Semiclassical limits for the hyperbolic plane

Abstract

We study the concentration properties of high-energy eigenfunctions for the Laplace-Beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions. At the center of our investigation is the microlocal lift of an eigenfunction to SL(2;ℝ) introduced by S. Zelditch. The microlocal lift is based on S. Helgason's Fourier transform and has a straightforward description in terms of the Lie algebra sl(2;ℝ). We begin with an elementary demonstration of Zelditch's exact differential equation for the microlocal lift. Further, for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity, we show that the limit of microlocal lifts is a geodesic-flow-invariant positive measure.

Our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the Macdonald-Bessel functions. We find that with a scaling of auxiliary parameters the corresponding high-energy limit converges to the positive Dirac measure for the lift to SL(2;ℝ) of a single geodesic on the upper half-plane. In particular, at high energy the microlocal lift of a Macdonald-Bessel function is concentrated along the lift of a single geodesic. We further find that the convergence is uniform in parameters, and thus our considerations can be applied to study automorphic eigenfunctions, in particular, the Maass cusp forms for certain cofinite subgroups of SL(2;ℝ). A formula is presented for the microlocal lift of a cusp form as an integral of a family of Dirac measures and a measure given directly by sums of products of the classical Fourier coefficients of the cusp form. The formula establishes that questions on "quantum chaos" for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients. In particular, we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums. We extend the considerations to the case of the nonholomorphic Eisenstein series for the modular group. We find that a result of W. Luo and P. Sarnak is equivalent to a limit-sum formula involving the Riemann zeta-function and the elementary divisor function. The formula is suggestive of a formula of S. Ramanujan.