24 research outputs found

### 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