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Spectral geometry and scattering theory for certain complete surfaces of finite volume. (English) Zbl 0772.58063

The author studies complete surfaces \((M,g)\) of finite area whose Gaussian curvature equals \(-1\) in the complement of some compact subset of \(M\). The spectrum of the Laplacian consists of a sequence of (nonnegative) eigenvalues and an absolutely continuous spectrum. The stationary approach to scattering theory gives rise to a scattering matrix \(C(s)\) i.e. a meromorphic matrix valued function of \(s \in C\). The poles and zeros of \(\phi(s)=C(s)\) are called resonances. \(\sigma(M)\) is the union of: (a) the resonances; (b) the set of all complex \(s_ j\) such that \(\lambda_ j=s_ j(1-s_ j)\) is an eigenvalue of \(\Delta\); (c) \(\{1/2\}\). The points \(\eta\) of \(\sigma(M)\) occur with a certain multiplicity \(m(\eta)\). It is shown that \(\phi(s)\) is a meromorphic function of order 2 and that moreover an analogue of the Weyl’s formula holds \[ \sum_{\eta\in\sigma(M); |\eta|\leq T}m(\eta) \sim {\text{Area}(M)\over 2\pi}T^ 2. \] Using a variant of Selberg’s trace formula the author can prove that \(\sigma(M)\) determines also the Euler characteristic \(\chi(M)\) and the number \(m\) of ends. The resonance zeta function \(\zeta_ B(s)\) is defined by \[ \zeta_ B(s)=\sum_{\eta \in\sigma(M); \eta\neq 1}(1-\eta)^{-s} \] while \(\zeta_ \Delta\) is the spectral zeta function of the Laplacian. Correspondingly we get two regularized determinants \[ det'(\Delta)=e^{-\zeta'_ \Delta(0)}\text{ and }det'B_ 1=e^{-\zeta'_ B(0)} \] which are related by \[ det'\Delta=exp\left({\text{Area}(M)\over 8\pi}-{3\pi\gamma\over 2}m\right)det'B_ 1 \] where \(\gamma\) is Euler’s constant. For \(\text{Re}(z) > 1\) the author introduces the regularized determinant \(\text{det}(B_ 1+(z-1))\). When \(M\) is hyperbolic this determinant is described in terms of \(\phi(s)\) and the Selberg zeta function.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35P25 Scattering theory for PDEs
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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References:

[1] [Bassr] Barnes, E.W.: The theory of the G-function. Q.J. Math.31, 264-314 (1900)
[2] [B1] Bateman, H.: Tables of integral transforms, vol. I. New York Toronto London: McGraw-Hill 1954
[3] [B2] Bateman, H.: Tables of integral transforms, vol. II New York Toronto London: McGraw-Hill 1954
[4] [B3] Bateman, H.: Higher transcendental functions, vol. I. New York Toronto London: McGraw-Hill 1953
[5] [Bau] Baumg?rtel, H.: Analytic Perturbation Theory of Matrices and Operators. Berlin: Akademie-Verlag 1984
[6] [B?r] B?rard, P.: Transplantation et isospectralit? I. (Preprint 1990)
[7] [Be] Bers, L.: A remark on Mumford’s compactness theorem. Isr. J. Math.12, 400-407 (1972) · Zbl 0249.30019 · doi:10.1007/BF02764631
[8] [Bo] Boas, R.P.: Entire Functions. New York: Academic Press 1954 · Zbl 0058.30201
[9] [CG] Cheeger, J., Gromov, M.: On the characteristic numbers of complete manifolds of bounded geometry and finite volume. In: Chavel I., Farkas, H.M. (eds.) Differential Geometry and Complex Analysis, pp. 115-154. Berlin Heidelberg New York: Springer 1985
[10] [Ch] Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal.12, 401-414 (1973) · Zbl 0263.35066 · doi:10.1016/0022-1236(73)90003-7
[11] [C1] Colin de Verdiere, Y.: Une nouvelle d?monstration du prolongement m?romorphe de s?ries d’Eisenstein. C.R. Acad. Sci. Paris, S?r. I293, 361-363 (1981)
[12] [C2] Colin de Verdiere, Y.: Pseudo-Laplacians II. Ann. Inst. Fourier33, 87-113 (1983)
[13] [DM] Davies, E.B., Mandouvalos, N.: Heat kernel bounds on manifolds with cusps. J. Funct. Analysis75, 311-322 (1987) · Zbl 0629.53044 · doi:10.1016/0022-1236(87)90098-X
[14] [DT] Deift, P., Trubowitz, E.: Inverse scattering on the line. Commun. Pure Appl. Math.32, 121-251 (1979) · doi:10.1002/cpa.3160320202
[15] [E1] Efrat, I.: Determinants of Laplacians on surfaces of finite volume. Commun. Math. Phys.119, 443-451 (1988) · Zbl 0661.10038 · doi:10.1007/BF01218082
[16] [E2] Efrat, I.: Erratum: Determinants of Laplacians on surfaces of finite volume. Commun. Math. Phys.138, 607 (1991) · Zbl 0722.11029 · doi:10.1007/BF02102044
[17] [FK] Fricke, R., Klein, F.: Vorlesungen ?ber die Theorie der automorphen Funktionen, vol. I. Leipzig: Teubner 1896/1912
[18] [G] Gelfand, I.M.: Automorphic functions and the theory of representations. In: Proc. Int. Cong. Math., Stockholm, pp. 74-85. Diursholm: Institute Mittag-Leffler 1963
[19] [Ha] Hayman, W.K.: Meromorphic functions. Oxford: Clarendon, Oxford University Press 1964
[20] [H1] Hejhal, D.A.: The Selberg trace formula and the Riemann zeta function. Duke Math. J.43, 441-482 (1976) · Zbl 0346.10010 · doi:10.1215/S0012-7094-76-04338-6
[21] [H2] Hejhal, D.A.: The Selberg trace formula for PSL (2,R), vol. I. (Lect. Notes Math., vol. 548) Berlin Heidelberg New York: Springer 1976
[22] [H3] Hejhal, D.A.: The Selberg trace formula for PSL (2,R), vol. II. (Lect. Notes Math., vol. 1001) Berlin Heidelberg New York: Springer 1983
[23] [Hu] Huber, H.: Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. Math. Ann.138, 1-26 (1959) · Zbl 0089.06101 · doi:10.1007/BF01369663
[24] [Hx] Huxley, M.N.: Scattering matrices for congruence subgroups. In: Rankin, R. (ed.) Modular forms, pp. 141-156. Chichester: Ellis Horwood 1984
[25] [J] Jorgensen, T.: Simple geodesics on Riemann surfaces. Proc. Am. Math. Soc.86, 120-122 (1982)
[26] [K] Kato, T.: Perturbation theory for linear operators. Berlin Heidelberg New York: Springer 1966 · Zbl 0148.12601
[27] [Ke] Keen, L.: Canonical polygons for finitely generated Fuchsian groups. Acta Math.115, 1-16 (1966) · Zbl 0144.34101 · doi:10.1007/BF02392200
[28] [Kr] Kra, I.: Automorphic Forms and Kleinian Groups. Reading, Mass.: Benjamin 1972 · Zbl 0253.30015
[29] [LP] Lax, P., Phillips, R.: Scattering theory for automorphic forms.l (Ann. Math. Stud., vol. 87) Princeton: Princeton University Press 1976
[30] [L] Lundelius, R.: Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume. Ph.D. Thesis, Stanford University, Stanford (1990) · Zbl 0790.58044
[31] [Mc] McKean, H.P.: Selberg’s trace formula as applied to a compact Riemann surface. Commun. Pure Appl. Math.25, 225-246 (1972) · doi:10.1002/cpa.3160250302
[32] [MO] Magnus, W., Oberhettinger, F., Soni, R.: Special functions of mathematical physics. Berlin Heidelberg New York: Springer 1966 · Zbl 0143.08502
[33] [M?1] M?ller, W.: Spectral theory for Riemannian manifolds with cusps and a related trace formula. Math. Nachr.111, 197-288 (1983) · Zbl 0529.58035 · doi:10.1002/mana.19831110109
[34] [M?2] M?ller, W.: The point spectrum and spectral geometry for Riemannian manifolds with cusps. Math. Nachr.125, 243-257 (1986) · Zbl 0593.58042
[35] [Mf] Mumford, D.: A remark on Mahler’s compactness theorem. Proc. Am. Math. Soc.28, 289-294 (1971) · Zbl 0215.23202
[36] [OPS1] Osgood, B., Phillips, R., Sarnak, P.: Extremals of determinants of Laplacians. J. Funct. Anal.80, 148-211 (1988) · Zbl 0653.53022 · doi:10.1016/0022-1236(88)90070-5
[37] [OPS2] Osgood, B., Phillips, R., Sarnak, P.: Compact isospectral sets of surfaces. J. Funct. Anal.80, 212-234 (1988) · Zbl 0653.53021 · doi:10.1016/0022-1236(88)90071-7
[38] [PS1] Phillips, R., Sarnak, P.: Perturbation theory for the Laplacian on automorphic functions. Am. Math. Soc.5, 1-32 (1992) · Zbl 0743.30039 · doi:10.1090/S0894-0347-1992-1127079-X
[39] [PS2] Phillips, R., Sarnak, P.: On cusp forms for cofinite subgroups of PSL (2,R). Invent. Math.80, 339-364 (1985) · Zbl 0558.10017 · doi:10.1007/BF01388610
[40] [P] Prachar, K.: Primzahlverteilung. Berlin Heidelberg New York: Springer 1957 · Zbl 0080.25901
[41] [Se1] Selberg, A.: Harmonic Analysis. In: Collected papers, vol. I., pp. 626-674. Berlin Heidelberg New York: Springer 1989
[42] [Se2] Selberg, A.: Remarks on the distribution of poles of Eisenstein series. In: Gelbart, S., Howe, R., Sarnak, P. (eds.) Israel Math. Conf. Proc., vol. 3, part II. Festschrift in honor of I.I. Piatetski-Shapiro pp. 251-278. Jerusalem: Weizman 1990
[43] [Su] Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. Math.121, 169-186 (1985) · Zbl 0585.58047 · doi:10.2307/1971195
[44] [T] Titchmarsh, E.C.: The theory of functions. London: Oxford University Press. 1950 · Zbl 0041.22102
[45] [V] Vign?ras, M.F.: Vari?t?s riemanniennes isospectrales et non isom?triques. Ann. Math.112, 21-32 (1980) · Zbl 0445.53026 · doi:10.2307/1971319
[46] [W] Wolpert, S.: The length spectra as moduli for compact Riemann surfaces. Ann. Math.109, 323-351 (1979) · Zbl 0441.30055 · doi:10.2307/1971114
[47] [Z] Zelditch, S.: Kuznecov sum formulae and Szeg? limit formulae on manifolds. (Preprint, Johns Hopkins University 1991) · Zbl 0749.58062
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