ABSTRACT Spectral Functions for Generalized Piston Configurations Pedro Fernando Morales-Almaz´an,Ph.D. Advisor: Klaus Kirsten, Ph.D. In this work we explore various piston configurations with different types of po- ij E E tentials. We analyze Laplace-type operators P = −g ri rj + V where V is the potential. First we study delta potentials and rectangular potentials as examples of non-smooth potentials and find the spectral zeta functions for these piston config- urations on manifolds I × N , where I is an interval and N is a smooth compact Riemannian d − 1 dimensional manifold. Then we consider the case of any smooth potential with a compact support and develop a method to find spectral functions by finding the asymptotic behavior of the characteristic function of the eigenvalues for P . By means of the spectral zeta function on these various configurations, we obtain the Casimir force and the one-loop effective action for these systems as the values at s = −1=2 and the derivative at s = 0. Information about the heat kernel coefficients can also be found in the spectral zeta function in the form of residues, which provide an indirect way of finding this geometric information about the manifold and the operator. Spectral Functions for Generalized Piston Configurations by Pedro Fernando Morales-Almaz´an,B.S., B.E., M.S. A Dissertation Approved by the Department of Mathematics Lance L. Littlejohn, Ph.D., Chairperson Submitted to the Graduate Faculty of Baylor University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved by the Dissertation Committee Klaus Kirsten, Ph.D., Chairperson Paul A. Hagelstein, Ph.D. Mark R. Sepanski, Ph.D. Ronald G. Stanke, Ph.D. Anzhong Wang, Ph.D. Accepted by the Graduate School August 2012 J. Larry Lyon, Ph.D., Dean Page bearing signatures is kept on file in the Graduate School. Copyright c 2012 by Pedro Fernando Morales-Almaz´an All rights reserved TABLE OF CONTENTS 1 Spectral Functions 1 1.1 Finite Dimensional Setting . 1 1.2 Infinite Dimensional Setting . 3 1.3 Heat Kernel . 6 1.4 Zeta Function . 8 1.4.1 Integral Representation . 10 1.5 Casimir Effect and One-loop Effective Action . 13 1.5.1 Van der Waals Forces . 13 1.5.2 Zero-point Energy . 14 1.5.3 Casimir Effect . 16 1.5.4 Piston Configuration . 18 1.5.5 Zeta Function Regularization . 20 1.5.6 One-loop Effective Action . 24 2 Delta Potentials 27 2.1 Differential Equation . 27 2.2 Associated Zeta Function . 29 2.3 Contour Deformation . 30 2.4 Analytic Continuation . 32 2.5 Functional Determinant and Casimir Force . 34 2.5.1 Functional Determinant . 34 2.5.2 Casimir Force . 36 3 Rectangular Potentials 39 3.1 One Dimension . 39 iii 3.1.1 Differential Equation . 39 3.1.2 Associated Zeta Function . 41 3.1.3 Analytic Continuation . 42 3.1.4 Functional Determinant and Casimir Force . 43 3.1.5 Functional Determinant . 43 3.1.6 Casimir Force . 44 3.2 Higher Dimensions . 46 3.2.1 Differential Equation . 46 3.2.2 Associated Zeta Function . 46 3.2.3 Analytic Continuation . 47 3.2.4 Functional Determinant and Casimir Force . 48 3.2.5 Functional Determinant . 49 3.2.6 Casimir Force . 49 4 Smooth Potentials 51 4.1 One Dimension . 51 4.1.1 Differential Equation . 51 4.1.2 Zeta Function . 52 4.1.3 Analytic Continuation . 52 4.2 Two Dimensions . 54 4.2.1 Differential Equation . 55 4.2.2 Zeta Function . 56 4.2.3 WKB Asymptotics . 56 4.2.4 Zero Case . 56 4.2.5 Non-zero Case . 57 4.2.6 Characteristic Function Asymptotic Expansion . 58 4.2.7 Analytic Continuation . 59 4.2.8 Functional Determinant and Casimir Force . 60 iv 4.2.9 Functional Determinant . 61 4.2.10 Casimir Force . 62 5 Smooth Potentials in Spherical Shells 63 5.1 Differential Equation . 63 5.2 Associated Zeta Function . 64 5.3 Characteristic Function Asymptotic Expansion . 65 5.4 Analytic Continuation . 66 5.5 First Asymptotic Terms . 67 + 5.5.1 Calculation of the First Si .................... 68 5.5.2 Calculation of the First Ai .................... 68 5.5.3 Residues . 70 5.6 Three Dimensional Spherical Shell . 71 5.6.1 Functional Determinant . 71 5.6.2 Casimir Force . 72 6 Smooth Potentials in Cylindrical Shells 73 6.1 Differential Equation . 73 6.2 Zeta Function . 74 6.3 Characteristic Function Asymptotic Expansion . 75 6.3.1 Zero Case . 75 6.3.2 Non-zero Case . 76 6.3.3 Asymptotics . 77 6.4 Analytic Continuation . 77 6.4.1 First Asymptotic Terms . 81 6.4.2 Zero Case Asymptotic Terms . 81 6.4.3 Non-zero Case Asymptotic Terms . 82 6.4.4 Integral Representation for Ai .................. 83 v 6.4.5 Analytic Continuation for Ai ................... 84 6.5 Three Dimensional Cylindrical Shell . 85 6.5.1 Functional Determinant . 85 6.5.2 Casimir Force . 86 APPENDICES . 87 A Basic Inequalities 89 1.1 Ratio of Hyperbolic Sines . 89 1.2 Product of positive functions . 90 B Basic Zeta Functions 91 2.1 Barnes Zeta Function . 91 2.2 Epstein zeta function . 92 BIBLIOGRAPHY 95 vi CHAPTER ONE Spectral Functions 1.1 Finite Dimensional Setting A central topic in the field of linear algebra is the study of invariant functions, for example, the trace and the determinant of a square matrix. Such functions go from the set of, say n × n matrices, into the field K, usually taken to be a closed algebraic field. Keeping in mind the basic properties of these functions, we state that an invariant function is such that it is constant on the orbits of the action of SU(n) on the set of matrices given by conjugation, i.e. g:A = gAg−1 (1.1) for A 2 MN (C) and g 2 SU(n). Another way of saying this is that the trace and the determinant are constant under a change of basis and their values are well defined in the sense that they do not depend on the particular representation of the matrix A. Now, since the determinant is an invariant function, we have that the charac- teristic polynomial of a matrix defined by pA(t) = det(Int − A) (1.2) where In is the n × n identity matrix, is also an invariant of A. Due to the ring homomorphism between K[t] and K[A] that formally exchanges the real variable t and the matrix variable A, we can make the characteristic polynomial to be real valued or matrix valued depending on the context. 1 If we take the field to be the complex numbers, we can factor the characteristic polynomial as a product of linear factors, pA(t) = (t − λ1)(t − λ2) ····· (t − λn) (1.3) for some λn 2 C not necessarily distinct. These values λi are called the eigenvalues of the matrix A. With the homomorphism between the polynomial rings C[t] and C[A] we have an important result due to Cayley and Hamilton, which states that the characteristic polynomial of a matrix is satisfied by the matrix itself [15]. Theorem 1.1 (Cayley-Hamilton). If A is an n × n matrix with entries in a commu- tative ring R, then pA(A) = 0 : (1.4) Letting A be an n × n matrix over C and using the previous result gives us n the existence of a vector vi 2 C such that (A − λiIn) vi = 0 ; (1.5) which is the usual notion of an eigenvalue and an eigenvector. Looking at this, we find that the notion of a characteristic equation or secular equation plays a central role in spectral theory, for it provides a way of linking the study of an operator with the study of its eigenvalues. By the relation obtained in (1:3), we find a characterization of the eigenvalues of A as being the solutions of pA(λ) = 0 : (1.6) Expanding the characteristic polynomial of a matrix A gives us that the co- efficients of pA are invariant, as pA itself is invariant. Getting the expansion of the characteristic polynomial, n n X k n−k pA(t) = t + (−1) ck(A)t ; (1.7) k=1 2 gives these new invariants of A, namely ck(A). If we consider the case of t = 0, we find that cn(A) = det(A) : (1.8) Via Jacobi's formula, d det(tI − A) = tr (adj(tI − A)) ; (1.9) dt n n n and considering the polynomial t PA(1=t), we have that setting t = 0 gives c1(A) = tr(A): (1.10) Exploiting the structure given by (1:3), we have that all the coefficients ck(A) can be expressed in terms of the eigenvalues λi by Vieta's formulas as the elementary symmetric polynomials in the eigenvalues, X ck(A) = λi1 λi2 ····· λik : (1.11) 1≤i1<i2<···<ik≤n Now, we can say that the functions ck : Mn ! C are spectral functions in the sense that they are defined using the eigenvalues of the matrix. Thus, when we n n realize a matrix as an operator A : C ! C , the functions ck are defined via the spectrum of the operator, i.e. by the eigenvalues of A. 1.2 Infinite Dimensional Setting On the other hand, when in the infinite dimensional setting, the idea is to mimic what happens for the finite dimensional case. Yet, in order to study infinite dimensions, we will no longer work with vector spaces but with Hilbert spaces H = (H; h; i) and with linear operators L(H) from H into itself.