Change search
Link to record
Permanent link

Direct link
BETA
Publications (10 of 11) Show all publications
Larson, S. (2019). Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization. (Doctoral dissertation). Stockholm, Sweden: KTH Royal Institute of Technology
Open this publication in new window or tab >>Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization
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
Spectral theory, shape optimization, semiclassical asymptotics, spectral inequalities, quantum mechanics
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-249837 (URN)978-91-7873-199-2 (ISBN)
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
Larson, S. (2019). Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains. Journal of Spectral Theory, 9(3), 857-895
Open this publication in new window or tab >>Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains
2019 (English)In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 9, no 3, p. 857-895Article in journal (Refereed) Published
Abstract [en]

For Omega subset of R-n, a convex and bounded domain, we study the spectrum of -Delta(Omega) the Dirichlet Laplacian on Omega. For Lambda >= 0 and gamma >= 0 let Omega(Lambda,gamma)(A) denote any extremal set of the shape optimization problem sup {Tr(-Delta(Omega) - Lambda)(gamma) : Omega is an element of .A, vertical bar Omega vertical bar = 1}, where A is an admissible family of convex domains in R-n. If gamma >= 1 and (1) and {Lambda(j)}(j >= 1) is a positive sequence tending to infinity we prove that {Omega Lambda j , gamma(A)}(j >= 1) is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on A we characterize the possible limits of such subsequences asminimizers of the perimeter among domains in A of unit measure. For instance if A is the set of all convex polygons with no more than m faces, then Omega(Lambda,gamma) converges, up to rotation and translation, to the regular m-gon.

Place, publisher, year, edition, pages
EUROPEAN MATHEMATICAL SOC, 2019
Keywords
Shape optimization, Riesz eigenvalue means, eigenvalue sums, Dirichlet-Laplace operator, Weyl asymptotics, convexity
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-261050 (URN)10.4171/JST/265 (DOI)000484709400004 ()2-s2.0-85071167293 (Scopus ID)
Note

QC 20191001

Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-01Bibliographically approved
Frank, R. L. & Larson, S. (2019). Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain. Journal für die Reine und Angewandte Mathematik
Open this publication in new window or tab >>Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain
2019 (English)In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345Article in journal (Refereed) Published
Abstract [en]

We prove a two-term Weyl-type asymptotic formula for sums of eigenvalues of the Dirichlet Laplacian in a bounded open set with Lipschitz boundary. Moreover, in the case of a convex domain we obtain a universal bound which correctly reproduces the first two terms in the asymptotics.

Place, publisher, year, edition, pages
De Gruyter, 2019
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-263261 (URN)10.1515/crelle-2019-0019 (DOI)2-s2.0-85071182075 (Scopus ID)
Note

QC 20191106

Available from: 2019-11-06 Created: 2019-11-06 Last updated: 2019-11-06
Gittins, K. & Larson, S. (2017). Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues. Integral equations and operator theory, 89(4), 607-629
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
Larson, S. (2017). On the remainder term of the Berezin inequality on a convex domain. Proceedings of the American Mathematical Society, 145(5), 2167-2181
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
Larson, S. (2016). A bound for the perimeter of inner parallel bodies. Journal of Functional Analysis, 271(3), 610-619
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
Larson, S. (2016). Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group. Bulletin of Mathematical Sciences, 6(3), 335-352
Open this publication in new window or tab >>Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group
2016 (English)In: Bulletin of Mathematical Sciences, ISSN 1664-3607, E-ISSN 1664-3615, Vol. 6, no 3, p. 335-352Article in journal (Refereed) Published
Abstract [en]

We prove geometric Lp versions of Hardy’s inequality for the sub-elliptic Laplacian on convex domains Ω in the Heisenberg group Hn, where convex is meant in the Euclidean sense. When p= 2 and Ω is the half-space given by ⟨ ξ, ν⟩ > d this generalizes an inequality previously obtained by Luan and Yang. For such p and Ω the inequality is sharp and takes the form (Formula presented.), where dist(·,∂Ω) denotes the Euclidean distance from ∂Ω.

Place, publisher, year, edition, pages
Springer, 2016
Keywords
35A23, 35H20
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-195306 (URN)10.1007/s13373-016-0083-4 (DOI)000385157400001 ()2-s2.0-84991388174 (Scopus ID)
Funder
Swedish Research Council, 2012-3864
Note

QC 20161110

Available from: 2016-11-10 Created: 2016-11-02 Last updated: 2017-11-29Bibliographically approved
Larson, S.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
Larson, S., Lundholm, D. & Nam, P. T.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
Larson, S.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
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-0057-8211

Search in DiVA

Show all publications