Spectra of hyperbolic surfaces

By Peter Sarnak. 2003

153 citations recorded by the source corpus.

These notes attempt to describe some aspects of the spectral theory of modular surfaces. They are by no means a complete survey.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6678fe9361f78b53cb5bf12586882919037c9977

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Norms of geodesic restrictions for eigenfunctions on hyperbolic surfaces and representation theory (2004, background): “as of mathematics and of mathematical physics. In particular, various questions concerning analytic properties of the eigenfunctions φi drew a lot of attention in recent years (see the surveys [Sa1], [Sa2] and the references therein). 1.2. Restrictions. In this paper we address the problem of estima”
Arithmetic Quantum Chaos (2006, background): “formula The statistical properties of the rescaled eigenvalues Xj (cf. (4)) of the Laplacian can be characterized by their distribution in small intervals (13) N (x, L) := # {j : x ≤ Xj ≤ x+ L} , where x is uniformly distributed, say, in the interval [X, 2X], X large.”
Sieving for mass equidistribution (2008, background): “The study of the inner products hφuj,uji as tj −→ ∞ provides connections between a classically chaotic dynamical system, Hamiltonian ﬂow on a surface of negative curvature, and its quantum model (see [Sa] and [Ze]). 2 1.2 Main results We will be analyzing the inner products seen in the statement of ”
On the Voronoĭ formula for GL(n) (2013, methodology): “Recent applications of the formula for GL ( 3 ) may be found in Miller-Schmid [28, 30], Sarnak-Watson [32] and Li [26, 27].”
The quantum unique ergodicity conjecture for thin sets (2013, background): “on the L2 norm of Urestricted to such a disc, namely (5.57) Z Dz0(A−1) |U(z)|2 dxdy y2 ≪z0 A −1Tε. This is curiously just as strong as what follows from Cauchy’s inequality and the SarnakWatson bound [Sa1] kUk4 ≪ Tε (conditional on Ramanujan). Here (5.55) comes from the triple product approach while”