The scattering matrix is defined on a perturbed stratified medium. For a class of perturbations, its main part at fixed energy is a Fourier integral operator on the sphere at infinity. Proving this is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix determines the asymptotics of a large class of perturbations.

Strong asymptotic completeness is shown for a pair of Schrödinger type operators on a cylindrical Lipschitz domain. A key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology. The results are based on a refined version of Mourre’s method within the context of pseudo-selfadjoint operators.

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show that the...

Let $A_{𝒩}$ be the symmetric operator given by the restriction of $A$ to $𝒩$, where $A$ is a self-adjoint operator on the Hilbert space $\mathscr{H}$ and $𝒩$ is a linear dense set which is closed with respect to the graph norm on $D\left(A\right)$, the operator domain of $A$. We show that any self-adjoint extension $A_{\Theta }$ of $A_{𝒩}$ such that $D\left(A_{\Theta }\right)\cap D\left(A\right)=𝒩$ can be additively decomposed by the sum $A_{\Theta }=\overline{A}+T_{\Theta }$, where both the operators $\overline{A}$ and $T_{\Theta }$ take values in the strong dual of $D\left(A\right)$. The operator $\overline{A}$ is the closed extension of $A$ to the whole $\mathscr{H}$ whereas $T_{\Theta }$ is explicitly written in terms...

We consider a version of the Weyl formula describing the asymptotic behaviour of the counting function of eigenvalues in the semiclassical approximation for self-adjoint elliptic differential operators under weak regularity hypotheses. Our aim is to treat Hölder continuous coefficients and to investigate the case of critical energy values as well.

An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of ${ℂ}^N$ are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as above is also studied, and it is characterized if the domain is ${ℂ}^N$. The results obtained extend or improve earlier work...

We extend the results in [5] to non-compactly supported perturbations for a class of symmetric first order systems.

In the context of the spaces of homogeneous type, given a family of operators that look like approximations of the identity, new sharp maximal functions are considered. We prove a good-λ inequality for Muckenhoupt weights, which leads to an analog of the Fefferman-Stein estimate for the classical sharp maximal function. As a consequence, we establish weighted norm estimates for certain singular integrals, defined on irregular domains, with Hörmander conditions replaced by some estimates which do...

We describe the asymptotic distribution of eigenvalues of self-adjoint elliptic differential operators, assuming that the first-order derivatives of the coefficients are Lipschitz continuous. We consider the asymptotic formula of Hörmander's type for the spectral function of pseudodifferential operators obtained via a regularization procedure of non-smooth coefficients.

We consider an abstract non-negative self-adjoint operator L acting on L²(X) which satisfies Davies-Gaffney estimates. Let $H_L^p\left(X\right)$ (p > 0) be the Hardy spaces associated to the operator L. We assume that the doubling condition holds for the metric measure space X. We show that a sharp Hörmander-type spectral multiplier theorem on $H_L^p\left(X\right)$ follows from restriction-type estimates and Davies-Gaffney estimates. We also establish a sharp result for the boundedness of Bochner-Riesz means on $H_L^p\left(X\right)$.

Let $M$ be a Riemannian manifold, which possesses a transitive Lie group $G$ of isometries. We suppose that $G$, and therefore $M$, are compact and connected. We characterize the Sobolev spaces $W_p^1\left(M\right)$$\left(1<p<...\; <p></p>\; <article>\; <h3>\; <a\; href="\; /doc/231196">\; Solution\; of\; a\; spectral\; problem\; for\; the\; curl\; and\; the\; Stokes\; operator\; with\; periodic\; boundary\; conditions.\; </a>\; </h3>\; <div\; class="meta\; meta-detail">\; <p\; class="author">\; Saks,\; R.S.\; (2004)\; </p>\; <p\; class="source">\; Zapiski\; Nauchnykh\; Seminarov\; POMI\; </p>\; </div>\; <div\; class="excerpt">\; <p>\; </p>\; </div>\; </article>\; <section\; class="unit\; unit-results-currently-displaying-subject">\; <h2>Currently\; displaying\; 1\; –\; 20\; of\; 57</h2>\; <div>\; <p\; class="pagination"\; aria-label="Pages">\; <strong>Page\; 1</strong>\; <a\; href="?pageNumber=2\&prefix=s">\; <span\; class="next">Next</span>\; </a>\; </p>\; </div>\; </section>\; <!--\; /\; end\; \#tabs-1\; -->\; <!--\; /\; end\; \#tabs\; -->\; <!--\; /\#primary-content\; -->\; <!--\; /\#page-content\; -->\; <!--\; \#container\; -->\right)$