Related papers: Trace singularities in obstacle scattering and the…
We develop relative oscillation theory for one-dimensional Dirac operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing…
We derive the asymptotic behavior of the radial distribution function $g(x)$ for one-dimensional (1D) hard-rod systems and related quasi-one-dimensional geometries at high packing fractions using Laplace transform techniques and pole…
We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive integer. In \cite{PSS} a trace formula for operator Taylor polynomials was obtained.…
In 1971 Fedi\u{i} proved the remarkable theorem that the linear second order partial differential operator in the plane with coefficients 1 and f^2 is hypoelliptic provided that f is smooth, vanishes at the origin and is positive otherwise.…
The relation $Tr \gamma_{5}= 0$ implies the contribution to the trace from unphysical (would-be) species doublers in lattice gauge theory. This statement is also true for the Pauli-Villars regularization in continuum theory. If one insists…
Starting with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$, one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the…
For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…
In this paper a wave is generated by an initial data whose support is localized at the outside of unknown obstacles and observed in a limited time on a known closed surface or the same position as the support of the initial data. The…
This paper is concerning the inverse conductive scattering of acoustic waves by a bounded inhomogeneous object with possibly embedded obstacles inside. A new uniqueness theorem is proved that the conductive object is uniquely determined by…
For continuous \gamma, g:[0,1]\to(0,\infty), consider the degenerate stochastic differential equation dX_t=[1-|X_t|^2]^{1/2}\gamma(|X_t|) dB_t-g(|X_t|)X_t dt in the closed unit ball of R^n. We introduce a new idea to show pathwise…
For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with…
We consider Schr\"odinger operators with complex decaying potentials (in general, not from trace class) on the lattice. We determine trace formulae and estimate of eigenvalues and singular measure in terms of potentials. The proof is based…
In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb…
Results from Direct Numerical Simulations of particle relative dispersion in three dimensional homogeneous and isotropic turbulence at Reynolds number $Re_\lambda \sim 300$ are presented. We study point-like passive tracers and heavy…
The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is…
This is a survey of recent results on zeta- and eta-function poles and values for realizations of Laplace- and Dirac-type operators defined by pseudodifferential projection boundary conditions (including the Atiyah-Patodi-Singer operator…
This paper is devoted to establish the most essential connections of the eigenvalue problems for the Laplace operator, Lam\'{e} operator, Stokes operator, buckling operator and clamped plate operator. We show that the $k$-th Stokes…
Now a final and maybe simplest formulation of the enclosure method applied to inverse obstacle problems governed by partial differential equations in a {\it spacial domain with an outer boundary} over a finite time interval is fixed. The…
An old problem in mathematical physics deals with the structure of the dispersion relation of the Schr\"odinger operator $-\Delta+V(x)$ in $R^n$ with periodic potential near the edges of the spectrum. A well known conjecture says that…
The structure of diagonal singularities of Green functions of partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian man ifold is studied. A special class of operators formed by the…