Variational Methods for Quantum Hamiltonians

Research output: Book/ReportPh.D. thesis

  • Sabiha Tokus
This thesis consists of two parts, each investigating variational approaches to study spectral properties of quantum Hamiltonians in different settings.
The first part is based on a paper with Lukas Schimmer and Jan Philip Solovej in which we construct a distinguished self-adjoint extension of symmetric operators that satisfy a certain gap condition. Additionally, we prove a corresponding variational principle which gives the eigenvalues of the self-adjoint extension in the gap. A prominent example of such a gapped operator is the Coulomb–Dirac operator, describing non-interacting relativistic fermions in a Coulomb field. Another example that falls into the class of gapped operators is the Dirac operator on a bounded cylinder. For such an operator we relate the self-adjoint extension to the non-local Atiyah–Patodi–Singer boundary conditions.
In priorwork, Esteban and Loss [EL] constructed a self-adjoint extension of certain gapped operators and Dolbeault, Esteban and Séré [DES00] presented a min-max principle for self-adjoint operators with a gap in the essential spectrum.
Our construction of a self-adjoint extension of gapped operators is in many ways inspired by the work of Esteban and Loss, but starts from a more general setting and shows some difference in the concrete implementation of the basic ideas. The min–max principle for the eigenvalues in the gap of such a self-adjoint extension has similarities to the one presented in [DES00]. The crucial difference and novelty in our setting, however, is that we do not need to determine the domain of the self-adjoint operator since our min–max principle can be formulated by specifying the domain of the symmetric operator only.
This characteristic feature of our min–max principle has an analogue in the case of lower semibounded symmetric operators, for which there is known to exist a distinguished self-adjoint extension, the Friedrichs extension. We call our self-adjoint extension a Friedrichs extension for gapped operators. In fact, the Friedrichs extension appears as a special case in our construction of a self-adjoint extension for a gapped operator.
The second part of the thesis is concerned with a variational approach to Bogoliubov’s approximation theory of bosonic systems interacting via two-body potentials. This approach was introduced and analysed by Napiórkowski, Reuvers and Solovej [NRS18a] as a variational reformulation of Bogliubov’s
approximation. We call it the Bogoliubov variational principle.
We test this approximation method by applying it to the Lieb–Liniger model of a Bose gas in one dimension interacting via a delta-potential. The Lieb–Liniger model as introduced by [LL63] is an exactly solvable model and therefore particularly suitable for testing approximation schemes, as already pointed out by Lieb and Liniger themselves. The model has effectively one parameter, ξ = c/%, where 2c is the coupling strength and % the particle density. The ground state energy in the Lieb–Liniger gas is not known explicitly, but it can be expanded for small and large ξ.
We compute the ground state energy of the Lieb–Liniger gas by the Bogoliubov variational method as expansions in ξ for ξ → ∞ and ξ → 0. For large ξ we find that the ground state energy diverges as ξ½. Our results thus show that the Bogoliubov variational principle is not suited to describe the model at large where the ground state energy is expected to approach an asymptotic value of π2/3. For ξ → 0 our result agrees with the rst two terms of an expansion of the exact Lieb–Liniger model. A third term for the exact model has been derived by Tracy and Widom [TW16]. Our third term diers by a squared logarithmic factor from Tracy and Widom’s result.
For ξ → 0 our result agrees with the rst two terms of an expansion of the exact Lieb–Liniger model. A third term for the exact model has been derived by Tracy and Widom [TW16]. Our third term diers by a squared logarithmic factor from Tracy and Widom’s result.For ξ → 0 our result agrees with the rst two terms of an expansion of the exact Lieb–Liniger model. A third term for the exact model has been derived by Tracy and Widom [TW16]. Our third term diers by a squared logarithmic factor from Tracy and Widom’s result.For ξ → 0 our result agrees with the rst two terms of an expansion of the exact Lieb–Liniger model. A third term for the exact model has been derived by Tracy and Widom [TW16]. Our third term diers by a squared logarithmic factor from Tracy and Widom’s result.For ξ → 0ξξξ For ξ → 0 our result agrees with the rst two terms of an expansion of the exact Lieb–Liniger model. A third term for the exact model has been derived by Tracy and Widom [TW16]. Our third term diers by a squared logarithmic factor from Tracy and Widom’s result.
Original languageEnglish
PublisherDepartment of Mathematical Sciences, Faculty of Science, University of Copenhagen
Number of pages116
Publication statusPublished - 2021

ID: 281602815