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Applications of Integral Transform Methods to the Schrodinger¨ Equation and Dynamical Systems

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Applications of Integral Transform Methods to the Schrodinger¨ Equation and Dynamical Systems APPLICATIONS OF INTEGRAL TRANSFORM METHODS TO THE SCHRODINGER¨ EQUATION AND DYNAMICAL SYSTEMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Min Huang, B.S., M. S. Graduate Program in Mathematics The Ohio State University 2009 Dissertation Committee: Professor Ovidiu Costin , Advisor Professor Saleh Tanveer Professor Rodica Costin ⃝c Copyright by Min Huang May, 2010 ABSTRACT Integral transform methods, in particular the generalized Borel summation meth- ods, have been employed in the study of ordinary and partial differential equations, difference equations, and dynamical systems. These methods are especially useful for analyzing long time asymptotic behaviors of physical systems, and for describing highly complicated behaviors of dynamical systems. In the first part of the dissertation we consider one dimensional Schr¨odingerequa- tions with (1) time-dependent damped delta potentials, (2) time-periodic delta po- tentials, and (3) time-independent compactly supported (finite-range) potentials. We obtain time-asymptotic expressions for the wave functions, and address several issues of physical interest, including ionization and resonance. In the second part of the dissertation we study certain types of lacunary series and obtain asymptotic formulas that describe the behaviors of such series at the natural boundary (a barrier of singularities). We then explore the connection between certain special lacunary series and Julia sets. ii ACKNOWLEDGMENTS First I would like to thank my advisor, Professor Ovidiu Costin, for introducing me to the field of asymptotic analysis, for his insightful guidance in our research projects, and for his constant support and encouragement throughout my graduate studies. I would like to thank Professor Saleh Tanveer for his outstanding teaching, for numerous invaluable research discussions, and for his support and help to me during my graduate studies. I would also like to thank my other committee member Professor Rodica Costin for agreeing to serve on my committee, and for many helpful research discussions. Thanks are also due to the Graduate School for providing me with the Distin- guished University Fellowship, and to the Department of Mathematics for various GTA and GRA supports. Finally I would like to extend my thanks to my friend and former colleague Dr. Zhi Qiu for research collaborations, and to my friend and fellow student Lizhi Zhang for many interesting discussions. iii VITA 2005-present . Graduate Teaching Associate and Grad- uate Fellow The Ohio State University 2008 . M.S. in Mathematics, The Ohio State University 2005 . B.S. in Mathematics, Peking University PUBLICATIONS On the geometry of Julia sets (O. Costin, M. Huang), submitted Borel summability in a class of quantum systems (O. Costin, M. Huang), submitted Gamow vectors in a periodically perturbed quantum system (M. Huang), Journal of Statistical Physics 137: 569-592 DOI 10.1007/s10955-009-9853-7 (2009) Behavior of lacunary series at the natural boundary (O. Costin, M. Huang), Advances in Mathematics Vol. 222, 4, pp 1370-1404 (2009) Ionization in damped time-harmonic fields (O. Costin, M. Huang, Z. Qiu), J. Phys. A: Math. Theor. 42 325202 (2009) iv FIELDS OF STUDY Major Field: Mathematics Specialization: Asymptotic Analysis, Mathematical Physics, Ordinary and Partial Differential Equations, Dynamical Systems v TABLE OF CONTENTS Abstract . ii Acknowledgments . iii Vita . iv List of Figures . ix CHAPTER PAGE 1 Introduction and preliminaries . 1 1.1 Introduction . 1 1.2 Preliminaries . 2 1.2.1 Asymptotic expansions . 2 1.2.2 Laplace transform and Watson's lemma . 4 1.2.3 Transseries and Borel summation . 6 1.2.4 A class of level one transseries . 8 1.2.5 Uniqueness of the transseries representation . 9 1.2.6 Some results in functional analysis . 12 1.2.7 Miscellaneous formulas . 13 1.3 The Schr¨odingerequation . 13 1.3.1 Ionization . 15 1.3.2 Metastable states and resonances . 15 1.4 Lacunary series and dynamical systems . 16 1.4.1 Lacunary series . 16 1.4.2 The Mandelbrot set and Julia sets . 17 2 Ionization in damped time-harmonic fields . 19 2.1 Introduction . 19 2.2 Main results . 22 2.2.1 ! =0 ............................... 23 vi 2.3 Proofs and further results . 24 2.3.1 The associated Laplace space equation . 24 2.3.2 Further transformations, functional space . 26 2.3.3 Equation for A .......................... 28 2.3.4 Positions and residues of the poles . 29 2.3.5 Infinite sum representation of Am;n . 34 2.4 Proof of Theorem 2.1 . 35 2.5 Proof of Theorem 2.2 . 36 2.5.1 Proof of Theorem 2.2, (i) . 37 2.5.2 Proof of Theorem 2.2, (ii) . 39 2.5.3 Numerical results . 40 2.6 Ionization rate under a short pulse . 41 2.7 Results for λ = 0 and ! =0.................6 41 2.7.1 Small λ behavior. 44 3 Gamow Vectors and Borel summation . 46 3.1 Setting and main results . 48 3.2 Proofs of Main Results . 50 3.2.1 Integral reformulation of the problem . .p . 50 3.2.2 Analyticity of ^ on the Riemann surface of p . 51 3.2.3 The poles for large p in the left half plane . 53 3.2.4 Asymptotics of ^ ......................... 58 3.2.5 The inverse Laplace transform . 61 3.2.6 Connection with Gamow Vectors . 67 3.3 Example: square barrier . 68 4 Gamow Vectors in a Periodically Perturbed Quantum System . 73 4.1 Introduction . 73 4.2 Setting and Main Results . 74 4.3 Proof of Main Results . 76 4.3.1 Integral reformulation of the equation . 76 4.3.2 Recurrence relation and analyticity of ^ . 79 4.3.3 The homogeneous equation . 85 4.3.4 Resonance for small r . 90 4.3.5 Resonances in general . 93 4.3.6 Proof of Theorem 1 . 96 4.4 Further Discussion and Numerical Results . 103 vii 4.4.1 Metastable states and multiphoton ionization . 103 4.4.2 Position of resonance: numerical results . 104 4.4.3 Delta potential barrier . 106 5 Boundary behavior of lacunary series and structure of Julia sets . 108 5.1 Introduction . 108 5.2 Results . 111 5.2.1 Results under general assumptions . 111 5.2.2 Results in specific cases . 114 5.2.3 Universal behavior near boundary in specific cases . 119 5.2.4 Fourier series of the B¨otcher map and structure of Julia sets . 122 5.3 Proofs . 126 5.3.1 Proof of Theorem 5.2 . 126 5.3.2 Proof of Theorem 5.3 . 126 5.3.3 Proof of Theorem 5.5 . 128 5.3.4 Proof of Theorem 5.11 . 129 5.3.5 Proof of Proposition 5.6 . 130 5.3.6 The case g(j) = aj . 130 5.3.7 Proof of Theorem 5.8 . 132 5.3.8 Details of the proof of Theorem 5.2 . 139 5.4 Proof of Theorem 5.12 . 142 5.5 Appendix . 145 5.5.1 Proof of Lemma 5.14 . 145 5.5.2 Proof of Theorem 5.3 part (i) . 146 5.5.3 Proof of Lemma 5.17 . 146 5.5.4 Direct calculations for b 2 N integer; the cases b = 3; b = 3=2 . 148 5.5.5 Notes about iterations of maps . 151 Bibliography . 154 viii LIST OF FIGURES FIGURE PAGE 2.1 Log-log plot of jR0j as a function of λ for ! = 0. R0;0 is the residue of the pole of ^(p; x) at p = i, see Corollary 2.11. 40 2.2 jR0;0j as a function of λ, with fixed ratio !/λ = 5. R0;0 is the residue of the pole of ^(p; x) at p = i, see Corollary 2.11. 42 2.3 jR0;0j as a function of λ, with fixed ratio !/λ = 10. 42 2.4 jR0;0j as a function of λ, with fixed ratio !/λ = 15. 43 2.5 jR0;0j as a function of λ, with fixed ratio !/λ = 20. 43 2.6 jR0;0j as a function of λ, with fixed ratio !/λ = 30. 44 2.7 jR0;0j, at λ = 0:01, as a function of !. R0;0 is the residue of the pole of ^(p; x) at p = i, see Corollary 2.11. 45 3.1 Curves pk(s) passing between poles (plotted with square barrier po- tential) . 61 3.2 Region of contractiveness . 64 3.3 A sketch of the contour deformation used. 67 3.4 Density graph of 1=Wp. Dark dots indicate poles from the second column of the table above. 70 3.5 Decay of j (x; t)j2 for x =8....................... 72 4.1 Contour C1 ................................ 101 4.2 Contour C2 ................................ 101 ix 4.3 Contour C3 ................................ 102 4.4 Real part of the resonance as a function of ! . 104 4.5 Position of resonances for different r. Dots are resonances for the usual branch, and \×" and \+" are those resonances continuing on the Riemann surface (they are not visible with the usual branch cut). The \×" and \+" curves in the middle are on different Riemann sheets.105 4.6 Position of resonances for different r for the delta potential barrier. 107 5.1 The standard function Q(x). The points above Q = 0:05 are the only ones present in the actual graph. 120 5.2 Point-plot graph of Q4;4, normalized to one. 120 5.3 Point-plot graph of Q3;3; Q3=2;1=2 follows from it through the transfor- mation (5.24). 121 5.4 The Julia set of xn+1 = λxn(1 − xn), for λ = 0:3 and λ = 0:3i, calculated from the Fourier series (5.30) discarding all o(λ2) terms.
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