In this paper, we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with one-dimensional regular Sturm-Liouville problems endowed with self-adjoint separated and coupled boundary conditions. The spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the well-known Liouville-Green (or WKB) asymptotic expansion of the eigenfunctions associated with the problem. The analytically continued expression of the spectral zeta function is then used to compute the functional determinant of the Sturm-Liouville operator and the coefficients of the asymptotic expansion of the associated heat kernel.

1.
Bailey
,
P. B.
,
Everitt
,
W. N.
, and
Zettl
,
A.
, “
Regular and singular Sturm-Liouville problems with coupled boundary conditions
,”
Proc.-R. Soc. Edinburgh, Sect. A: Math.
126
,
505
(
1996
).
2.
Beauregard
,
M.
,
Fucci
,
G.
,
Kirsten
,
K.
, and
Morales
,
P.
, “
Casimir effect in the presence of external fields
,”
J. Phys. A: Math. Theor.
46
,
115401
(
2013
).
3.
Bender
,
C. M.
and
Orszag
,
S. A.
,
Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory
(
Springer
,
New York
,
2010
).
4.
Birrell
,
N. D.
and
Davies
,
P. C. W.
,
Quantum Fields in Curved Space
(
Cambridge University Press
,
1984
).
5.
Blau
,
S. K.
,
Visser
,
M.
, and
Wipf
,
A.
, “
Zeta functions and the Casimir energy
,”
Nucl. Phys. B
310
,
163
(
1988
).
6.
Bordag
,
M.
,
Klimchitskaya
,
G. L.
,
Mohideen
,
U.
, and
Mostepanenko
,
V. M.
,
Advances in the Casimir Effect
(
Oxford Science Publications
,
2009
).
7.
Bordag
,
M.
,
Mohideen
,
U.
, and
Mostepanenko
,
V. M.
, “
New developments in the Casimir effect
,”
Phys. Rep.
353
,
1
(
2001
).
8.
Coleman
,
S.
,
Aspects of Symmetry: Selected Lectures of Sidney Coleman
(
Cambridge University Press
,
Cambridge
,
1985
).
9.
Dreyfuss
,
T.
and
Dym
,
H.
, “
Product formulas for the eigenvalues of a class of boundary value problems
,”
Duke Math. J.
45
,
15
37
(
1978
).
10.
Elizalde
,
E.
,
Odintsov
,
S. D.
,
Romeo
,
A.
,
Bytsenko
,
A.
, and
Zerbini
,
S.
,
Zeta Regularization Techniques with Applications
(
World Scientific
,
Singapore
,
1994
).
11.
Forman
,
R.
, “
Functional determinants and geometry
,”
Invent. Math.
88
,
447
493
(
1987
);
Forman
,
R.
,
Invent. Math.
Erratum,
108
,
453
454
(
1992
).
12.
Forman
,
R.
, “
Determinants, finite-difference operators and boundary value problems
,”
Commun. Math. Phys.
147
,
485
526
(
1992
).
13.
Fucci
,
G.
and
Kirsten
,
K.
, “
The Casimir effect for generalized piston geometries
,”
Int. J. Mod. Phys. A
27
,
1260008
(
2012
).
14.
Fucci
,
G.
and
Kirsten
,
K.
, “
The spectral zeta function for Laplace operators on warped product manifolds of the type I×fN
,”
Commun. Math. Phys.
317
,
635
(
2013
).
15.
Fucci
,
G.
,
Kirsten
,
K.
, and
Morales
,
P.
, “
Pistons modelled by potentials
,” in
Cosmology, Quantum Vacuum, and Zeta Functions
, edited by
Odintsov
S.
,
Sáez-Gómez
D.
, and
Xambó
S.
(
Springer-Verlag
,
Berlin
,
2011
), p.
313
.
16.
Fulling
,
S. A.
,
Aspects of Quantum Field Theory in Curved Spacetime
(
Cambridge University Press
,
1989
).
17.
Gelfand
,
I. M.
and
Yaglom
,
A. M.
, “
Integration in functional spaces and its applications in quantum physics
,”
J. Math. Phys.
1
,
48
69
(
1960
).
18.
Gilkey
,
P. B.
, “
The spectral geometry of a Riemannian manifold
,”
J. Differ. Geom.
10
,
601
(
1975
).
19.
Gilkey
,
P. B.
,
Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem
(
Boca Raton
,
CRC Press
,
1995
).
20.
Gilkey
,
P. B.
,
Asymptotic Formulae in Spectral Geometry
(
CRC Press
,
Boca Raton
,
2004
).
21.
Greiner
,
P.
, “
An asymptotic expansion for the heat equation
,”
Arch. Ration. Mech. Anal.
41
,
163
(
1971
).
22.
Hawking
,
S. W.
, “
Zeta function regularization of path integrals in curved space-time
,”
Commun. Math. Phys.
55
,
133
(
1977
).
23.
Kirsten
,
K.
,
Spectral Functions in Mathematics and Physics
(
CRC Press
,
Boca Raton
,
2001
).
24.
Kirsten
,
K.
and
McKane
,
A. J.
, “
Functional determinants by contour integration methods
,”
Ann. Phys.
308
,
502
(
2003
).
25.
Kirsten
,
K.
and
McKane
,
A. J.
, “
Functional determinants for general Sturm-Liouville problems
,”
J. Phys. A: Math. Gen.
37
,
4649
(
2004
).
26.
Lesch
,
M.
, “
Determinants of regular singular Sturm-Liouville operators
,”
Math. Nachr.
194
,
139
170
(
1998
).
27.
Lesch
,
M.
and
Vertman
,
B.
, “
Regular singular sturm-liouville operators and their zeta-determinants
,”
J. Funct. Anal.
261
,
408
450
(
2011
).
28.
Levit
,
S.
and
Smilansky
,
U.
, “
A theorem on infinite products of eigenvalues of Sturm-Liouville type operators
,”
Proc. Am. Math. Soc.
65
,
299
302
(
1977
).
29.
Miller
,
P. D.
, “
Applied asymptotic analysis
,” in
American Mathematical Society
(
Rhode Island
,
Providence
,
2006
).
30.
Minakshisundaram
,
S.
, “
Eigenfunctions on Riemannian manifolds
,”
J. Indian Math. Soc.
17
,
158
(
1953
).
31.
Minakshisundaram
,
S.
and
Pleijel
,
A.
, “
Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds
,”
Can. J. Math.
1
,
242
(
1949
).
32.
Ray
,
D. B.
and
Singer
,
I. M.
, “
R-torsion and the Laplacian on Riemannian manifolds
,”
Adv. Math.
7
,
145
(
1971
).
33.
Sarnak
,
P.
, “
Determinants of Laplacians
,”
Commun. Math. Phys.
110
,
113
(
1987
).
34.
Seeley
,
R. T.
, “
Complex powers of an elliptic operator, Singular Integrals, Chicago 1966
,”
Proceedings of Symposia in Pure Mathematics
10
,
288
(
1968
).
35.
Srivastava
,
H. M.
and
Choi
,
J.
,
Zeta and q-Zeta Functions and Associated Series and Products
(
Elsevier
,
2011
).
36.
Vassilevich
,
D. V.
, “
Heat kernel expansion: User’s manual
,”
Phys. Rep.
388
,
279
(
2003
).
37.
Voros
,
A.
, “
Spectral functions, special functions and Selberg zeta function
,”
Commun. Math. Phys.
110
,
439
(
1987
).
38.
Zettl
,
A.
,
Sturm-Liouville Theory, Mathematical Surveys and Monographs
(
American Mathematical Society
,
2005
), Vol.
121
.
You do not currently have access to this content.