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Spectral Edge Properties of Periodic Elliptic Operators View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Texas A&M Repository SPECTRAL EDGE PROPERTIES OF PERIODIC ELLIPTIC OPERATORS A Dissertation by MINH TUAN KHA Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Chair of Committee, Peter Kuchment Committee Members, Gregory Berkolaiko Kenneth Dykema Valery Pokrovsky Head of Department, Emil Straube August 2017 Major Subject: Mathematics Copyright 2017 Minh Tuan Kha ABSTRACT In this dissertation, we study some spectral problems for periodic elliptic operators arising in solid state physics, material sciences, and differential geometry. More precisely, we are interested in dealing with various effects near and at spectral edges of such opera- tors. We use the name “threshold effects” for the features that depend only on the infinites- imal structure (e.g., a finite number of Taylor coefficients) of the dispersion relation at a spectral edge. We begin with an example of a threshold effect by describing explicitly the asymp- totics of the Green’s function near a spectral edge of an internal gap of the spectrum of a periodic elliptic operator of second-order on Euclidean spaces, as long as the disper- sion relation of this operator has a non-degenerate parabolic extremum there. This result confirms the expectation that the asymptotics of such operators resemble the case of the Laplace operator. Then we generalize these results by establishing Green’s function asymptotics near and at gap edges of periodic elliptic operators on abelian coverings of compact Riemannian manifolds. The interesting feature we discover here is that the torsion-free rank of the deck transformation group plays a more important role than the dimension of the covering manifold. Finally, we provide a combination of the Liouville and the Riemann-Roch theorems for periodic elliptic operators on abelian co-compact coverings. We obtain several results in this direction for a wide class of periodic elliptic operators. As a simple application of our Liouville-Riemann-Roch inequalities, we prove the existence of non-trivial solutions of polynomial growth of certain periodic elliptic operators on noncompact abelian coverings with prescribed zeros, provided that such solutions grow fast enough. ii DEDICATION To my father Xuong Kha, my mother Sang Lam, and my sister Loan Kha. iii ACKNOWLEDGMENTS First, it is my great pleasure to express my deepest gratitude to my advisor Professor Peter Kuchment for his endless support of my research during my Ph.D study. I am in- debted to him for shaping me as a mathematician as well as helping and encouraging me through my tough times in life. I also thank Professors Yehuda Pinchover and Andy Raich for their collaboration, discussion, support, and advice. I would like to thank the rest of my Ph.D committee. Professor Ken Dykema’s valu- able support began when I just entered the department and has continued all these years. Thank you for the insight gained from our discussions. I also thank Professor Gregory Berkolaiko for many interesting and very helpful mathematical conversations. Professor Valery Pokrovsky was very kind to agree to be in my commitee. Thank you for the time and effort spent looking into my work. I am also grateful to many people from the department of Mathematics at Texas A&M University. Special thanks go to Professor Peter Howard, whose helpfulness, patience, and encouragement went beyond any expectation during my entire six years here; Ms. Monique Stewart for her tireless work and help; Professor Ciprian Foias for his inspiring lectures and precious advices in research; Professor Dean Baskin for many enthusiastic conversations and his great teaching microlocal analysis. I also feel thankful to have many great friends who have made my life enjoyable and comfortable in College Station; among them are Yeong Chyuan Chung, Wai-Kit Chan, Guchao Zheng, Adrian Barquero Sanchez, Mahishanka Withanachchi, Fatma Terzioglu, Jimmy Corbin. Thank you to Hanh Nguyen, for all her love and support. iv I don’t even know how to begin thanking my family. Thank you, Mom and Dad, for giving birth to me and your unconditional love, endless care, and spiritual support throughout my life. There is no way I could have accomplished this without you. This dissertation is partly based on the research carried out under the support of the NSF grant DMS-1517938. I express my gratitude to the NSF for this support. v CONTRIBUTORS AND FUNDING SOURCES Contributors This work was supported by a thesis (or) dissertation committee consisting of Professor Peter Kuchment [advisor], Professor Gregory Berkolaiko, and Professor Kenneth Dykema of the Department of Mathematics and Professor Valery Pokrovsky of the Department of Physics. The results in Chapter 3 were obtained from a collaboration of the student with Pro- fessor Peter Kuchment (Texas A&M University) and Professor Andrew Raich (University of Arkansas), and this will be published in an article listed in the Biographical Sketch (see [40]). The results in Chapter 5 were obtained together with Professor Peter Kuchment. All other work conducted for the dissertation was completed by the student indepen- dently (see [41, 42]). Funding Sources Graduate study was supported by a Graduate Teaching Assistantship from the De- partment of Mathematics at Texas A&M University and also partially supported by the National Science Foundation. vi TABLE OF CONTENTS Page ABSTRACT . ii DEDICATION . iii ACKNOWLEDGMENTS . iv CONTRIBUTORS AND FUNDING SOURCES . vi TABLE OF CONTENTS . vii LIST OF FIGURES . x 1. INTRODUCTION . 1 2. FLOQUET-BLOCH THEORY FOR PERIODIC ELLIPTIC OPERATORS . 9 2.1 Periodic elliptic operators . 9 2.2 Floquet-Bloch theory . 12 3. GREEN’S FUNCTION ASYMPTOTICS NEAR THE INTERNAL EDGES OF SPECTRA OF PERIODIC ELLIPTIC OPERATORS. SPECTRAL GAP INTE- RIOR. 21 3.1 Introduction . 21 3.2 Assumptions, notation and the main result . 22 3.3 Proof of the main theorem 3.2.5 and some remarks . 30 3.4 On local geometry of the resolvent set . 32 3.5 A Floquet reduction of the problem . 32 3.5.1 The Floquet reduction . 33 3.5.2 Singling out the principal term in Rs,λ . 34 3.5.3 A reduced Green’s function. 38 3.6 Asymptotics of the Green’s function . 39 3.6.1 The asymptotics of the leading term of the Green’s function . 40 3.6.2 Estimates of the integral J ..................... 53 3.7 The full Green’s function asymptotics . 57 3.7.1 Parameter-dependent periodic pseudodifferential operators . 58 3.7.2 Decay of the Schwartz kernel of Ts . 62 vii 3.8 Some results on parameter-dependent toroidal ΨDOs . 71 3.9 Some auxiliary statements . 75 3.9.1 A lemma on the principle of non-stationary phase . 75 3.9.2 The Weierstrass preparation theorem . 76 3.9.3 Proofs of Proposition 3.4.1 and Lemma 3.5.1 . 77 3.9.4 Regularity of eigenfunctions φ(z; x) . 84 3.10 Concluding remarks . 89 4. GREEN’S FUNCTION ASYMPTOTICS OF PERIODIC ELLIPTIC OPERA- TORS ON ABELIAN COVERINGS OF COMPACT MANIFOLDS. 90 4.1 Introduction . 90 4.2 Notions and preliminary results . 92 4.2.1 Group actions and abelian coverings . 92 4.2.2 Additive and multiplicative functions on abelian coverings . 96 4.2.3 Some notions and assumptions . 100 4.3 The main results . 105 4.4 A Floquet-Bloch reduction of the problem . 109 4.4.1 The Floquet transforms on abelian coverings and a Floquet reduc- tion of the problem . 109 4.4.2 Isolating the leading term in Rs,λ and a reduced Green’s function . 113 4.5 Some auxiliary statements . 117 4.6 Proofs of the main results . 120 4.7 Proofs of technical statements . 123 4.7.1 Proof of Proposition 4.2.8 . 123 4.7.2 Proof of Proposition 4.2.10 . 124 4.7.3 Proof of Theorem 4.4.6 . 125 4.8 Concluding remarks . 140 5. A LIOUVILLE-RIEMANN-ROCH THEOREM ON ABELIAN COVERINGS. 142 5.1 Introduction . 142 5.2 Some preliminaries for Liouville type results . 143 5.2.1 Some preliminaries for periodic elliptic operators on abelian cov- erings . 143 5.2.2 Floquet-Bloch solutions and Liouville theorem on abelian coverings 147 1 5.2.3 Explicit formulas for dimensions of spaces VN (A) . 150 p 5.2.4 A characterization of the spaces VN (A) . 152 5.3 The Gromov-Shubin version of the Riemann-Roch theorem for elliptic op- erators on noncompact manifolds . 154 5.3.1 Some notions and preliminaries . 154 5.3.2 Point divisors . 157 5.3.3 Rigged divisors . 158 viii 5.3.4 Gromov-Shubin theorem on noncompact manifolds . 160 5.4 The main results . 166 5.4.1 Non-empty Fermi surface . 166 5.4.2 Empty Fermi surface . 174 5.5 Proofs of the main results . 179 5.6 Applications of Liouville-Riemann-Roch theorems to specific operators . 200 5.6.1 Periodic operators with nondegenerate spectral edges . 201 5.6.2 Periodic operators with Dirac points . 206 5.6.3 Non-self-adjoint second order elliptic operators . 209 5.7 Some auxiliary statements and proofs of technical lemmas . 213 5.7.1 Some properties of Floquet functions on abelian coverings . 213 5.7.2 Some basic facts about the family fA(k)gk2Cd . 215 5.7.3 Some properties of Floquet transforms on abelian coverings . 217 5.7.4 A Schauder type estimate . 220 5.7.5 A variant of Dedekind’s lemma . 221 5.7.6 Proofs of technical statements . 223 5.8 Concluding remarks . 234 6. SUMMARY AND CONCLUSIONS .
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