Hamid Hezari

Inverse Spectral Problems for Schrödinger Operators

In this article we find some explicit formulas for the semi-classical wave invariants at the bottom of the well of a Schrödinger operator. As an application of these new formulas for the wave invariants, we improve the inverse spectral results proved by Guillemin and Uribe in [GU]. They proved that under some symmetry assumptions on the potential V(x), the Taylor expansion of V(x) near a non-degenerate global minimum can be recovered from the knowledge of the low-lying eigenvalues of the associated Schrödinger operator in

Resonant Uniqueness of Radial Semiclassical Schrödinger Operators

{\mathbb R^n}

The Neumann Isospectral Problem for Trapezoids

. We prove similar inverse spectral results using fewer symmetry assumptions. We also show that in dimension 1, no symmetry assumption is needed to recover the Taylor coefficients of V(x).

Robin spectral rigidity of nearly circular domains with a reflectional symmetry

The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius

Lp norms, nodal sets, and quantum ergodicity

r(\lambda) \to 0

A natural lower bound for the size of nodal sets

, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of

Inverse problems in spectral geometry

r(\lambda)

C∞ spectral rigidity of the ellipse

. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of Han and Hezari-Rivi\`ere small scale QE holds on negatively curved manifolds at logarithmically shrinking rates, we get logarithmic improvements on such manifolds for the above measurements of eigenfunctions. We also get