Teloni, Daniele
(2019)

*Semiclassical analysis of systems of Schrödinger equations.*
[Laurea magistrale], Università di Bologna, Corso di Studio in

Matematica [LM-DM270]

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## Abstract

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for x<a or x>b. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Abstract

The scalar Schrödinger equation models the probability density distribution for a particle to be found in a point x given a certain potential V(x) forming a well with respect to a fixed energy level E_0. Formally two real inversion points a,b exist such that V(a)=V(b)=E_0, V(x)<0 in (a,b) and V(x)>0 for x<a or x>b. Following the work made by D.Yafaev and performing a WKB approximation we obtain solutions defined on specific intervals. The aim of the first part of the thesis is to find a condition on E, which belongs to a neighbourhood of E_0, such that it is an eigenvalue of the Schrödinger operator, obtaining in this way global and linear dependent solutions in L2. In quantum mechanics this condition is known as Bohr-Sommerfeld quantization. In the second part we define a Schrödinger operator referred to two potential wells and we study the quantization conditions on E in order to have a global solution in L2xL2 with respect to the mutual position of the potentials. In particular their wells can be disjoint,can have an intersection, can be included one into the other and can have a single point intersection. For these cases we refer to the works of A.Martinez, S. Fujiié, T. Watanabe, S. Ashida.

Tipologia del documento

Tesi di laurea
(Laurea magistrale)

Autore della tesi

Teloni, Daniele

Relatore della tesi

Scuola

Corso di studio

Indirizzo

Curriculum A: Generale e applicativo

Ordinamento Cds

DM270

Parole chiave

Schrödinger equation WKB approximation Born-Oppenheimer approximation semiclassical analysis potential wells

Data di discussione della Tesi

25 Ottobre 2019

URI

## Altri metadati

Tipologia del documento

Tesi di laurea
(NON SPECIFICATO)

Autore della tesi

Teloni, Daniele

Relatore della tesi

Scuola

Corso di studio

Indirizzo

Curriculum A: Generale e applicativo

Ordinamento Cds

DM270

Parole chiave

Schrödinger equation WKB approximation Born-Oppenheimer approximation semiclassical analysis potential wells

Data di discussione della Tesi

25 Ottobre 2019

URI

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