# Project Details

### Description

This proposal is to continue the PI's research in Global Harmonic Analysis, relating asymptotic properties of eigenfunctions of the Laplacian of a Riemannian manifold (X; g) to the dynamics of the geodesic

flow. There is a parallel theory for holomorphic sections of powers Lk ! M of

a positive Hermitian line bundle over a Kahler manifold. The two settings come together in the theory of Grauert tubes X� of real analytic (X; g), where g determines a natural Kahler metric on the complexification XC of X. Grauert tubes are canonically isomorphic to ball bundles in T�X and may be viewed as phase spaces. In the area of eigenfunctions of (X; g), the PI proposes to study a number of `restriction problems' to hypersurfaces H � X. One problem involves the relations between weak* limits of matrix elements of restricted eigenfunctions to the global matrix elements on M. Toth and the

PI have recently proved a uniform lower bound on L2 norms of restrictions for a proportion 1� of an orthonormal basis, and the PI proposes methods to strengthen the result. Sogge and the PI obtained upper bounds on sup norms of restrictions (Cauchy data) for non-positively curved manifolds with concave boundary, and the PI proposes several ways to improve these.

Motivation to obtain improved bounds comes from recent work of J. Jung and the author showing that the number of nodal domains of almost all eigenfunctions �j increases to infinity with the eigenvalue on surfaces of non-positive curvature and concave boundary. The PI seeks to obtain a logarithmic lower bound on the number of nodal domains using improved sup norm bounds. The PI also proposes to study analytic perturbations of nodal sets.

Another project is to study nodal sets of analytic continuations of eigenfunctions. In earlier work the PI showed that normalized currents of integration over complex nodal sets tend almost surely to one limit current when the geodesic flow is ergodic. The PI plans to study the limit problems when the geodesic flow is completely integrable and to show that many singular limits exist. The same technique gives upper bounds on nodal volumes and the PI proposes that they might work for Carleman smooth metrics.

In recent work with Zhou, the PI has obtained `Erf-asymptotics' of `partial Bergman kernels' on Kahler manifolds M, i.e. spectral projections for Toeplitz operators, across the interface between the allowed and forbidden regions. The PI plans to study related uncertainty principle problems for Toeplitz operatos quantizing characteristic functions of subsets of M, ranging from smooth domains to fractal subsets, parallel to results on pseudo-differential operators.

1.1. Intellectual merit of the proposed activity. The inverse spectral problem, `Can you hear the shape of a drum', is a classical unsolved problem that remains actively studied today. The PI's result on analytic domains with one symmetry is the strongest positive known to date. Eigenfunctions of Schrodinger operators are the stationary states of quantum mechanics, and their sizes and shapes are important in both physics and chemistry.

1.2. Broader impacts resulting from the proposed activity. The PI has given many lecture series on global harmonic analysis, notably a one-hour address at the 2016 national AMS meeting in Seattle, and at distinguished lecture series at UC Irvine and Univ. Indiana. The PI has also given mini-courses at the SNAP program in 2017 at Northwestern and at Cargese Institute in 2016.

flow. There is a parallel theory for holomorphic sections of powers Lk ! M of

a positive Hermitian line bundle over a Kahler manifold. The two settings come together in the theory of Grauert tubes X� of real analytic (X; g), where g determines a natural Kahler metric on the complexification XC of X. Grauert tubes are canonically isomorphic to ball bundles in T�X and may be viewed as phase spaces. In the area of eigenfunctions of (X; g), the PI proposes to study a number of `restriction problems' to hypersurfaces H � X. One problem involves the relations between weak* limits of matrix elements of restricted eigenfunctions to the global matrix elements on M. Toth and the

PI have recently proved a uniform lower bound on L2 norms of restrictions for a proportion 1� of an orthonormal basis, and the PI proposes methods to strengthen the result. Sogge and the PI obtained upper bounds on sup norms of restrictions (Cauchy data) for non-positively curved manifolds with concave boundary, and the PI proposes several ways to improve these.

Motivation to obtain improved bounds comes from recent work of J. Jung and the author showing that the number of nodal domains of almost all eigenfunctions �j increases to infinity with the eigenvalue on surfaces of non-positive curvature and concave boundary. The PI seeks to obtain a logarithmic lower bound on the number of nodal domains using improved sup norm bounds. The PI also proposes to study analytic perturbations of nodal sets.

Another project is to study nodal sets of analytic continuations of eigenfunctions. In earlier work the PI showed that normalized currents of integration over complex nodal sets tend almost surely to one limit current when the geodesic flow is ergodic. The PI plans to study the limit problems when the geodesic flow is completely integrable and to show that many singular limits exist. The same technique gives upper bounds on nodal volumes and the PI proposes that they might work for Carleman smooth metrics.

In recent work with Zhou, the PI has obtained `Erf-asymptotics' of `partial Bergman kernels' on Kahler manifolds M, i.e. spectral projections for Toeplitz operators, across the interface between the allowed and forbidden regions. The PI plans to study related uncertainty principle problems for Toeplitz operatos quantizing characteristic functions of subsets of M, ranging from smooth domains to fractal subsets, parallel to results on pseudo-differential operators.

1.1. Intellectual merit of the proposed activity. The inverse spectral problem, `Can you hear the shape of a drum', is a classical unsolved problem that remains actively studied today. The PI's result on analytic domains with one symmetry is the strongest positive known to date. Eigenfunctions of Schrodinger operators are the stationary states of quantum mechanics, and their sizes and shapes are important in both physics and chemistry.

1.2. Broader impacts resulting from the proposed activity. The PI has given many lecture series on global harmonic analysis, notably a one-hour address at the 2016 national AMS meeting in Seattle, and at distinguished lecture series at UC Irvine and Univ. Indiana. The PI has also given mini-courses at the SNAP program in 2017 at Northwestern and at Cargese Institute in 2016.

Status | Active |
---|---|

Effective start/end date | 9/1/18 → 8/31/21 |

### Funding

- National Science Foundation (DMS-1810747-002)