Abstract
One of the most interesting problems in the Selberg trace formalism from the standpoint of computation is the explicit determination of discrete eigenfunctions of the automorphic Laplacian. Cf. [6, 7, 17, 20] for the necessary theoretical background.
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References
N. L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Reports 143(3) (1986) 109 – 240.
M. Berry, Quantum chaology, Proc. Royal Soc. London A413(1987) 183 — 198. See also: Proc. Royal Soc. London A400(1985) 229 – 251.
O. Bohigas, M. J. Giannoni, and Ch. Schmit, Spectral fluctuations, random matrix theories, and chaotic motion, Springer Lecture Notes in Physics 262(1986) 118 – 138.
P. Cartier, Some numerical computations relating to automorphic functions, in Conputers in Nuni,er Theory (ed. by A. O. L. Atkin and B. J. Birch), Academic Press, 1971, pp. 37 – 48.
G. Golub and C. Van Loan, Matrix Conputations,Johns Hopkins Univ. Press, 1983, especially pages 25 – 27 and 71 – 72.
D. A. Hejhal, The Selberg trace formula and the Riemann zeta function, Duke Math. J. 43 (1976) 441 –482.
D. A. Hejhal, The Selberg Trace Fomula for PSL (2,),volume 2, Springer Lecture Notes 1001(1983).
D. A. Hejhal, Some observations concerning eigenvalues of the Laplacian and Dirichlet L-series, in Recent Progress in Analytic Number Theory (ed. by H. Halberstam and C. Hooley), volume 2, Academic Press, 1981, pp. 95 – 110.
D. A. Hejhal and E. Bombieri, Sur les zéros des fonctions zêta d ‘Epstein, Comptes Rendus Acad. Sci. Paris 304(1987)213– 217.
D. A. Hejhal, Zeros of Epstein zeta functions and supercomputers, in Proceedings of the International Congress of Mathematicians, Berkeley, 1986, pp. 1362 – 1384.
D. A. Hejhal, Some remarks about cusp forms: holomorphic and non-holomorphic, Technical Report No. 1984–26, Chalmers Univ. of Tech. ( Sweden ), 1984, 33 pp.
D. A. Hejhal and B. Berg, Some new results concerning eigenvalues of the non-Euclidean Laplacian for PSL (2, 77), Technical Report No. 82 — 172, University of Minnesota, 1982, 7 pp.
H. Iwaniec, Non-holomorphic modular forms and their applications, in Modular Forms (ed. by R. A. Rankin), Ellis-Horwood Ltd., 1984, pp. 157 — 196.
N. V. Kuznecov, Petersson’s conjecture for cusp forms of weight zero and Linnik’s conjecture; sums of Kloosterman sums, Math. USSR Sbomik 39(1981) 299 — 342.
A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. of Comp. 48 (1987) 273 — 308.
G. Pôlya, Bemerkung über die Integraldarstellung der Riemannschen -Funktion, Acta Math. 48(1926) 305 — 317.
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20(1956) 47 — 87.
C. Se ries, Some geometrical models of chaotic dynamics, Proc. Royal Soc. London A413 (1987) 171 — 182.
H. Stark, Fourier coefficients of Maass waveforms, in Modular Foras (ed. by R. A. Rankin), EllisHorwood Ltd., 1984, pp. 263 — 269.
A. B. Venkov, Spectral Theory of Automorphic Functions,Proc. Steklov Inst. of Math. 153(1982). (English Translation)
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2’d edition, Cambridge Univ. Press, 1944.
M. Wilkinson, Random matrix theory in semiclassical quantum mechanics of chaotic systems, J. Phys. A: Math. Gen. 21(1988) 1173 — 1190.
A. Winkler, Cusp forms and Hecke groups, J. Reine Angew. Math. 386(1988) 187 — 204.
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Hejhal, D.A. (1991). Eigenvalues of the Laplacian for PSL (2, ℤ): Some New Results and Computational Techniques. In: Gong, S., Lu, QK., Wang, Y., Yang, L. (eds) International Symposium in Memory of Hua Loo Keng. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07981-2_5
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