A Rellich type theorem for discrete Schrödinger operators
Hiroshi Isozaki Hisashi Morioka
An analogue of Rellich's theorem is proved for discrete Laplacians on square lattices, and applied to show unique continuation properties on certain domains as well as non-existence of embedded eigenvalues for discrete Schrödinger operators.
keywords: Schrödinger operator Rellich's theorem. square lattice
Probing for inclusions in heat conductive bodies
Patricia Gaitan Hiroshi Isozaki Olivier Poisson Samuli Siltanen Janne Tamminen
This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1--3 for probing the discontinuous part of the conductivity from local temperature and heat flow measurements at the boundary. The approach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is possible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering locations of discontinuities approximately from noisy data.
keywords: Inverse problem heat equation interface reconstruction.
Inverse boundary value problems in the horosphere - A link between hyperbolic geometry and electrical impedance tomography
Hiroshi Isozaki
We consider a boundary value problem for the Schrödinger operator $- \Delta + q(x)$ in a ball $\Omega : (x_1 + R)^2 + x_2^2 + (x_3 - r)^2 < r^2$, whose boundary we regard as a horosphere in the hyperbolic space $ H^3$ realized in the upper half space $ R^3_+$. Let $S = \{|x| = R, x_3 > 0\}$ be a hemisphere, which is generated by a family of geodesics in $ H^3$. By imposing a suitable boundary condition on $\partial\Omega$ in terms of a pseudo-differential operator, we compute the integral mean of $q(x)$ over $S\cap\Omega$ from the local knowledge of the associated (generalized) Robin-to-Dirichlet map for $- \Delta + q(x)$ around $S\cap\partial\Omega$. The potential $q(x)$ is then reconstructed by virtue of the inverse Radon transform on hyperbolic space. If the support of $q(x)$ has a positive distance from $\partial\Omega$, one can construct this generalized Robin-to-Dirichlet map from the usual Dirichlet-to-Neumann map. These results explain the mathematical background of the well-known Barber-Brown algorithm in electrical impedance tomography.
keywords: Horosphere Inverse boundary value problems Barber-Brown algorithm. Electrical Impedance Tomography

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