Functional Analysis and Its Applications最新文献

A new linear extension operator which extends (generalized) functions on a hyperplane in a Euclidean space to the whole space is introduced. It is shown that this operator is continuous as an operator between appropriate function spaces for a large class of Sobolev–Slobodetsky, Besov, and Triebel–Lizorkin spaces.

We review Yafaev’s approach to asymptotic completeness for systems of particles mutually interacting with short-range potentials. The resulting theory is based on computation of commutators with time-independent (mostly bounded) observables yielding a sufficient supply of Kato smoothness bounds.

We develop scattering theory for the Laplace operator in the half-space with Robin type boundary conditions on the boundary plane. In particular, we show that, in addition to usual space waves living in cones and described by standard wave operators, surface waves may arise in this problem. They are localized in parabolic neighbourhoods of the boundary. We find conditions on the boundary coefficient ensuring the existence of surface waves. Several open problems are formulated.

We obtain sufficient conditions for the perturbation of the power of the resolvent of the polyharmonic operator under a perturbation by a highly singular potential to belong to Schatten classes.

Given a self-adjoint operator (H_0) bounded from below in a complex, separable Hilbert space (mathcal H), the corresponding scale of spaces (mathcal H_{+1}(H_0) subset mathcal H subset mathcal H_{-1}(H_0)=[mathcal H_{+1}(H_0)]^*), and a fixed (Vin mathcal B(mathcal H_{+1}(H_0),mathcal H_{-1}(H_0))), we define the operator-valued map (A_V(,cdot,)colon rho(H_0)to mathcal B(mathcal H)) by

where (rho(H_0)) denotes the resolvent set of (H_0). Assuming that (A_V(z)) is compact for some (z=z_0in rho(H_0)) and has norm strictly less than one for some (z=E_0in (-infty,0)), we employ an abstract version of Tiktopoulos’ formula to define an operator (H) in (mathcal H) that is formally realized as the sum of (H_0) and (V). We then establish a Birman–Schwinger principle for (H) in which (A_V(,cdot,)) plays the role of the Birman–Schwinger operator: (lambda_0in rho(H_0)) is an eigenvalue of (H) if and only if (1) is an eigenvalue of (A_V(lambda_0)). Furthermore, the geometric (but not necessarily the algebraic) multiplicities of (lambda_0) and (1) as eigenvalues of (H) and (A_V(lambda_0)), respectively, coincide.

As a concrete application, we consider one-dimensional Schrödinger operators with (H^{-1}(mathbb{R})) distributional potentials.

In (L_2(mathbb R^d)), we consider a selfadjoint operator ({mathbb A}_varepsilon), (varepsilon >0), of the form

where (0< alpha < 2). It is assumed that a function (mu(mathbf{x},mathbf{y})) is bounded, positive definite, periodic in each variable, and is such that (mu(mathbf{x},mathbf{y})=mu(mathbf{y},mathbf{x})). A rigorous definition of the operator ({mathbb A}_varepsilon) is given in terms of the corresponding quadratic form. It is proved that the resolvent (({mathbb A}_varepsilon+I)^{-1}) converges in the operator norm on (L_2(mathbb R^d)) to the operator (({mathbb A}^0+I)^{-1}) as (varepsilonto 0). Here, ({mathbb A}^0) is an effective operator of the same form with the constant coefficient (mu^0) equal to the mean value of (mu(mathbf{x},mathbf{y})). We obtain an error estimate of order (O(varepsilon^alpha)) for (0< alpha < 1), (O(varepsilon (1+| operatorname{ln} varepsilon|)^2)) for ( alpha=1), and (O(varepsilon^{2- alpha})) for (1< alpha < 2). In the case where (1< alpha < 2), the result is refined by taking the correctors into account.

For a wide class of unbounded integral Hankel operators on the positive half-line, we prove essential self-adjointness on the set of smooth compactly supported functions.

Consider a bound state (an eigenfunction) (psi) of an atom with (N) electrons. We study the spectra of the one-particle density matrix (gamma) and the one-particle kinetic energy density matrix (tau) associated with (psi). The paper contains two results. First, we obtain the bounds (lambda_k(gamma)le C k^{-8/3}) and (lambda_k(tau)le C k^{-2}) with some positive constants (C) that depend explicitly on the eigenfunction (psi). The sharpness of these bounds is confirmed by the asymptotic results obtained by the author in earlier papers. The advantage of these bounds over the ones derived by the author previously is their explicit dependence on the eigenfunction. Moreover, their new proofs are more elementary and direct. The second result is new, and it pertains to the case where the eigenfunction (psi) vanishes at the particle coalescence points. In particular, this is true for totally antisymmetric (psi). In this case, the eigenfunction (psi) exhibits enhanced regularity at the coalescence points, which leads to the faster decay of the eigenvalues: (lambda_k(gamma)le C k^{-10/3}) and (lambda_k(tau)le C k^{-8/3}).

The proofs rely on estimates for the derivatives of the eigenfunction (psi) that depend explicitly on the distance to the coalescence points. Some of these estimates are borrowed directly from, and some are derived using the methods of, a recent paper by S. Fournais and T. Ø. Sørensen.

Using the well known approach developed in the papers of B. Davies and his co-authors, we obtain inequalities for the location of possible complex eigenvalues of non-selfadjoint functional difference operators. When studying the sharpness of the main result, we discovered that complex potentials can create resonances.