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Mathematical Analysis of Quantum Mechanics with Non-Self-Adjoint Operators Radek Novak

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Mathematical Analysis of Quantum Mechanics with Non-Self-Adjoint Operators Radek Novak Mathematical analysis of quantum mechanics with non-self-adjoint operators Radek Novak To cite this version: Radek Novak. Mathematical analysis of quantum mechanics with non-self-adjoint operators. Func- tional Analysis [math.FA]. Université de Nantes; Czech Technical University in Prague, 2018. English. tel-01903520 HAL Id: tel-01903520 https://hal.archives-ouvertes.fr/tel-01903520 Submitted on 24 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L'UNIVERSITE DE NANTES ET L’UNIVERSITE TECHNIQUE DE PRAGUE COMUE UNIVERSITE BRETAGNE LOIRE ECOLE DOCTORALE N° 601 Mathématiques et Sciences et Technologies de l'Information et de la Communication Spécialité : Mathématiques et leurs Interactions Par Radek NOVÁK Mathematical analysis of quantum mechanics with non-self-adjoint operators Thèse présentée et soutenue à Nantes, le 19 octobre 2018 Unité de recherche : Laboratoire de Mathématiques Jean Leray Rapporteurs avant soutenance : Michael LEVITIN Professeur, Université de Reading Thierry RAMOND Maître de Conférences, Université Paris Sud Composition du Jury : Président : Examinateurs : Frédéric HÉRAU Professeur, Université de Nantes Miloslav ZNOJIL Chercheur, Académie tchèque des sciences Dir. de thèse : Xue-Ping WANG Professeur, Université de Nantes David KREJČIŘÍK Professeur, Université technique de Prague University of Nantes Jean Leray Mathematics Institute and Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Mathematical analysis of quantum mechanics with non-self-adjoint operators Doctoral thesis August 15, 2018 Ing. Radek Nov´ak University of Nantes Jean Leray Mathematics Institute and Czech Technical University in Prague Faculty of Nuclear Sciences and Physical Engineering Mathematical analysis of quantum mechanics with non-self-adjoint operators Doctoral thesis Author: Ing. Radek Nov´ak Study programme: Mathematics (UN) Mathematical engineering (CTU) Supervisors: Xue Ping Wang David Krejˇciˇr´ık Date: August 15, 2018 Abstracts Abstract in English The importance of non-self-adjoint operators in modern physics increases every day as they start to play more prominent role in quantum mechanics. However, the significance of their examination is much more recent than the interest in the examination of their self-adjoint counterparts. Thus, since many self-adjoint techniques fail to be general- ized to this context, there are not many well-developed methods for examining their properties. This thesis aims to contribute to filling this gap and demonstrates several non-self-adjoint models and the means of their study. The topics include pseudospec- trum as a suitable analogue of the spectrum, a model of a quantum layer with balanced gain and loss at the boundary, and the Kramers-Fokker-Planck equation with a short- range potential. Keywords: quantum mechanics, non-self-adjoint operator, quantum waveguide, pseu- dospectrum, Kramers-Fokker-Planck equation vii Abstracts Abstrakt v ˇceˇstinˇe D˚uleˇzitostnesamosdruˇzen´ych oper´ator˚uv modern´ıfyzice se zvyˇsujekaˇzd´ymdnem jak zaˇc´ınaj´ıhr´atst´alepodstatnˇejˇs´ıroli v kvantov´emechanice. Avˇsakd˚uleˇzitostjejich zk- oum´an´ıje mnohem v´ıceˇcerst´aneˇzz´ajemo zkoum´an´ıjejich samosdruˇzen´ych protˇejˇsk˚u. Jelikoˇzmnoho samosdruˇzen´ych technik nelze b´ytzobecnˇenodo tohoto kontextu, nen´ızde mnoho dobˇrevypracovan´ych metod pro vyˇsetˇrov´an´ıjejich vlastnost´ı.Tato teze se snaˇz´ı trochu zaplnit tuto mezeru a prezentuje nˇekolik nesamosdruˇzen´ych model˚ua zp˚usoby jejich studia. T´ematazahrnuj´ı pseudospektrum jako vhodn´yanalog spektra, model kvantov´evrstvy s vyv´aˇzenouztr´atoua ziskem na hranici, a Kramers-Fokker-Planckovu rovnici s kr´atkodosahov´ympotenci´alem. Kl´ıˇcov´aslova: kvantov´amechanika, nesamosdruˇzen´yoper´ator,kvantov´yvlnovod, pseu- dospektrum, Kramers-Fokker-Planck rovnice viii Abstracts R´esum´een fran¸cais La m´ecaniquequantique est sans conteste l'une des parties les plus ´etabliesde la physique moderne avec des milliers d'exp´eriences`al'appui de ses revendications et des applica- tions allant des transistors `al'imagerie par r´esonancemagn´etique.L'une des ´equations fondamentales de la th´eorieest l'´equationde Schr¨odinger [31], @ H = i : @t Il d´ecritl'´etatde la particule quantique comme un vecteur dans un espace de Hilbert et donne sa d´ependance `al'´energie totale du syst`emeexprim´eevia un op´erateurauto- adjoint lin´eaire H, le Hamiltonien. Un outil fondamental pour l'´etudedes points λ de 1 son spectre (les ´energiesadmissibles de la particule) est l'op´erateurr´esolvante (H λ)− itH − et la solution de l'´equationest alors donn´eepar le propagateur e− appliqu´e`aun ´etatinitial. Aucun de ces deux op´erateursn'est n´ecessairement auto-adjoint, en fait, la r´esolvante est non auto-adjointe pour λ avec une partie complexe non nulle et le propagateur forme un groupe unitaire. On peut trouver des op´erateursnon auto-adjoints aussi bien dans d'autres outils techniques - mentionnons l'utilisation de la m´ethode de mise `al'´echelle complexe pour l'analyse des r´esonances ou l'utilisation d'op´erateursde cr´eationet d'annihilation pour l'´etude des oscillateurs harmoniques. Tous ces exemples ont une chose en commun: les op´erateursnon auto-adjoints n'apparaissent qu'en fonction de l'op´erateurauto-adjoint ou en tant qu'instrument math´ematiquepour son ´etude, ils n'apparaissent jamais comme observables. En effet, outre les diff´erents mod`elesph´enom´enologiqueset efficaces, la m´ecanique quantique exige toujours que son op´erateurcentral soit auto-adjoint. Ce n'est qu'alors que la r´ealit´edu spectre (c’est-`a-direles valeurs observables des grandeurs physiques) et l'´evolution unitaire des solutions de Schr¨odingersont garanties. Clairement, toute extension de la m´ecaniquequantique doit traiter de ces questions et cette th`esevise `afournir un bref r´esum´ede la recherche dans ce sens et pr´esente les contributions de l'auteur au sujet. L'une des difficult´esfondamentales que l'on rencontre lorsqu'on traite avec des op´erateurs non auto-adjoints en m´ecaniquequantique est le manque de techniques math´ematiques rigoureuses pour leur ´etude.Bien que les premi`eresmentions des probl`emes de valeurs limites non auto-adjoints dans le contexte de la m´ecanique quantique se soient produites il y a plus de cent ans dans les travaux de G. D. Birkhoff [2], les probl`emesn'ont pas suscit´eun int´er^etsuffisant de la communaut´e. En effet, il a fallu plus de quarante ans pour que les premiers r´esultatsabstraits apparaissent dans les travaux de Keldyˇs[16] en 1951. A` cette ´epoque, la th´eoriedes op´erateursauto-adjoints ´etaitbien ´etablieet four- nissait des r´esultatssolides. s'aventurer en dehors de son cadre et essayer de contester ses ix Abstracts fondamentaux. A ce jour, l'´etude des op´erateursauto-adjoints peut b´en´eficierde divers outils puissants tels que le th´eor`emespectral ou le principe variationnel, alors que ces outils ne sont pas applicables aux op´erateursnon auto-adjoints. De plus, les op´erateurs non auto-adjoints sont plus divers que leurs homologues auto-adjoints, un scientifique habitu´eau cadre auto-adjoint typique peut rencontrer des ph´enom`enes ´etrangeset inat- tendus. Il est difficile de trouver un th´eor`emesuffisamment g´en´eralpour les englober et l'´etuded'exemples sp´ecifiqueset la recherche d'attributs communs semblent ^etreune approche fructueuse. La notion d'observateurs non auto-adjoints a ´et´esugg´er´eeen 1992 dans le domaine de la physique nucl´eairepar F. G. Scholtz, H. B. Geyer et F. J. W. Hahne [30]. Ils ont observ´eque l'un peut construire une th´eoriequantique coh´erente avec un op´erateur non auto-adjoint H, `acondition qu'il s'agisse d'un quasi-hermitien (quasi-auto-adjoint), c’est-`a-dire 1 H∗ = ΘHΘ− : Ici, Θ est un op´erateurpositif, born´eavec un inverse born´e, ´egalement appel´em´etrique. (La quasi-Hermicit´esemble appara^ıtrepour la premi`erefois en 1962 dans les travaux de J. Dieudonn´e[9].) Tous les principes fondamentaux de la m´ecaniquequantique sont valables si l'on consid`ereun changement du produit int´erieurde l'espace Hilbert { on prend ; Θ au lieu de ; . Ceci peut sembler `apremi`erevue une extension de la h· i h· ·i m´ecaniquequantique standard (car elle suit dans le cas particulier Θ = I) mais on peut voir que la notion de quasi-auto-adjointe est ´equivalente `al'op´erateur H similaire `aun op´erateurauto-adjoint h, c.-`a-d. 1 h = ΩHΩ− avec un op´erateurborn´eΩ avec un inverse born´e, nous avons Ω = Θ1=2. Ainsi, la pro- pri´et´ede quasi-auto-adjointe repr´esente simplement une reformulation de la m´ecanique quantique avec des op´erateursauto-adjoints. Notez qu'il peut encore apporter des avan- tages en tant que repr´esentation potentiellement plus simple des probl`emesauto-adjoints. Cependant, la morale ici est que la quasi-auto-adjonction constitue un crit`erefonda- mental pour d´eterminersi un op´erateurnon auto-adjoint peut produire une dynamique quantique coh´erente. L'`erede l'int´er^etpour les op´erateursnon-autoadjoints a commenc´een 1998, lorsque C. M. Bender et P. N. Boettcher ont remarqu´equ'une grande classe de l'op´erateurnon- autoadjoints poss`edespectres r´eel [1]. Cette propri´et´ea ´et´eattribu´ee`aune sym´etrie espace-temps physique du syst`eme,appel´eesym´etrie . Bien que leur premi`ereob- PT servation ait ´et´euniquement num´erique,une preuve rigoureuse a ´et´efaite plusieurs ann´eesplus tard par P. Dorey, C. Dunning et R.
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