In this paper we study the Cauchy problem and boundary-value of general form in exterior a compact set for hyperbolic operators L whose coefficients depend only on x are constant near infinity. Assuming that wave fronts Green's matrix go off to infinity as , determine asymptotic behaviour solutions . For corresponding stationary obtain short-wave real complex frequencies.

CONTENTSIntroductionChapter I. Equations with constant coefficients § 1. Auxiliary statements 2. Construction of fundamental solutions 3. Asymptotic behaviour 4. Conditions for existence and uniqueness a solutionChapter II. Principle limit absorption Generalized principle Sharpening Theorem 2.2Chapter III. variable solution An example A sufficient condition generalized equations coefficientsChapter IV. amplitudeReferences

3R15. Linear Water Waves: A Mathematical Approach. - N Kuznetsov (Russian Acad of Sci, St Petersburg, Russia), V Maz’ya (Univ Linkoping, Sweden), B Vainberg Carolina, Charlotte, NC, Sweden). Cambridge UP, Cambridge, UK. 2002. 513 pp. ISBN 0-521-80853-7. $100.00.Reviewed by J Miles (Inst Geophysics and Planetary Phys, UCSD, La Jolla CA 92093-0225).As its subtitle suggests preface proclaims, “The aim the present book is to give a self-contained up-to-date account mathematical results in linear...

The paper concerns the discreteness of eigenvalues and solvability interior transmission problem for anisotropic media. Conditions ellipticity are written explicitly, it is shown that they do not guarantee eigenvalues. Some simple sufficient conditions found. They expressed in terms values anisotropy matrix refraction index at boundary domain. if a small perturbation applied to index.

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using Dirichlet-to-Neumann map, it can be reduced to an elliptic one. leads discreteness of spectrum as well certain results on a possible location eigenvalues. If index refraction real, then obtain result existence infinitely many positive ITEs and Weyl-type lower bound its counting function. All are obtained under assumption that n(x) − 1 does vanish at boundary...

We investigate the nonstationary parabolic Anderson problem ∂u∂t=ϰLu(t,x)+ξt(x)u(t,x),u(0,x)≡1,(t,x)∈[0,∞)×Zd where ϰL denotes a nonlocal Laplacian and ξt(x) is correlated white-noise potential. The irregularity of solution linked to upper spectrum certain multiparticle Schrödinger operators that govern moment functions mp(t,x1,x2,⋯,xp)=⟨u(t,x1)u(t,x2)⋯u(t,xp)⟩. First, we establish weak form intermittency under broad assumptions on L positive-definite noise correlator B=B(x). then examine...

Abstract The problem of determining a unique solution the Schrödinger equation on lattice is considered, where Δ difference Laplacian and both f q have finite supports. It shown that there an exceptional set So points for which limiting absorption priciple fails, even unperturbed operator (q(x)=0). This consists when d odd. For all Values , radiation conditions are found single out same solutions as ones determined by principle. These conbinations several waves propagating with different...

This paper contains the Weyl formula for counting function of interior transmission problem when latter is parameter elliptic. Branching billiard trajectories are constructed, and second term asymptotics estimated from above under some conditions on set periodic trajectories.

This paper contains lower bounds on the counting function of positive eigenvalues interior transmission problem when latter is elliptic. In particular, these justify existence an infinite set and provide asymptotic estimates from above for large values wave number. They also lead to certain important upper first few eigenvalues. We consider classical as well case inhomogeneous medium obstacle.

In this paper elliptic problems in exterior domains polynomially depending on a spectral parameter are considered. These obtained from mixed problem for hyperbolic equations by substituting . For such analytic properties of the resolvent studied neighborhood point , which permits, corresponding nonstationary problem, complete asymptotic expansion solutions as .Bibliography: 11 items.

We establish a long time soliton asymptotics for nonlinear system of wave equation coupled to charged particle. The has six-dimensional manifold solutions. show that in the large approximation, any solution, with an initial state close solitary manifold, is sum and dispersive which solution free equation. It assumed charge density satisfies Wiener condition version Fermi Golden Rule, momenta distribution vanish up fourth order. proof based on development general strategy introduced by...

Existence of a spectral singularity (SS) in the spectrum Schr\"odinger operator with non-Hermitian potential requires exact matching parameters potential. We provide necessary and sufficient condition for to have SS at given wavelength. It is shown that potentials SSs prescribed wavelengths can be obtained by simple effective procedure. In particular, developed approach allows one obtain several second order, as well obeying symmetry, say, $\mathcal{PT}$-symmetric potentials. illustrate all...

We investigate the behavior of waves in a periodic medium containing small soft inclusions or cavities arbitrary shape, such that homogeneous Dirichlet conditions are satisfied at boundary. The leading terms Bloch waves, their dispersion relations, and low frequency cutoff rigorously derived. Our approach reveals existence exceptional wave vectors for which comprised clusters perturbed plane propagate different directions. demonstrate these vectors, no any one specific direction.