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The Laplace-Beltrami Operator, Quadratic Formsand Symmetry

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The Laplace-Beltrami Operator, Quadratic Formsand Symmetry UNIVERSIDAD CARLOS III DE MADRID TESIS DOCTORAL ON THE THEORY OF SELF-ADJOINT EXTENSIONS OF THE LAPLACE-BELTRAMI OPERATOR, QUADRATIC FORMS AND SYMMETRY Autor: Juan Manuel Perez´ Pardo Directores: Prof. Alberto Ibort Latre Prof. Fernando Lledo´ Macau DEPARTAMENTO DE MATEMATICAS´ Leganes,´ 2013 Firma del Tribunal Calificador Firma Presidente: Vocal: Secretario: Calificacion´ : Leganes,´ de de ON THE THEORY OF SELF-ADJOINT EXTENSIONS OF THE LAPLACE-BELTRAMI OPERATOR, QUADRATIC FORMS AND SYMMETRY PhD Thesis Author: Juan Manuel Perez´ Pardo Advisors: Prof. Alberto Ibort Latre Prof. Fernando Lledo´ Macau Por un mundo feliz CONTENTS CONTENTS iii AGRADECIMIENTOS v RESUMEN ix 1 INTRODUCTION 1 2 PRELIMINARIES 9 2.1 General theory of symmetric operators in Hilbert space . 9 2.2 Closable quadratic forms . 14 2.3 Laplace-Beltrami operator on Riemannian manifolds and Sobolev spaces . 18 2.4 Scales of Hilbert spaces . 22 3 CLOSABLE AND SEMI-BOUNDED QUADRATIC FORMS ASSOCIATED TO THE LAPLACE-BELTRAMI OPERATOR 25 3.1 A class of closable quadratic forms on a Riemannian manifold . 26 3.1.1 Isotropic subspaces . 27 3.1.2 Admissible unitaries and closable quadratic forms . 32 3.2 Closable and semi-bounded quadratic forms . 34 3.2.1 Functions and operators on collar neighbourhoods . 34 3.2.2 Quadratic forms and extensions of the minimal Laplacian 40 iv CONTENTS 3.3 Examples . 45 4 NUMERICAL SCHEME TO SOLVE THE SPECTRAL PROBLEM OF THE LAPLACE-BELTRAMI OPERATOR 53 4.1 Convergence of the numerical scheme . 55 4.2 Finite element method for the eigenvalue problem in dimension 1 . 57 4.2.1 Finite elements for general self-adjoint boundary conditions 59 4.2.2 Conditioning of the boundary matrix . 64 4.2.3 The spectral pencil . 67 N 4.2.4 The family fS gN is an approximating family of QU . 69 4.3 Numerical experiments . 71 5 SELF-ADJOINT EXTENSIONS WITH SYMMETRY AND REPRESENTA- TION OF QUADRATIC FORMS 83 5.1 General theory of self-adjoint extensions with symmetry . 85 5.2 Invariant quadratic forms . 88 5.3 A class of invariant self-adjoint extensions of the Laplace-Beltrami operator . 91 5.4 Examples . 95 5.5 Representation of generic unbounded quadratic forms . 99 CONCLUSIONS AND FURTHER WORK 111 LIST OF PUBLICATIONS 117 LIST OF SYMBOLS 119 REFERENCES 123 AGRADECIMIENTOS Despues´ de varios anos˜ embarcado en esta aventura ha llegado el momento de escribir los agradecimientos. Aunque para algunas personas escribir esta parte de la tesis pueda ser poco mas´ que un tramite´ o una obligacion,´ yo me veo en la ne- cesidad de agradecer verdaderamente todo el apoyo que he recibido y el esfuerzo que mucha gente me ha dedicado durante todo este tiempo. En primer lugar quiero citar a mis flamantes directores de tesis Alberto Ibort y Fernando Lledo.´ No tengo palabras para describir el gran respeto que me merecen como personas y como cient´ıficos. La cualidad que mas´ aprecio en Alberto (co- mo cient´ıfico, como persona no sabr´ıa por donde´ empezar) es su capacidad para vislumbrar la solucion´ de un problema con la m´ınima informacion,´ por muy com- plejo que este´ sea , “¡Que s´ı que sale hombre, que s´ı!” Por poner una analog´ıa de como´ es Alberto a nivel cient´ıfico aqu´ı teneis´ el siguiente acertijo: Se abre el telon´ y aparecen un tornillo, una goma elastica´ y un lapiz.´ Se cierra el telon.´ Se abre el telon´ y aparece un formula´ 1. ¿Quien´ ha pasado por all´ı?1 De Fernando me gustar´ıa destacar su atencion´ al detalle, “Es cuestion´ de esti- lo”, que mas´ de una vez nos ha costado una discusion´ pero que sin lugar a dudas ha mejorado notablemente los contenidos que se encuentran en esta memoria. Much´ısimas gracias a los dos y a vuestras maravillosas familias. Tambien´ quer´ıa agradecer a los profesores Beppe Marmo, “It is better not to 1 er. MacGyv vi AGRADECIMIENTOS increase Entropy”; A.P. Balachandran, “Why is that so?”; Manolo Asorey, “¿Yo he dicho eso?” y Franco Ventriglia, “Scorzette d’arancia ricoperte di cioccolato” todos sus comentarios, ensenanzas˜ y hospitalidad. Las discusiones con ellos han dado lugar a muchos de los resultados que aparecen en esta memoria. Otros profesores y miembros del departamento de matematicas´ de la UCIIIM que han contribuido en mayor o menor medida a mi formacion´ predoctoral y a la realizacion´ de esta tesis son Olaf Post, Froilan´ Mart´ınez, Daniel Peralta, Julio Moro, Bernardo de la Calle y Alberto Calvo. A lo largo de estos anos˜ he conocido muchos companeros˜ y amigos y, aunque va a ser dif´ıcil citarlos a todos, voy a hacer un intento. Todos ellos han contribuido a crear un ambiente agradable donde poder trabajar. Gio, por todos los ratos del cafe.´ Sveto, - Hoy das tu´ el seminario intergrupos, ¿no? - ¿De verdad? Ay, Ay, Ay. Sergio, que se puede hablar con el´ a traves´ de las paredes. Edo y Daniel, los com- paneros˜ silenciosos. Espero que piensen lo mismo de mi. Jele, con su peculiar vision´ del mundo, “¡En Londres se come bien! Hay ensaladas, comida china, tai- landesa, filipina,. ” Alberto Lopez,´ “Esto es un calculillo”. Leonardo, “No puedo dormir en esa cama”. Esta ultima´ frase explica algunos acontecimientos recientes. Y, en definitiva, a todos con los que he tenido el placer de trabajar estos anos.˜ Esta lista es larga y si me olvido a alguien me lo va a recordar toda la vida. Os pido por favor a los que leais´ esta lista que no le deis mucha publicidad, podeis´ herir la sensibilidad de alguien: Javi, Elyzabeth, Pedro, Julio, Kenier, Alfredo, Mar´ıa, Ulises, Hector, Mari Francis, Natalia, Walter y Yadira. No me olvido del grupo de los seminarios informales Alejandro, Borja, Car- los, Edu, Emilio y Marco. Por cierto, Borja, nos debes uno. A los amigotes de Madrid y Valencia, esa fuerza poderosa que ha empujado en contra de la finalizacion´ de esta memoria. “Solo´ una cana˜ y a casa.” “¿Ya te vas? ¡Si acabo de llegar!” vii “¿Bo que pacha?” “Ataco Ucrania desde Ural.” “¿Cuando´ vienes?” “¿Manana˜ piscina?” “¿Una partidita?” “Tienes que comprarte la Play.” Ya sabeis´ quienes sois, no empiezo a citaros que corro el riesgo de acabar como Almodovar. Tambien´ le quiero agradecer a toda mi familia, incluyendo la pol´ıtica, todo su apoyo. En especial a mi madre y a mi hermano. “Hijo, ¿en que´ piensas?” “Hijo, ¿en que´ piensas?” “Hijo, ¿en que´ estas´ pensando?” “Hijo, me preocupas.” Los que compartan profesion´ conmigo entenderan´ lo que quiero expresar. ¿Que´ se puede decir de una madre que te lo ha dado todo? Pues eso. A mi hermano, “Los hipsters son como nuestros modernetes pero con un toque lumpen muy calcula- do”, le deseo lo mejor en estos momentos de cambio y le tengo que agradecer haberme ensenado˜ que siempre hay otra manera de hacer las cosas. Y por supuesto a Lara, “Juanmi. ”, “¡Ay Galan!´ ”, “¿Y mi sorpresa?” Si hay una persona que sabe que este trabajo no acaba cuando sales del despacho, es ella. Por eso, a parte de corresponder su amor, carino˜ y comprension´ tambien´ quiero disculparme por todas las veces que no puedo dedicarle el tiempo que se merece. En resumen, a todos, mil gracias. Juan Manuel Perez´ Pardo Leganes,´ 2013 This work was partly supported by the project MTM2010-21186-C02-02 of the spanish Min- isterio de Ciencia e Innovacion´ , QUITEMAD programme P2009 ESP-1594 and the 2011 and 2012 mobility grants of “Universidad Carlos III de Madrid”. RESUMEN El objetivo principal de esta memoria es analizar en detalle tanto la construc- cion´ de extensiones autoadjuntas del operador de Laplace-Beltrami definido sobre una variedad Riemanniana compacta con frontera, como el papel que juegan las formas cuadraticas´ a la hora de describirlas. Mas´ aun,´ queremos enfatizar el papel que juegan las formas cuadraticas´ a la hora de describir sistemas cuanticos.´ Es bien conocido que −∆min, la extension´ minimal del operador de Laplace- Beltrami, es autoadjunta cuando la variedad Riemanniana no tiene frontera. Sin embargo, cuando la variedad Riemanniana tiene frontera este operador es simetri-´ co pero no autoadjunto. Esta situacion´ es comun´ en el el analisis´ de sistemas cuanticos´ con frontera. Por ejemplo, el operador anterior describe la dinamica´ de una part´ıcula libre confinada a desplazarse por la variedad. El analisis´ de sistemas cuanticos´ con frontera esta´ recibiendo atencion´ creciente por parte de la comuni- dad ya que hay un gran numero´ de situaciones f´ısicas en las que la frontera juega un papel prominente. Por citar algunos ejemplos: el efecto Hall cuantico,´ los ais- lantes topologicos,´ el efecto Casimir, los grafos cuanticos,´ etc. Otras situaciones f´ısicas en las que la frontera juega un papel importante es en el diseno˜ de mode- los efectivos que describen las interacciones con impurezas o con interfases entre materiales, vease´ [A+05] y las referencias all´ı citadas. Definir un operador auto- adjunto en un sistema con frontera requiere que ciertas condiciones de contorno sean especificadas. Resulta por lo tanto de gran importancia fijar las condiciones de contorno apropiadas, ya que la dinamica´ de un sistema cuantico´ no queda bien determinada hasta que se haya elegido un operador autoadjunto que la describa. La importancia de los operadores autoadjuntos en Mecanica´ Cuantica´ radica en que, x RESUMEN de acuerdo con los postulados de la Mecanica´ Cuantica,´ son interpretados como las magnitudes observables de los sistemas que describen. Su espectro se interpre- ta como los posibles resultados de un proceso de medida. Mas´ aun,´ el teorema de Stone [Sto32] establece una correspondencia un´ıvoca entre grupos uniparametri-´ cos fuertemente continuos y operadores autoadjuntos. Por ello, los operadores au- toadjuntos son los principales objetos que caracterizan la evolucion´ unitaria de un sistema cuantico.´ Vale la pena mencionar en este punto que aunque el espectro de los operadores autoadjuntos constituya el conjunto de posibles resultados para un proceso de medida, es la forma cuadratica´ asociada a ellos, hΦ ;T Φi, siendo T el operador autoadjunto, la que describe el valor esperado resultado de una medida cuando el estado del sistema queda descrito por el vector de estado Φ .
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