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Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-0057-8211
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of eight papers primarily concerned with the quantitative study of the spectrum of certain differential operators. The majority of the results split into two categories. On the one hand Papers B-E concern questions of a spectral-geometric nature, namely, the relation of the geometry of a region in d-dimensional Euclidean space to the spectrum of the associated Dirichlet Laplace operator. On the other hand Papers G and H concern kinetic energy inequalities arising in many-particle systems in quantum mechanics.

Paper A falls outside the realm of spectral theory. Instead the paper is devoted to a question in convex geometry. More precisely, the main result of the paper concerns a lower bound for the perimeter of inner parallel bodies of a convex set. However, as is demonstrated in Paper B the result of Paper A can be very useful when studying the Dirichlet Laplacian in a convex domain.

In Paper B we revisit an argument of Geisinger, Laptev, and Weidl for proving improved Berezin-Li-Yau inequalities. In this setting the results of Paper A allow us to prove a two-term Berezin-Li-Yau inequality for the Dirichlet Laplace operator in convex domains. Importantly, the inequality exhibits the correct geometric behaviour in the semiclassical limit.

Papers C and D concern shape optimization problems for the eigenvalues of Laplace operators. The aim of both papers is to understand the asymptotic shape of domains which in a semiclassical limit optimize eigenvalues, or eigenvalue means, of the Dirichlet or Neumann Laplace operator among classes of domains with fixed measure. Paper F concerns a related problem but where the optimization takes place among a one-parameter family of Schrödinger operators instead of among Laplace operators in different domains. The main ingredients in the analysis of the semiclassical shape optimization problems in Papers C, D, and F are combinations of asymptotic and universal spectral estimates. For the shape optimization problem studied in Paper C, such estimates are provided by the results in Papers B and E.

Paper E concerns semiclassical spectral asymptotics for the Dirichlet Laplacian in rough domains. The main result is a two-term asymptotic expansion for sums of eigenvalues in domains with Lipschitz boundary.

The topic of Paper G is lower bounds for the ground-state energy of the homogeneous gas of R-extended anyons. The main result is a non-trivial lower bound for the energy per particle in the thermodynamic limit.

Finally, Paper H deals with a general strategy for proving Lieb-Thirring inequalities for many-body systems in quantum mechanics. In particular, the results extend the Lieb-Thirring inequality for the kinetic energy given by the fractional Laplace operator from the Hilbert space of antisymmetric (fermionic) wave functions to wave functions which vanish on the k-particle coincidence set, assuming that the order of the operator is sufficiently large.

Place, publisher, year, edition, pages
Stockholm, Sweden: KTH Royal Institute of Technology, 2019. , p. 57
Series
TRITA-SCI-FOU ; 2019:24
Keywords [en]
Spectral theory, shape optimization, semiclassical asymptotics, spectral inequalities, quantum mechanics
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-249837ISBN: 978-91-7873-199-2 (print)OAI: oai:DiVA.org:kth-249837DiVA, id: diva2:1313048
Public defence
2019-06-05, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Note

QC 20190502

Available from: 2019-05-02 Created: 2019-05-02 Last updated: 2019-05-02Bibliographically approved
List of papers
1. A bound for the perimeter of inner parallel bodies
Open this publication in new window or tab >>A bound for the perimeter of inner parallel bodies
2016 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 271, no 3, p. 610-619Article in journal (Refereed) Published
Abstract [en]

We provide a sharp lower bound for the perimeter of the inner parallel sets of a convex body Ω. The bound depends only on the perimeter and inradius r of the original body and states that. |∂Ωt|≥(1-tr)+n-1|∂Ω|. In particular the bound is independent of any regularity properties of ∂Ω. As a by-product of the proof we establish precise conditions for equality. The proof, which is straightforward, is based on the construction of an extremal set for a certain optimization problem and the use of basic properties of mixed volumes.

Place, publisher, year, edition, pages
Academic Press, 2016
Keywords
Convex geometry, Inner parallel sets, Perimeter, Primary, Secondary
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186837 (URN)10.1016/j.jfa.2016.02.022 (DOI)000378013400006 ()
Funder
Swedish Research Council, 2012-3864
Note

QC 20160530

Available from: 2016-05-30 Created: 2016-05-13 Last updated: 2019-05-02Bibliographically approved
2. On the remainder term of the Berezin inequality on a convex domain
Open this publication in new window or tab >>On the remainder term of the Berezin inequality on a convex domain
2017 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 145, no 5, p. 2167-2181Article in journal (Refereed) Published
Abstract [en]

We study the Dirichlet eigenvalues of the Laplacian on a convex domain in R-n, with n >= 2. In particular, we generalize and improve upper bounds for the Riesz means of order sigma >= 3/2 established in an article by Geisinger, Laptev and Weidl. This is achieved by refining estimates for a negative second term in the Berezin inequality. The obtained remainder term reflects the correct order of growth in the semi-classical limit and depends only on the measure of the boundary of the domain. We emphasize that such an improvement is for general Omega subset of R-n not possible and was previously known to hold only for planar convex domains satisfying certain geometric conditions. As a corollary we obtain lower bounds for the individual eigenvalues lambda(k), which for a certain range of k improves the Li-Yau inequality for convex domains. However, for convex domains one can use different methods to obtain even stronger lower bounds for lambda(k)

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2017
Keywords
Dirichlet-Laplace operator, semi-classical estimates, Berezin-Li-Yau inequality
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-205425 (URN)10.1090/proc/13386 (DOI)000395809900031 ()2-s2.0-85013627273 (Scopus ID)
Note

QC 20170522

Available from: 2017-05-22 Created: 2017-05-22 Last updated: 2019-05-02Bibliographically approved
3. Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains
Open this publication in new window or tab >>Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-249754 (URN)
Note

QC 20190502

Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-05-02Bibliographically approved
4. Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues
Open this publication in new window or tab >>Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues
2017 (English)In: Integral equations and operator theory, ISSN 0378-620X, E-ISSN 1420-8989, Vol. 89, no 4, p. 607-629Article in journal (Refereed) Published
Abstract [en]

We prove that in dimension n≥ 2 , within the collection of unit-measure cuboids in Rn (i.e. domains of the form ∏i=1n(0,an)), any sequence of minimising domains RkD for the Dirichlet eigenvalues λk converges to the unit cube as k→ ∞. Correspondingly we also prove that any sequence of maximising domains RkN for the Neumann eigenvalues μk within the same collection of domains converges to the unit cube as k→ ∞. For n= 2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for n= 3 was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as k→ ∞. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

Place, publisher, year, edition, pages
Springer Basel, 2017
Keywords
Asymptotics, Cuboids, Eigenvalues, Laplacian, Spectral optimisation
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-227117 (URN)10.1007/s00020-017-2407-5 (DOI)000416537600008 ()2-s2.0-85032668674 (Scopus ID)
Funder
Swedish Research Council
Note

QC 20180508

Available from: 2018-05-08 Created: 2018-05-08 Last updated: 2019-10-18Bibliographically approved
5. Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain
Open this publication in new window or tab >>Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-249757 (URN)
Note

QC 20190502

Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-05-02Bibliographically approved
6. Maximizing Riesz means of anisotropic harmonic oscillators
Open this publication in new window or tab >>Maximizing Riesz means of anisotropic harmonic oscillators
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-249755 (URN)
Note

QC 20190502

Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-05-02Bibliographically approved
7. Exclusion Bounds for Extended Anyons
Open this publication in new window or tab >>Exclusion Bounds for Extended Anyons
2017 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 227, no 1, p. 309-365Article in journal (Refereed) Published
Abstract [en]

We introduce a rigorous approach to the many-body spectral theory of extended anyons, that is quantum particles confined to two dimensions that interact via attached magnetic fluxes of finite extent. Our main results are many-body magnetic Hardy inequalities and local exclusion principles for these particles, leading to estimates for the ground-state energy of the anyon gas over the full range of the parameters. This brings out further non-trivial aspects in the dependence on the anyonic statistics parameter, and also gives improvements in the ideal (non-extended) case.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2017
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-219743 (URN)10.1007/s00205-017-1161-9 (DOI)000419121700007 ()2-s2.0-85028757256 (Scopus ID)
Funder
Swedish Research Council, 2012-3864Swedish Research Council, 2013-4734
Note

QC 20171213

Available from: 2017-12-11 Created: 2017-12-11 Last updated: 2019-05-02Bibliographically approved
8. Lieb-Thirring inequalities for wave functions vanishing on the diagonal set
Open this publication in new window or tab >>Lieb-Thirring inequalities for wave functions vanishing on the diagonal set
(English)Manuscript (preprint) (Other academic)
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:kth:diva-249771 (URN)
Note

QC 20190502

Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-05-02Bibliographically approved

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