body quantum systems. First, we define a self-adjoint Schrödinger operator with Dirac delta interactions (point interactions) using a sesquilinear form. We specify the operator’s domain and determine its essential spectrum, the part of the spectrum that is not made of isolated eigenvalues. To identify the essential spectrum, we prove a special case of the HVZ theorem tailored to this setting. We also establish results about the resolvent (the operator (H − z)^{-1}), a standard tool for analyzing spectra. In a final study of a related three-body model, we apply perturbation theory to show that a discrete eigenvalue (a bound state) exists and to describe how it depends on the coupling constant κ. For small κ, this eigenvalue scales like O(κ^4).
[This abstract was generated with the help of AI]
Documents