Quantum Ergodicity for a Class of Mixed Systems

We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is the Liouville measure. This proves part of a conjecture of Percival.Comment: 16 pages, 2 figure

A Quantitative Vainberg Method for Black Box Scattering

We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $\tau$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $\tau$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $\tau$, then there are resonances in logarithmic strips whose width is given by $\tau$. As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.Comment: 22 pages, 1 figur

Resonances for Thin Barriers on the Circle

We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V$ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ is a Hamiltonian used to model both quantum corrals and leaky quantum graphs. Since highly frequency dependent delta potentials are out of reach of the more general techniques in previous work, we study the special case where $\Omega=B(0,1)\subset \mathbb{R}^2$ and $V\equiv h^{-\alpha }V_0>0$ with $\alpha\leq 1$. Here $h^{-1}\sim \Re \lambda$ is the frequency. We give sharp bounds on the size of resonance free regions for $\alpha\leq 1$ and the location of bands of resonances when $5/6\leq \alpha\leq 1$. Finally, we give a lower bound on the number of resonances in logarithmic size strips: $-M\log \Re \lambda\leq \Im \lambda \leq 0$.Comment: 23 pages, 6 figur

A microlocal approach to eigenfunction concentration

We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$\Big|\int _H u_hd\sigma_H\Big|$$ where $H\subset M$ is a submanifold and $\sigma_H$ the induced by the Riemannian metric on $M$. This approach can be applied uniformly to submanifolds of codimension $1\leq k\leq n$ and in particular, gives a new approach to understanding $\|u_h\|_{L^\infty(M)}$. The method, developed in the author's recent work together with Y. Canzani and J. Toth, relies on estimating averages by the behavior of $u_h$ microlocally near the conormal bundle to $H$. By doing this, we are able to obtain quantitative improvements on eigenfunction averages under certain uniform non-recurrent conditions on the conormal directions to $H$. In particular, we do not require any global assumptions on the manifold $(M,g)$.Comment: 16 pages, 7 figure

Pseudospectra of Semiclassical Boundary Value Problems

We consider operators $-\Delta + X$ where $X$ is a constant vector field, in a bounded domain and show spectral instability when the domain is expanded by scaling. More generally, we consider semiclassical elliptic boundary value problems which exhibit spectral instability for small values of the semiclassical parameter h, which should be thought of as the reciprocal of the Peclet constant. This instability is due to the presence of the boundary: just as in the case of $-\Delta + X$, some of our operators are normal when considered on R^d. We characterize the semiclassical pseudospectrum of such problems as well as the areas of concentration of quasimodes. As an application, we prove a result about exit times for diffusion processes in bounded domains. We also demonstrate instability for a class of spectrally stable nonlinear evolution problems that are associated to these elliptic operators.Comment: 43 pages, 6 figure

Improvements for eigenfunction averages: An application of geodesic beams

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_\lambda \}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda =\lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$\Big|\int_H\phi_\lambda d\sigma_H\Big|=O\Big(\frac{\lambda^{\frac{k-1}{2}}}{\sqrt{\log \lambda}}\Big)\qquad \text{or}\qquad |\phi_\lambda(x)|=O\Big(\frac{\lambda ^{\frac{n-1}{2}}}{\sqrt{\log \lambda}}\Big),$$ as $\lambda\to \infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $H\subset M$. We also find weaker conditions than having no conjugate points which guarantee $\sqrt{\log \lambda}$ improvements for the $L^\infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.Comment: 70 pages, 4 figures. The new version includes a major revision of Appendix A, parts of which have been replaced by section

Eigenfunction concentration via geodesic beams

In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $\lambda^{-\frac{1}{2}}$. We control $\phi_\lambda(x)$ by the $L^2$-mass of $\phi_\lambda$ on each geodesic tube and derive a purely dynamical statement through which $\phi_\lambda(x)$ can be studied. In particular, we obtain estimates on $\phi_\lambda(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds.Comment: 61 pages, 2 figures. Improved exposition and includes new explanatory material in the introduction as well as an examples section (1.5) and a full section on comparison with previous work (1.6). Appendices A.1 (Index of notation) and B were also adde

Fractal Weyl laws and wave decay for general trapping

We prove a Weyl upper bound on the number of scattering resonances in strips for manifolds with Euclidean infinite ends. In contrast with previous results, we do not make any strong structural assumptions on the geodesic flow on the trapped set (such as hyperbolicity) and instead use propagation statements up to the Ehrenfest time. By a similar method we prove a decay statement with high probability for linear waves with random initial data. The latter statement is related heuristically to the Weyl upper bound. For geodesic flows with positive escape rate, we obtain a power improvement over the trivial Weyl bound and exponential decay up to twice the Ehrenfest time.Comment: 36 pages, 5 figures; minor revision

Restriction Bounds for the Free Resolvent and Resonances in Lossy Scattering

We establish high energy $L^2$ estimates for the restriction of the free Green's function to hypersurfaces in $\mathbb{R}^d$. As an application, we estimate the size of a logarithmic resonance free region for scattering by potentials of the form $V\otimes \delta_{\Gamma}$, where $\Gamma \subset \mathbb{R}^d$ is a finite union of compact subsets of embedded hypersurfaces. In odd dimensions we prove a resonance expansion for solutions to the wave equation with such a potential.Comment: 24 page