Resonance regimes of scattering by small bodies with impedance boundary conditions

The paper concerns scattering of plane waves by a bounded obstacle with complex valued impedance boundary conditions. We study the spectrum of the Neumann-to-Dirichlet operator for small wave numbers and long wave asymptotic behavior of the solutions of the scattering problem. The study includes the case when $k=0$ is an eigenvalue or a resonance. The transformation from the impedance to the Dirichlet boundary condition as impedance grows is described. A relation between poles and zeroes of the scattering matrix in the non-self adjoint case is established. The results are applied to a problem of scattering by an obstacle with a springy coating. The paper describes the dependence of the impedance on the properties of the material, that is on forces due to the deviation of the boundary of the obstacle from the equilibrium position

Difference factorizations and monotonicity in inverse medium scattering for contrasts with fixed sign on the boundary

We generalize the factorization method for inverse medium scattering using a particular factorization of the difference of two far field operators. While the factorization method has been used so far mainly to identify the shape of a scatterer's support, we show that factorizations based on Dirichlet-to-Neumann operators can be used to compute bounds for numerical values of the medium on the boundary of its support. To this end, we generalize ideas from inside-outside duality to obtain a monotonicity principle that allows for alternative uniqueness proofs for particular inverse scattering problems (e.g., when obstacles are present inside the medium). This monotonicity principle indeed is our most important technical tool: It further directly shows that the boundary values of the medium's contrast function are uniquely determined by the corresponding far field operator. Our particular factorization of far field operators additionally implies that the factorization method rigorously characterizes the support of an inhomogeneous medium if the contrast function takes merely positive or negative values on the boundary of its support independently of the contrast's values inside its support. Finally, the monotonicity principle yields a simple algorithm to compute upper and lower bounds for these boundary values, assuming the support of the contrast is known. Numerical experiments show feasibility of a resulting numerical algorithm

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Time reversal in thermoacoustic tomography - an error estimate

The time reversal method in thermoacoustic tomography is used for approximating the initial pressure inside a biological object using measurements of the pressure wave made on a surface surrounding the object. This article presents error estimates for the time reversal method in the cases of variable, non-trapping sound speeds.Comment: 16 pages, 6 figures, expanded "Remarks and Conclusions" section, added one figure, added reference

Resonance-free Region in scattering by a strictly convex obstacle

We prove the existence of a resonance free region in scattering by a strictly convex obstacle with the Robin boundary condition. More precisely, we show that the scattering resonances lie below a cubic curve which is the same as in the case of the Neumann boundary condition. This generalizes earlier results on cubic poles free regions obtained for the Dirichlet boundary condition.Comment: 29 pages, 2 figure

On completeness of description of an equilibrium canonical ensemble by reduced s-particle distribution function

In this article it is shown that in a classical equilibrium canonical ensemble of molecules with $s$-body interaction full Gibbs distribution can be uniquely expressed in terms of a reduced s-particle distribution function. This means that whenever a number of particles $N$ and a volume $V$ are fixed the reduced $s$-particle distribution function contains as much information about the equilibrium system as the whole canonical Gibbs distribution. The latter is represented as an absolutely convergent power series relative to the reduced $s$-particle distribution function. As an example a linear term of this expansion is calculated. It is also shown that reduced distribution functions of order less than $s$ don't possess such property and, to all appearance, contain not all information about the system under consideration.Comment: This work was reported on the International conference on statistical physics "SigmaPhi2008", Crete, Greece, 14-19 July 200

On occurrence of spectral edges for periodic operators inside the Brillouin zone

The article discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics (e.g., Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the (reduced) Brillouin zone only, rather than the whole zone? Or, do the edges of the spectrum occur necessarily at the set of ``corner'' high symmetry points? This is known to be true in 1D, while no apparent reasons exist for this to be happening in higher dimensions. In many practical cases, though, this appears to be correct, which sometimes leads to the claims that this is always true. There seems to be no definite answer in the literature, and one encounters different opinions about this problem in the community. In this paper, starting with simple discrete graph operators, we construct a variety of convincing multiply-periodic examples showing that the spectral edges might occur deeply inside the Brillouin zone. On the other hand, it is also shown that in a ``generic'' case, the situation of spectral edges appearing at high symmetry points is stable under small perturbations. This explains to some degree why in many (maybe even most) practical cases the statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in the new versio

High orders of the perturbation theory for hydrogen atom in magnetic field

The states of hydrogen atom with principal quantum number $n\le3$ and zero magnetic quantum number in constant homogeneous magnetic field ${\cal H}$ are considered. The coefficients of energy eigenvalues expansion up to 75th order in powers of ${\cal H}^2$ are obtained for these states. The series for energy eigenvalues and wave functions are summed up to ${\cal H}$ values of the order of atomic magnetic field. The calculations are based on generalization of the moment method, which may be used in other cases of the hydrogen atom perturbation by a polynomial in coordinates potential.Comment: 16 pages, LaTeX, 6 figures (ps, eps