Asian Journal of Mathematics
Volume 24 (2020)
Lower bounds for the number of nodal domains for sums of two distorted plane waves in non-positive curvature
Pages: 417 – 436
In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set.
We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees an optimal lower bound, of the same order as Courant’s upper bound.
As an application, we will obtain optimal lower bounds for the number of nodal domains of distorted plane waves on some families of surfaces of non-positive curvature.
nodal domains, semiclassical analysis, scattering theory, quantum chaos, distorted plane waves
2010 Mathematics Subject Classification
Primary 58J50. Secondary 58J37, 58J51, 60G60.
Received 21 October 2018
Accepted 4 July 2019
Published 9 October 2020