Absolute continuity of the periodic Schr\"odinger operator in transversal geometry

We show that the spectrum of a Schr\"odinger operator on $\mathbb{R}^n$, $n\ge 3$, with a periodic smooth Riemannian metric, whose conformal multiple has a product structure with one Euclidean direction, and with a periodic electric potential in $L^{n/2}_{\text{loc}}(\mathbb{R}^n)$, is purely absolutely continuous. Previously known results in the case of a general metric are obtained in [12], see also [8], under the assumption that the metric, as well as the potential, are reflection symmetric

Inverse boundary problems for polyharmonic operators with unbounded potentials

We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(-\Delta)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator

Determining a first order perturbation of the biharmonic operator by partial boundary measurements

We consider an operator $\Delta^2 + A(x)\cdot D+q(x)$ with the Navier boundary conditions on a bounded domain in $R^n$, $n\ge 3$. We show that a first order perturbation $A(x)\cdot D+q$ can be determined uniquely by measuring the Dirichlet--to--Neumann map on possibly very small subsets of the boundary of the domain. Notice that the corresponding result does not hold in general for a first order perturbation of the Laplacian

Inverse boundary value problems for the perturbed polyharmonic operator

We show that a first order perturbation $A(x)\cdot D+q(x)$ of the polyharmonic operator $(-\Delta)^m$, $m\ge 2$, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in $R^n$, $n\ge 3$. Notice that the corresponding result does not hold in general when $m=1$

Inverse spectral problems on a closed manifold

In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $\Sigma$ into two components and we know the eigenvalues $\lambda_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $\Sigma$, of the corresponding eigenfunctions $\phi_j$, i.e. $\phi_j|_{\Sigma},\partial_\nu\phi_j|_{\Sigma}$, where $\nu$ is the normal to $\Sigma$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $\lambda_j$ and $\phi_j|_{\Sigma}$ only. However, if $\Sigma$ consists of at least two components, $\Sigma_1, \Sigma_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $\Sigma$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $\Sigma_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described

Reconstruction of Betti numbers of manifolds for anisotropic Maxwell and Dirac systems

We consider an invariant formulation of the system of Maxwell's equations for an anisotropic medium on a compact orientable Riemannian 3-manifold $(M,g)$ with nonempty boundary. The system can be completed to a Dirac type first order system on the manifold. We show that the Betti numbers of the manifold can be recovered from the dynamical response operator for the Dirac system given on a part of the boundary. In the case of the original physical Maxwell system, assuming that the entire boundary is known, all Betti numbers of the manifold can also be determined from the dynamical response operator given on a part of the boundary. Physically, this operator maps the tangential component of the electric field into the tangential component of the magnetic field on the boundary