اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSingularity traces in obstacle scattering and the Poisson relation for the relative trace
37
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the case of scattering of several obstacles in $mathbb{R}^d$ for $d geq 2$ for the Laplace operator $Delta$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $Delta_1$ and $Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative trace operator $g(Delta) - g(Delta_1) - g(Delta_2) + g(Delta_0)$ was introduced in [18] and shown to be trace-class for a large class of functions $g$, including certrain functions of polynomial growth. When $g$ is sufficiently regular at zero and fast decaying at infinity then, by the Birman-Krein formula, this trace can be computed from the relative spectral shift function $xi_{rel}(lambda) = -frac{1}{pi} Im(Xi(lambda))$, where $Xi(lambda)$ is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $xi_{rel}$. In particular we prove that $hatxi_{rel}$ is real-analytic near zero and we relate the decay of $Xi(lambda)$ along the imaginary axis to the first wave-trace invariant of the shortest bounding ball orbit between the obstacles. The function $Xi(lambda)$ is important in physics as it determines the Casimir interactions between the objects.
قيم البحث
اقرأ أيضاً
Let $Sigmasubsetmathbb{R}^d$ be a $C^infty$-smooth closed compact hypersurface, which splits the Euclidean space $mathbb{R}^d$ into two domains $Omega_pm$. In this note self-adjoint Schrodinger operators with $delta$ and $delta$-interactions supporte
d on $Sigma$ are studied. For large enough $minmathbb{N}$ the difference of $m$th powers of resolvents of such a Schrodinger operator and the free Laplacian is known to belong to the trace class. We prove trace formulae, in which the trace of the resolvent power difference in $L^2(mathbb{R}^d)$ is written in terms of Neumann-to-Dirichlet maps on the boundary space $L^2(Sigma)$.
We show that for a one-dimensional Schrodinger operator with a potential whose (j+1)th moment is integrable the jth derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to im
prove the known dispersive estimates with integrable time decay for the one-dimensional Schrodinger equation in the resonant case.
We consider the Landau Hamiltonian perturbed by a long-range electric potential $V$. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. First, we obtain an estimate of the rate of the shrinki
ng of these clusters to the Landau levels as the number of the cluster $q$ tends to infinity. Further, we assume that there exists an appropriate $V$, homogeneous of order $-rho$ with $rho in (0,1)$, such that $V(x) = V(x) + O(|x|^{-rho - epsilon})$, $epsilon > 0$, as $|x| to infty$, and investigate the asymptotic distribution of the eigenvalues within a given cluster, as $q to infty$. We obtain an explicit description of the asymptotic density of the eigenvalues in terms of the mean-value transform of $V$.
We apply the Painleve Test for the Benney and the Benney-Gjevik equations which describe waves in falling liquids. We prove that these two nonlinear 1+1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic so
lutions in terms of Laurent expansions are presented.
Let $H$ signify the free non-negative Laplacian on $mathbb{R}^2$ and $H_Y$ the non-negative Dirichlet Laplacian on the complement $Y$ of a nonpolar compact subset $K$ in the plane. We derive the low-energy expansion for the Krein spectral shift funct
ion (scattering phase) for the obstacle scattering system ${,H_Y,,H,}$ including detailed expressions for the first three coefficients. We use this to investigate the large time behaviour of the expected volume of the pinned Wiener sausage associated to $K$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
