Eigenfunction asymptotics and quantum chaos

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***My basic research interests involve the high-energy behaviour of quantum wavefunctions that model the probabilistic distribution of electron density.  I am most interested in the problem  of  estimating  wavefunction amplitudes and frequencies in the high-energy limit.*** *** More precisely, let $(M,g)$ be compact, Riemannian manifold with Laplace-Beltrami operator $\Delta_g.$ My proposed research program is focused on the asymptotics of the associated Laplace  eigenfunctions and Weyl asymptotics of the corresponding eigenvalues.  A substantial part of my research proposal is focused on asymptotic  restriction bounds (both upper and lower) for eigenfunctions. Eigenfunction restriction bounds have become an  an extremely active area in quantum chaos over the past several years, not only because of their intrinsic interest, but also because of their wide-ranging applications to the asymptotics of eigenfunction nodal domains and critical sets.**** In recent work with Zelditch and with El-Hajj,  I showed that indeed asymptotic eigenfunction restriction lower bounds along a real-analytic curve (ie. "goodness estimates") directly controls the intersection number of the eigenfunction nodal set with the curve. By making a judicious choice of curve,  in the case of arithmetic surfaces, Ghosh, Reznikov and Sarnak have recently shown one can inturn associated nodal domains with pairs of such intersection points. This leads to a very exciting new development in the field: a partial converse to the Courant nodal theorem. However, there are few cases where such quantitative lower bounds have been established.  Consequently, I propose to study the central questions:***Question 1: Given a curve $H$ on a Riemann surface $(M,g)$, under what conditions is $H$ necessarily a good curve?***Question 2: Under what conditions can one establish a quantitative lower bound in the Courant  theorem?***There are special cases that are of exceptional interest. Recently, with El-Hajj we answered Question 1 in the affirmative in the case where $H$ is strictly convex and the ambient manifold is a piecewise-smooth planar domain with ergodic bliiard dynamics. However, other cases remain completely open.  I propose to investigate both Questions 1 and 2 in other cases where the eigenfunction sequence is quantum ergodic. Finally, due to the high spectral multiplicity and the associated freedom in choosing eigenfunction bases, the case of general spherical harmonics should prove to be particularly fascinating and rich testing ground. With several of my students, I propose to investigate both questions in the latter case as well.*** *** *** **