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Asymptotic and Universal Spectral Estimates with Applications in Many-Body Quantum Mechanics and Spectral Shape Optimization

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Asymptotic and Universal Spectral Estimates with Applications in Many-Body Quantum Mechanics and Spectral Shape Optimization Asymptotic and universal spectral estimates with applications in many-body quantum mechanics and spectral shape optimization SIMON LARSON Doctoral Thesis Stockholm, Sweden 2019 KTH Institutionen för Matematik TRITA-SCI-FOU 2019:24 100 44 Stockholm ISBN 978-91-7873-199-2 SWEDEN Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan fram- lägges till offentlig granskning för avläggande av teknologie doktorsexamen i ma- tematik onsdagen den 5 juni 2019 kl 10.00 i sal F3, Kungliga Tekniska högskolan, Lindstedtsvägen 26, Stockholm. c Simon Larson, 2019 Tryck: Universitetsservice US AB iii Abstract This thesis consists of eight papers primarily concerned with the quanti- tative 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 associ- ated Dirichlet Laplace operator. On the other hand Papers G and H concern kinetic energy inequalities arising in many-particle systems in quantum me- chanics. 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 eigen- value 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 spec- tral 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 Lapla- cian 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. iv Sammanfattning Denna avhandling utgörs av åtta artiklar vars huvudsakliga tema är kvan- titativa resultat om spektrumet av differentialoperatorer. Merparten av resul- taten faller i en av två kategorier. Å ena sidan handlar Artikel B till och med E om frågor av spektralgeometrisk karaktär, specifikt relationen mellan formen av ett område i d-dimensionellt Euklidiskt rum och spektrumet av den associerade Dirichlet-Laplaceoperatorn. Å andra sidan handlar Artiklarna G och H om begränsningar för den kinetiska energin av mångpartikelsystem inom kvantmekanik. Artikel A faller utanför spektralteori och handlar istället om ett problem inom konvex geometri. Mer precist, handlar artikeln om en undre begränsning för måttet av ytan av de inre parallela kropparna av ett konvext område. Hur som helst är artikelns huvudresultatet användbart för att studera Dirichlet- Laplaceoperatorn i konvexa områden, vilket demonstreras i Artikel B. I Artikel B återvänder vi till ett argument av Geisinger, Laptev, och Weidl för att bevisa förbättrade Berezin–Li–Yau-olikheter. I detta samman- hang tillåter resultaten i Artikel A oss att bevisa en tvåterms-Berezin–Li–Yau- olikhet för Dirichlet-Laplaceoperatorn på konvexa områden. Av stor vikt för de tillämpningar vi har i åtanke är att oliheten uppvisar korrekt geometriskt beteende i den semiklassiska gränsen I Artiklarna C och D studeras geometriska optimeringsproblem för egen- värden av Laplaceoperatorer. Specifikt handlar båda artiklarna om den asymp- totiska formen av de områden som i en semiklassisk gräns optimerar egenvär- den, eller medelvärden av egenvärden, av Dirichlet- eller Neumann-Laplace- operatorn inom klasser av områden med fixerat mått. Artikel F behandlar ett likartat optimeringsproblem men där optimieringen sker över en en-parameter familj av Schrödingeroperatorer istället för bland Laplaceoperatorer i olika områden. De huvudsakliga ingredienserna i analysen av de semiklassiska op- timeringsproblem som studeras i Artiklarna C, D och F är kombinationer av asymptotiska och universella spektraluppskattningar. För det problem som studeras i Artikel C bevisas spektraluppskattningar av denna form i Arti- kel B och Artikel E. Artikel E handlar om semiklassisk asymptotik för Dirichlet-Laplacianen i icke-reguljära områden. Huvudresultatet är en asymptotisk utveckling till andra ordning för summan av egenvärden i områden med Lipschitz rand. Temat för Artikel G är undre begränsningar för grundtillståndsenergin av den homogena gasen av R-utvidgade anyoner. Huvudresultatet av artikeln är en icke-trivial undre begränsning för energin per partikel i den termodyna- miska gränsen. Slutligen handlar Artikel H om en generell metod för att bevisa Lieb– Thirring-olikheter för mångpartikelsystem i kvantmekanik. Som en tillämp- ning av den generella metoden utvidgas Lieb–Thirring-olikheten för den frak- tionella Laplaceoperatorn från Hilbertrummet av antisymmetriska (fermions- ka) vågfunktioner till vågfunktioner som försvinner på k-partikeldiagonaler, under antagandet att ordningen av operatorn är tillräckligt hög. Contents Contents v Acknowledgements vii Part I: Introduction and summary 1 Introduction 3 2 Background 7 2.1 Spectral theory . 7 2.2 Weyl’s law and semiclassical asymptotics . 14 2.3 Universal spectral inequalities . 17 3 Spectral shape optimization 21 4 Mathematical aspects of many-body quantum mechanics 29 5 Summary of results 35 References 51 Part II: Scientific papers Paper A A bound for the perimeter of inner parallel bodies Journal of Functional Analysis, vol. 271 (2016), no. 3, 610–619. Paper B On the remainder term of the Berezin inequality on a convex domain Proceedings of the American Mathematical Society, vol. 145 (2017), no. 5, 2167–2181. v vi CONTENTS Paper C Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains Journal of Spectral Theory, published online. Paper D Asymptotic behaviour of cuboids optimising Laplacian eigenvalues (joint with K. Gittins) Integral Equations and Operator Theory, vol. 89 (2017), no. 4, 607–629. Paper E Two-term spectral asymptotics for the Dirichlet Laplacian in a Lipschitz domain (joint with R. L. Frank) Preprint: arXiv (2019). Paper F Maximizing Riesz means of anisotropic harmonic oscillators Arkiv för Matematik, to appear. Paper G Exclusion bounds for extended anyons (joint with D. Lundholm) Archive for Rational Mechanics and Analysis, vol. 227 (2018), no. 1, 309–365. Paper H Lieb–Thirring inequalities for wave functions vanishing on the diagonal set (joint with D. Lundholm and P. T. Nam) Preprint: arXiv (2019). Acknowledgements First and foremost, I would like to express my deep gratitude to my advisor Ari Laptev for his constant support and encouragement. Throughout the past five years he has been an unending source of inspirational questions and beautiful mathemat- ical problems. Without him, this thesis would not have existed. I will forever be grateful for his willingness and enthusiasm in introducing me to the world of spectral theory and the freedom he has given me in exploring it. A great deal of thanks is also owed to my co-advisors, Tomas Ekholm and Douglas Lundholm. Their support and friendship has been invaluable. In partic- ular, I am deeply grateful to Douglas for all the lunches, dinners, conferences, and uncountable discussions we’ve had over the years. In the work leading up to this thesis I have also had the pleasure of collaborating with Rupert Frank, Katie Gittins, and Phan Thành Nam. I have greatly enjoyed working and discussing with all of them. I would also like to thank my colleagues and friends in the spectral theory and mathematical physics community. In particular, I would like to mention Richard Laugesen, Per Moosavi, Fabian Portmann, Bartosch Ruskowski, and Timo Weidl. My time as a Ph.D. student at KTH has been an incredibly fun and interesting period of my life. Largely this is due to all my friends at the department. I would like to thank them for providing a highly stimulating
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