## Final Report Summary - DSTAQC (Dynamics, Spectral Theory, and Arithmetic in Quantum Chaos)

This project aimed to study the relationship between the spectrum of the Laplacian on a compact hyperbolic surface, and the geometry and dynamics of the geodesic flow on the surface, in the context of the Quantum Unique Ergodicity (QUE) Conjecture--- which asks for the eigenfunctions to become equidistributed in the large eigenvalue limit. It is thought that this question is related to the problem of bounding multiplicities in the spectrum, and that large degeneracies should cause QUE to fail. We investigated this aspect by studying quasimodes, or approximate eigenfunctions, where we have more control over the "multiplicities" by adjusting the order of approximation to true eigenfunctions. Our objectives were roughly divided into two categories: first, to solidify the connection between multiplicities and non-equidistribution-- in particular, by showing that when the order of the quasimodes is weakened to the appropriate level, the QUE property fails; and second, in contrast, to show that there is variation across different dynamical "models" of quantum chaos, in the relationship between large spectral multiplicities and types of localization phenomena. This program was intended to shed light on the role of spectral multiplicities in the QUE problem in particular, and on the mysterious relationship between spectral data of the Laplacian and the geometry and dynamics of the underlying system in general. I am happy to report that significant progress was made in these directions.

First, we made significant progress in linking the failure of QUE with degeneracies in the spectrum. In a paper recently published ("Logarithmic-scale Quasimodes that do not Equidistribute", in Int. Math. Res. Not.), we showed that quasimodes can concentrate positive mass on any closed geodesic, right up to the conjectured threshold for QUE. This is significant not only in that it shows that QUE can fail for quasimodes that are weak enough, but that it can fail spectacularly-- yielding the most singular of ergodic components, the length measure on a closed geodesic. This also puts into context the well-known examples of concentration on periodic orbits for toy models of quantum chaos. The impact of this result is further evidenced by the interest in generated, as it has been extended to variable curvature by Eswarathasan-Nonnenmacher and to higher dimensions by Eswarathasan-Silberman.

The relationship between degeneracies and QUE has been further strengthened in current work in progress demonstrating the connection between QUE and long-logarithmic time propagation, which is exactly the time scale at which the conjectured quasimode threshold emerges.

Moreover, we have greatly extended our understanding of the role of arithmetic in questions of QUE. In a paper ("Eisenstein Quasimodes and QUE", in Annales Henri Poincare), we show that the analog of QUE for Eisenstein quasimodes persists well beyond the conjectured logarithmic threshold-- at least to first order-- due to arithmetic considerations uncovered in prior joint work with E. Lindenstrauss on joint quasimodes of the Laplacian and Hecke operators. On the other hand, we show that second-order approximations include enhancement on cusp-bound geodesics, in analogy with the failure of QUE for log-scale quasimodes. This interplay between arithmetic and dynamics is responsible for further surprising phenomena of Eisenstein quasimodes currently being prepared for publication, and which are being further investigated in ongoing joint work with R. Schubert.

Moreover, we have studied the interplay between QUE on the round sphere and the dynamics of embedded symmetries, such as the Hecke correspondence, which is the subject of open conjectures. In joint work with E. Le Masson and E. Lindenstrauss ("Quantum Ergodicity and Averaging Operators on the Sphere", in Int. Math. Res. Not.) we show that in each space of spherical harmonics of large eigenvalue, the quantum ergodicity conjecture is satisfied by any basis that are also eigenfunctions of an averaging operator over a finite set of rotations (satisfying mild, natural conditions). These averaging operators need not be arithmetic in nature, but show the impact of imposing additional symmetries on the basis of eigenfunctions. Ongoing work in progress with E. Le Masson seeks a deeper understanding of L^p norms of eigenfunctions on graphs, and their embedded cousins in the sphere and elsewhere.

Thus we have managed to both to elucidate the relationship between spectral degeneracies and QUE, and managed to understand its extensions and limitations in other settings. The impact of this work is already evident in a number of papers building on top of this research, with the promise of much more in the near future.

First, we made significant progress in linking the failure of QUE with degeneracies in the spectrum. In a paper recently published ("Logarithmic-scale Quasimodes that do not Equidistribute", in Int. Math. Res. Not.), we showed that quasimodes can concentrate positive mass on any closed geodesic, right up to the conjectured threshold for QUE. This is significant not only in that it shows that QUE can fail for quasimodes that are weak enough, but that it can fail spectacularly-- yielding the most singular of ergodic components, the length measure on a closed geodesic. This also puts into context the well-known examples of concentration on periodic orbits for toy models of quantum chaos. The impact of this result is further evidenced by the interest in generated, as it has been extended to variable curvature by Eswarathasan-Nonnenmacher and to higher dimensions by Eswarathasan-Silberman.

The relationship between degeneracies and QUE has been further strengthened in current work in progress demonstrating the connection between QUE and long-logarithmic time propagation, which is exactly the time scale at which the conjectured quasimode threshold emerges.

Moreover, we have greatly extended our understanding of the role of arithmetic in questions of QUE. In a paper ("Eisenstein Quasimodes and QUE", in Annales Henri Poincare), we show that the analog of QUE for Eisenstein quasimodes persists well beyond the conjectured logarithmic threshold-- at least to first order-- due to arithmetic considerations uncovered in prior joint work with E. Lindenstrauss on joint quasimodes of the Laplacian and Hecke operators. On the other hand, we show that second-order approximations include enhancement on cusp-bound geodesics, in analogy with the failure of QUE for log-scale quasimodes. This interplay between arithmetic and dynamics is responsible for further surprising phenomena of Eisenstein quasimodes currently being prepared for publication, and which are being further investigated in ongoing joint work with R. Schubert.

Moreover, we have studied the interplay between QUE on the round sphere and the dynamics of embedded symmetries, such as the Hecke correspondence, which is the subject of open conjectures. In joint work with E. Le Masson and E. Lindenstrauss ("Quantum Ergodicity and Averaging Operators on the Sphere", in Int. Math. Res. Not.) we show that in each space of spherical harmonics of large eigenvalue, the quantum ergodicity conjecture is satisfied by any basis that are also eigenfunctions of an averaging operator over a finite set of rotations (satisfying mild, natural conditions). These averaging operators need not be arithmetic in nature, but show the impact of imposing additional symmetries on the basis of eigenfunctions. Ongoing work in progress with E. Le Masson seeks a deeper understanding of L^p norms of eigenfunctions on graphs, and their embedded cousins in the sphere and elsewhere.

Thus we have managed to both to elucidate the relationship between spectral degeneracies and QUE, and managed to understand its extensions and limitations in other settings. The impact of this work is already evident in a number of papers building on top of this research, with the promise of much more in the near future.