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Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-0057-8211
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. Vol. 89, no 4, p. 607-629
Keywords [en]
Asymptotics, Cuboids, Eigenvalues, Laplacian, Spectral optimisation
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:kth:diva-227117DOI: 10.1007/s00020-017-2407-5ISI: 000416537600008Scopus ID: 2-s2.0-85032668674OAI: oai:DiVA.org:kth-227117DiVA, id: diva2:1204560
Funder
Swedish Research Council
Note

QC 20180508

Available from: 2018-05-08 Created: 2018-05-08 Last updated: 2019-10-18Bibliographically approved
In thesis
1. Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization
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

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