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Larson, Simonorcid.org/0000-0002-0057-8211

Open this publication in new window or tab >>Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization### Larson, Simon

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); 2019 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### 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

### Steinerberger, Stefan

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##### Supervisors

### Laptev, Ari

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#####

##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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.

Yale University.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

QC 20190502

Available from: 2019-05-02 Created: 2019-05-02 Last updated: 2019-05-02Bibliographically approvedOpen this publication in new window or tab >>Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains### Larson, Simon

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##### Abstract [en]

##### 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)
#####

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#####

##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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.

QC 20191001

Available from: 2019-10-01 Created: 2019-10-01 Last updated: 2019-10-01Bibliographically approvedOpen this publication in new window or tab >>Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain### Frank, R. L.

### Larson, Simon

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##### Abstract [en]

##### 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)
#####

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#####

##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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.

QC 20191106

Available from: 2019-11-06 Created: 2019-11-06 Last updated: 2019-11-06Open this publication in new window or tab >>Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues### Gittins, K.

### Larson, Simon

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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#####

##### Funder

Swedish Research Council
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

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.

QC 20180508

Available from: 2018-05-08 Created: 2018-05-08 Last updated: 2019-10-18Bibliographically approvedOpen this publication in new window or tab >>On the remainder term of the Berezin inequality on a convex domain### Larson, Simon

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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)

QC 20170522

Available from: 2017-05-22 Created: 2017-05-22 Last updated: 2019-05-02Bibliographically approvedOpen this publication in new window or tab >>A bound for the perimeter of inner parallel bodies### Larson, Simon

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##### Abstract [en]

##### 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 ()
#####

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##### Funder

Swedish Research Council, 2012-3864
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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.

QC 20160530

Available from: 2016-05-30 Created: 2016-05-13 Last updated: 2019-05-02Bibliographically approvedOpen this publication in new window or tab >>Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group### Larson, Simon

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 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]

##### 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)
#####

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##### Funder

Swedish Research Council, 2012-3864
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

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.

QC 20161110

Available from: 2016-11-10 Created: 2016-11-02 Last updated: 2017-11-29Bibliographically approvedOpen this publication in new window or tab >>Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains### Larson, Simon

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##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-249754 (URN)
#####

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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

QC 20190502

Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-05-02Bibliographically approvedOpen this publication in new window or tab >>Lieb-Thirring inequalities for wave functions vanishing on the diagonal set### Larson, Simon

### Lundholm, Douglas

### Nam, Phan Thành

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##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-249771 (URN)
#####

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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

QC 20190502

Available from: 2019-04-23 Created: 2019-04-23 Last updated: 2019-05-02Bibliographically approvedOpen this publication in new window or tab >>Maximizing Riesz means of anisotropic harmonic oscillators### Larson, Simon

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); (English)Manuscript (preprint) (Other academic)
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-249755 (URN)
#####

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#####

##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

QC 20190502

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