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Exactly solved quantum many-body systems in one dimension
KTH, School of Engineering Sciences (SCI), Theoretical Physics, Mathematical Physics.
2005 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.

Place, publisher, year, edition, pages
Stockholm: KTH , 2005. , vii, 40 p.
Series
Trita-FYS, ISSN 0280-316X ; 2005:57
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:kth:diva-564ISBN: 91-7178-224-9 (print)OAI: oai:DiVA.org:kth-564DiVA: diva2:14443
Presentation
2005-12-14, seminarierum 112:028, AlbaNova hus 11, Roslagstullsbacken 11, Stockholm, 10:15
Opponent
Supervisors
Note
QC 20101130Available from: 2005-12-28 Created: 2005-12-28 Last updated: 2010-11-30Bibliographically approved
List of papers
1. Exact solutions of two complementary one-dimensional quantum many-body systems on the half-line
Open this publication in new window or tab >>Exact solutions of two complementary one-dimensional quantum many-body systems on the half-line
2005 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 46, no 5Article in journal (Refereed) Published
Abstract [en]

We consider two particular one-dimensional quantum many-body systems with local interactions related to the root system C-N. Both models describe identical particles moving on the half-line with nontrivial boundary conditions at the origin, but in the first model the particles interact with the delta interaction while in the second via a particular momentum dependent interaction commonly known as delta-prime interaction. We show that the Bethe ansatz solution of the delta-interaction model is consistent even for the general case where the particles are distinguishable, whereas for the delta-prime interaction it only is consistent and nontrivial in the fermion case. We also establish a duality between the bosonic delta- and the fermionic delta-prime model, and we elaborate on the physical interpretations of these models.

Keyword
point interactions, lie-algebras, bose-gas, potentials, particles
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7251 (URN)10.1063/1.1865320 (DOI)000229155700001 ()2-s2.0-18844379385 (Scopus ID)
Note
QC 20101130Available from: 2007-05-31 Created: 2007-05-31 Last updated: 2017-12-14Bibliographically approved
2. Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles
Open this publication in new window or tab >>Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles
2005 (English)In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 38, no 22, 4957-4974 p.Article in journal (Refereed) Published
Abstract [en]

As is well known, there exists a four-parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum-dependent terms, and determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta-interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta and (the so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the other model we write down explicit formulae for all eigenfunctions.

Keyword
nonlinear schrodinger model, point interactions, one-dimension, bethe-ansatz, potentials, operators
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-7252 (URN)10.1088/0305-4470/38/22/018 (DOI)000230980100019 ()2-s2.0-19944394240 (Scopus ID)
Note
QC 20101130Available from: 2007-05-31 Created: 2007-05-31 Last updated: 2017-12-14Bibliographically approved
3. Explicit formulae for the eigenfunctions of the N-body Calogero model
Open this publication in new window or tab >>Explicit formulae for the eigenfunctions of the N-body Calogero model
2006 (English)In: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 39, no 14, 3511-3533 p.Article in journal (Refereed) Published
Abstract [en]

We consider the quantum Calogero model, which describes N non-distinguishable quantum particles on the real line confined by a harmonic oscillator potential and interacting via two-body interactions proportional to the inverse square of the inter-particle distance. We elaborate a novel solution algorithm which allows us to obtain fully explicit formulae for its eigenfunctions, arbitrary coupling parameter and particle number. We also show that our method applies, with minor changes, to all Calogero models associated with classical root systems.

Keyword
generalized hermite-polynomials, sutherland model, quantum, systems, operator
National Category
Physical Sciences
Identifiers
urn:nbn:se:kth:diva-26941 (URN)10.1088/0305-4470/39/14/002 (DOI)000237152500005 ()2-s2.0-33645134516 (Scopus ID)
Note
QC 20101130Available from: 2010-11-30 Created: 2010-11-30 Last updated: 2017-12-12Bibliographically approved

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