arXiv research

Sharp bounds on quasimode norms on compact space forms.

Estimates for eigenfunctions and quasimodes on compact manifolds.

Optimal estimates for spectral projection norms on compact manifolds.

Researchers solve an inverse problem for a semilinear elliptic equation on complex manifolds.

Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory.

Analyzes tunneling effects for Schrödinger operators on vector bundles.

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density fun…

In the limit $\hbar\to 0$, we analyze a class of Schrödinger operators $H_\hbar = \hbar^2 L + \hbar W + V\cdot \mathrm{id}$ acting on sections of a vector bundle $\mathcal{Eh}$ over a Riemannian manifold $M$ where $L$ is a Laplace type operator, $W$ is an endomorphism field and the potential energy $V$ has a non-degene…

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(λ)$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is …

We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flat…

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …

The paper constructs instanton complexes on stratified pseudomanifolds.