Quantum Leaks

Abstract

Abstract. We show that eigenfunctions of the Laplacian on certain non-compact domains with finite area may localize at infinity-provided there is no extreme level clustering-and thus rule out quantum unique ergodicity for such systems. The construction is elementary and based on 'bouncing ball' quasimodes whose discrepancy is proved to be significantly smaller than the mean level spacing.

Localized eigenfunctions in Šeba billiards

“…[4][5][6][7][8][9] One reason for this is that quasimodes can often be used to approximate eigenfunctions. In general ͑see the introduction to Sec.…”

“…17,[35][36][37] Under Assumption 4.1 for the set of values ͑86͒, Theorem 4.4 asserts the existence of a subsequence ͑j n ͒ ʕ N such that…”

“…The construction of families of quasimodes is a key step in Hassell's proof [3], as well as the proofs of many recent results on localisation of quantum eigenfunctions [4,5,6,7,8,9]. One reason for this is that quasimodes can often be used to approximate eigenfunctions.…”

“…We are interested in the situation when the quasi-eigenvalue µ and quasi-eigenfunction ψ approximate true eigenvalues λ j and eigenfunctions φ j of H. In this direction, the following classical results apply (see, e.g. [32,9]) For a quasimode with discrepancy d, the interval [µ − d, µ + d] contains at least one eigenvalue of H.…”