Spectral functions, special functions and the Selberg zeta function. (English) Zbl 0631.10025

The author starts from an arbitrary sequence \((\lambda_k)_{k\geq 1}\) of positive real numbers such that \(\lambda_k\to \infty\) which is subject to suitable regularity conditions. (Typically, \((\lambda_k)_{k\geq 1}\) will be the spectrum of a differential operator or a number-theoretically defined sequence.) Then he forms the associated \(\theta\)-series, the Fredholm determinant, the zeta-function and the functional determinant and he discusses the relations between these functions by means of regularization techniques. In particular, he establishes a very general relation between the functional determinant and the Fredholm determinant.
The results apply to a wide variety of examples covering many classical situations from mathematics and theoretical physics. In particular, the spectral sequence of the Laplacian on the two-dimensional sphere is related with the Barnes \(G\)-function. The main application is an explicit factorization of the Selberg zeta-function into two functional determinants, one of which is expressible in terms of the Barnes \(G\)-function. Relations with some recent results on the Selberg zeta-function are also established.


11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
35P99 Spectral theory and eigenvalue problems for partial differential equations
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J52 Determinants and determinant bundles, analytic torsion
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
Full Text: DOI


[1] D’Hoker, E., Phong, D. H.: (a) Multiloop amplitudes for the bosonic Polyakov string. Nucl. Phys.B269, 205-234 (1986)
[2] (b) On determinants of Laplacians on Riemann surfaces. Commun. Math. Phys.104, 537-545 (1986) · Zbl 0599.30073
[3] Ray, D., Singer, I. M.: Ann. Math.98, 154-177 (1973) · Zbl 0267.32014
[4] Donnelly, H.: Am. J. Math.101, 1365-1379 (1979) · Zbl 0425.53034
[5] Fried, D.: Invent. Math.84, 523-540 (1986) · Zbl 0621.53035
[6] Widder, D.: The Laplace Transform (Chap. V), Princeton. NJ: Princeton University Press 1946 · Zbl 0060.24801
[7] Duistermaat, H., Guillemin, V. W.: Invent. Math.29, 39-79 (1975) · Zbl 0307.35071
[8] Voros, A.: in: The Riemann problem.... Chudnovsky, D., Chudnovsky, G. (eds.) Lecture Notes in Mathematics Vol.925. Berlin, Heidelberg, New York: Springer 1982
[9] Voros, A.: The return of the quartic oscillator. The complex WKB method. Ann. Inst. H. Poincaré39A, 211-338 (1983) (especially Sects. 4, 10 and Appendices A, C, D) · Zbl 0526.34046
[10] Hille, E.: Analytic function theory, Vol. I, Chap. 8.7 and Vol.II, Chap. 14, Blaisdell 1962-1963 · Zbl 0102.29401
[11] Gelfand, I. M., Shilov, G. E.: Generalized functions Vol.1. New York: Academic Press 1964
[12] Seeley, R.: AMS Proc. Symp. Pure Math.10, 288-307 (1966)
[13] Gelfand, I. M., Levitan, B. M.: Dokl. Akad. Nauk. SSSR88, 593-596 (1953), Dikii, L. A.: Usp. Math. Nauk13, 111-143 (1958) (Translations AMS Series 218, 81-115)
[14] Barnes, E. W.: Q. J. Math.31, 264-314 (1900)
[15] Whittaker, E. T., Watson, G. N.: A course of modern analysis, Cambridge: Cambridge University Press 1965 · JFM 45.0433.02
[16] Erdelyi et al.: Higher transcendental functions Vol.1, Chap. 1 (Bateman Manuscript Project), New York: McGraw Hill 1953
[17] Selberg, A.: J. Ind. Math. Soc.20, 47-87 (1956)
[18] Hejhal, D. A.: Duke Math. J.43, 441-482 (1976) · Zbl 0346.10010
[19] Balazs, N. L., Voros, A.: Chaos on the Pseudosphere. Phys. Rep.143, 109-240 (1986)
[20] Huber, H.: Math. Anal.138, 1-26 (1959) · Zbl 0089.06101
[21] Belavin, A., Knizhnik, V.: JETP91, 364-390 (1986); Manin, YU.: JETP Lett.43, 161-163 (1986)
[22] Fried, D.: Invent. Math.84, 523-540 (1986) · Zbl 0621.53035
[23] Kinkelin,: J. Reine Angew. Math. (Crelle)57, 122-138 (1860), Glaisher, J. W. L.: Messenger of Math.6, 71-76 (1877) and24, 1-16 (1894) · ERAM 057.1509cj
[24] Cartier, P.: Analyse numérique d’un problème de valeurs propres à haute précision (Application aux fonctions automorphes), IHES preprint (1978); Vigneras, M-F.: Astérique61, 235-249 (1979)
[25] Widom, H.: Indian Univ. Math. J.21, 277-283 (1971) · Zbl 0223.33015
[26] Widom, H.: Am. J. Math.95, 333-383 (1973); McCoy, B., Wu, T. T.: The two-dimensional Ising model, Cambridge, MA; Harvard University Press 1973 (page 264 and Appendix B); Dyson, F. J.: Fredholm determinants and inverse scattering problems. Commun. Math. Phys.47, 171-183 (1976) · Zbl 0275.45006
[27] Hardy, G. H.: Divergent Series, Clarendon Press, Oxford 1949 · Zbl 0032.05801
[28] Gradshteyn, I. S., Ryzhik, I. M.: Tables of integrals, series and products (Corrected and Enlarged Edition prepared by A. Jeffrey), New York: Academic Press 1980 · Zbl 0521.33001
[29] Lenard, A.: Pacific J. Math.42, 137-145 (1972)
[30] Olver, F. W. J.: Asymptotics and special functions (Chap. 8, Sects. 2.2 and 3.3). New York: Academic Press 1974
[31] Vardi, I.: Determinants of Laplacians and multiple gamma functions. Stanford Math. preprint (Sept. 1986), submitted to SIAM J. Math. Anal.; Weisberger, W. I.: Normalization of the path integral measure and the coupling constants for basonic strings, Nucl. Phys. B (in press) (1987)
[32] Elstrodt, J.: Jber. d. Dt. Math. Verein83, 45-77 (1981), Eq. (10.5)
[33] Randol, B.: Trans. AMS201, 241-246 (1975)
[34] Selberg, A.: Lectures 1953-1954 (unpublished; private communication of J. Elstrodt); Randol, B.: Trans. AMS233, 241-247 (1977); Elstrodt, J., Grunewald, F., Mennicke, J.: Elementary and analytic theory of numbers. Banach center publications17, 83-120 (1985) (Warsaw)
[35] Fischer, J.: Dissertation, Univ. Münster 1985 (and ?An Approach to the Selberg trace formula via the Selberg zeta function?, submitted to Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer)
[36] Sarnak, P.: Determinants of Laplacians, Comm. Math. Phys. (in press) (1987) · Zbl 0618.10023
[37] Balazs, N. L., Schmit, C., Voros, A.: Spectral fluctuations and zeta functions. Saclay preprint PhT/86-156. J. Stat. Phys. (to appear) (M. Kac memorial issue) · Zbl 0692.58027
[38] Voros, A.: Phys. Lett.B180, 245-246 (1986)
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