Eigenfunction Asymptotics and Quantum Chaos

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Summary of Proposal (required) Broadly speaking, my research is focused on the study of eigenfunctions of Schrodinger operators in the semiclassical limit. I am primarily interested in the asymptotic growth and accumulation properties as well as the behaviour of the nodal and critical sets. I propose to continue my work in this field and more specifically, I plan to focus on two different (but related) lines of research: 1) EIGENFUNCTION RESTRICTION BOUNDS: Let $(M,g)$ be a compact manifold with Laplace-Beltrami operator $-\Delta$ and $\phi_{\lambda}$ be an $L^2$-normalized eigenfunction with eigenvalue $\lambda^2$ and $H \subset M$ a smooth hypersurface. The problem of estimating $L^2$-restriction bounds $ \| \phi_{\lambda} \|_{L^2(H)}$ from above and below has many important applications in the study of eigenfunction nodal (i.e. zero) and critical sets [TZ1, ET, JZ, TZ2] (a) Lower bounds: Proving unique continuation (i.e. ``goodness" bounds) for eigenfunction restrictions of the form $ \| \phi_{\lambda} \|_{L^2(H)} \geq e^{-C \lambda}$ for all $\phi_{\lambda}$ with $\lambda \geq \lambda_0$ is a very important ingredient in establishing upper bounds for nodal intersections with the hypersurface H. Recently, in joint work with S. Zelditch [TZ2], we have made important progress on this problem. However, the question of whether such estimates are generically satisfied in a general setting remains open. I propose to investigate this in the case of Dirichlet eigenfunctions on a domain $\Omega$ in the case of hypersurfaces $H$ close to the boundary $\partial \Omega. (b) Upper bounds: Improvements in universal upper bounds [BGT] for $\| \phi_{\lambda} \|_{L^2(H)|$ are also central to the study of eigenfunction oscillations. In particular, the question of obtaining improvements for hypersurfaces along the boundary is of particular importance. Recently in [CT], we have established improvements in the case where $H \subset \partial \Omega$ is totally-geodesic and $\Omega$ is a piecewise-smooth convex planar domain. I propose to extend these results to more general manifolds with boundary. 2) NODAL STRUCTURE OF INTERIOR STEKLOV EIGENFUNCTIONS Let $\Omega$ be a compact, smooth manifold with boundary $\partial \Omega = M.$ Recently, there has a great deal of activity related to the spectral asymptotics of the associated Dirichlet-to-Neumann (DtN)  or Steklov operator and the study of corresponding eigenfunction nodal sets.  In joint work with Polterovich and Sher [PST], we have recently proved the sharp analogue of the Yau conjecture for nodal sets for interior Steklov eigenfunctions in the case when $\Omega$ is a Riemann surface with real-anaytic boundary. In higher dimensions, very little  is known in the case of lower bounds. I propose to investigate this question using recent results with Galkowski [GT] on sharp pointwise bounds for Steklov eigenfunctions.