Spectral Functions for Regular Sturm-Liouville Problems

Transcription

1 Spectral Functions for Regular Sturm-Liouville Problems Guglielmo Fucci Department of Mathematics East Carolina University May 15, 13

2 Regular One-dimensional Sturm-Liouville Problems Let I = [, 1 R, and let L be the following symmetric second order differential operator L = d ( p(x) d ) + V (x), dx dx with p(x) > for x I, and p(x) and V (x) in L 1 (I, R). For the differential operator L we consider the differential equation Lϕ λ = λ ϕ λ, (1) where λ C and ϕ λ C (I). The differential equation (1) endowed with self-adjoint boundary conditions imposed on ϕ λ is called a regular Sturm-Liouville problem. Furthermore, the parameter λ R denotes the eigenvalues of the SL problem. Self-adjoint boundary conditions can be divided in two mutually excluding classes: Separated Boundary Conditions Coupled boundary conditions.

3 Separated Boundary Conditions Separated boundary conditions have the following general form A 1ϕ λ () A p()ϕ λ() =, B 1ϕ λ (1) + B p(1)ϕ λ(1) =, with A 1, A, B 1, B R and (A 1, A ) (, ), and (B 1, B ) (, ). Eigenvalues. For each λ we choose a solution ϕ λ satisfying the initial conditions ϕ λ () = A, and p()ϕ λ() = A 1. The eigenfunctions of the Sturm-Liouville problem are, then, those that also satisfy the condition Ω(λ) = B 1ϕ λ (1) + B p(1)ϕ λ(1) =, which represents an implicit equation for the eigenvalues λ. Remarks: Well-known examples For A 1 = B 1 = and A = B = we get Neuman and Dirichlet boundary conditions, respectively. When A 1 = B 1 and A = B we have Robin Boundary conditions. By setting A 1 = B = or A = B 1 = we obtain mixed or hybrid boundary conditions.

4 Coupled Boundary Conditions Coupled boundary conditions can be expressed in general as ( ) ( ) ϕλ (1) p(1)ϕ = e iγ ϕλ () K λ(1) p()ϕ, λ() where π < γ or γ < π and K SL (R). For γ = and K = I we have periodic boundary conditions. Eigenvalues. We write the solution as ϕ λ (x) = αu λ (x) + βv λ (x), where for each λ, u λ (x) and v λ (x) are defined by the initial conditions ϕ λ () = β and p()ϕ λ() = α. By imposing coupled boundary conditions and by denoting [k ij = K we obtain the linear system α [u λ (1) e iγ k 1 + β [v λ (1) e iγ k 11 = α [p(1)u λ(1) e iγ k + β [p(1)v λ(1) e iγ k 1 =, which has a non-trivial solution if and only if (λ) = cos γ [ k v λ (1) k 1u λ (1) + k 11p(1)u λ(1) k 1p(1)v λ(1) =.

5 Spectral Zeta Function The implicit equations for the eigenvalues provide an integral representation of the spectral zeta function valid for R(s) > 1/ as ζ { { } S C } 1 Ω(λ) (s) = πi C { S C } dλλ s λ ln. (λ) By deforming the contour to the imaginary axis and by changing variables iλ z one obtains the representation ζ { S C } (s) = sin πs π { } dzz s Ω(z) z ln, (z) valid for 1/ < R(s) < 1. To perform the analytic continuation to the left of the strip 1/ < R(s) < 1 we subtract and add from the integrand a suitable number of terms from the asymptotic expansion of ln Ω(z) and ln (z) for z. The desired asymptotic expansion is obtained through a WKB analysis of the solutions of Sturm-Liouville problem. Remark: For a general one-dimensional Sturm-Liouville differential operator with general self-adjoint boundary conditions, Ω(z) and (z) are not known explicitly.

6 WKB Expansion In the parameter z, the Sturm-Liouville differential equation reads [ d ( p(x) d ) + V (x) ϕ z(x) = z ϕ z(x). dx dx By introducing the auxiliary function S(x, z) = ln ϕz(x), x the equation above is equivalent to the following [p(x)s(x, z) = V (x) + z p(x)s (x, z). As z we assume that S(x, z) has the asymptotic expansion S(x, z) zs 1(x) + S (x) + i=1 S i(x) z i. Once the asymptotic expansion of S(x, z) is known, the one for the solution ϕ z(x) will immediately follow.

7 WKB Expansion By substituting the asymptotic form of S(x, z) in the previous non-linear differential equation we obtain S 1(x) ± = ± 1, S ± (x) = 1 d p(x) dx ln ( p(x)s 1(x) ± ) = p (x) 4p(x), S ± 1 (x) = and for i 1 1 [ p(x)s ± 1 (x) V (x) p(x) ( S ± ) ( (x) p(x)s ± (x) ), S i+1(x) ± 1 = p(x)s ± 1 (x) [ (p(x)s ± i (x) ) + p(x) i S m(x)s ± i m(x) ±. m= The terms S + i (x) and S i (x) provide the exponentially growing and decaying parts of the solution ϕ z(x) as { x } { x } ϕ z(x) = A exp S + (t, z)dt + B exp S (t, z)dt, with A and B uniquely determined by the initial conditions.

8 Asymptotic Expansion: Separated Boundary Conditions For separated boundary conditions the implicit equation for the eigenvalues is ln Ω(z) ln [ A p()s (, z) + A 1 + ln [ Bp(1)S + (1, z) + B 1 ln [ p() ( S + (, z) S (, z) ) + 1 S + (t, z)dt. By introducing the function δ(x) = 1 for x =, and δ(x) = for x one can further expand ln Ω(z) to obtain ln Ω(z) = 1 4 ln p()p(1) + [1 δ(a) ln A p() + [1 δ(b) ln B p(1) Remark: + δ(a ) ln A 1 + δ(b ) ln B 1 ln z + [ δ(a ) δ(b ) ln z 1 + z S 1(t)dt + M i +. z i i=1 The terms M i, i 1, are expressed only in terms of p (n) (x) and V (n) (x) with n i + 1 and their powers.

9 Asymptotic Expansion: Coupled Boundary Conditions For coupled boundary conditions the implicit equation for the eigenvalues is ln (z) ln [ p() ( S + (, z) S (, z) ) + 1 S + (t, z)dt + ln [ k 1 k p()s (, z) + k 11p(1)S + (1, z) + k 1p(1)p()S (, z)s + (1, z) The large-z asymptotic behavior depends on whether k 1 vanishes or not. Both cases are described by the expression Remark: ln (z) = 1 4 ln p()p(1) + [1 δ(k1) ln k1 p()p(1) ) + δ(k 1) ln (k p() + k11 p(1) + [ δ(k 1) ln z ln z + z 1 S + 1(t)dt + i=1 N i z i. The terms N i, i 1, are expressed only in terms of p (n) (x) and V (n) (x) with n i + 1 and their powers.

10 Analytic Continuation of the Spectral Zeta Function From the integral representation of ζ { S C } (s) we add and subtract L leading terms of the respective asymptotic expansions to obtain ζ { S C } (s) = Z { S C } (s) + L i= 1 A { S C } i (s), with Z { S C } { S (s) an analytic function for Rs > (L + 1)/, and A C } i (s) meromorphic functions for s C. In particular we have ζ S (s) = Z S sin πs (s)+ π ζ C (s) = Z C (s) + Remarks: sin πs π [ 1 δ(a ) δ(b ) + 1 s s 1 [ 1 δ(k 1) + 1 s s S + 1(t)dt S + 1(t)dt L i=1 i Mi s + i L N i i. s + i ζ S (s) and ζ C (s) are meromorphic functions of s C with only a simple pole at s = 1/. i=1,

11 Functional Determinant and Heat Kernel Coefficients From the analytically continued expression of the spectral zeta function one can compute The functional determinant, det(l) = exp{ ζ ()}. The coefficients of the asymptotic expansion of θ(t) = Tr L e tl. For the HKC, by using the Mellin transform one has a 1 s = Γ(s)Res ζ(s), a 1 ( 1)n +n = ζ( n). n! when s = 1/ and s = (n + 1)/ with n N. In our case we have a S = a C = 1 1 dt, π p(t) a S 1 = 1 δ(a) δ(b), a S m+1 1 = (m 1)! Mm, as n+1 = n n! M n+1, π(n)! a C 1 = 1 δ(k1), a C m+1 1 = (m 1)! Nm, ac n+1 = n n! N n+1, π(n)! with m N + and n N.

12 Further Research The analysis outlined above represents the foundation for further research Analysis of the Casimir energy and force for a one-dimensional piston modeled by a compact potential with separated or coupled boundary conditions. Study of the behavior of the force as the boundary conditions change. Generalize the technique presented here to study spectral functions for Laplace operator on manifolds of the type I N or I f N with N being a compact Riemannian manifold, and I = [a, b R. These results could be applied to the analysis of the Casimir effect for potential pistons with arbitrary cross-section. It would be particularly interesting to develop a method similar to the one presented in this paper to obtain the analytic continuation of the spectral zeta function for one-dimensional singular Sturm-Liouville problems: The functions p(x) and V (x) become unbounded in the neighborhood of the endpoints of I. The interval I = R is unbounded and the potential V (x) +, as x, is confining.