We prove that the spectrum of certain non-self-adjoint Schrödinger operators is unstable in the semi-classical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semi-classical modes of the operator by the JWKB method for energies far from the spectrum.

. This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type \Psi 1;fl with 2 IR and 0 fl 1 in the weighted function spaces B s p;q (IR n ; w(x)) and F s p;q (IR n ; w(x)) treated in [17]. Furthermore we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2 (x)b(x; D)b1(x), where b1 (x) and b2(x) are appropriate functions and b(x; D) 2 \Psi 1;fl . Finally, on the basis of the Birman-Schwinger principle, we deal with the "negative spectrum" (bound states) of related symmetric operators in L2 . Math. Subject Classification: 46E35, 47G30, 35S05 1. Introduction The spaces B s p;q and F s p;q with s 2 IR; 0 ! p 1 (p ! 1 for the F - spaces) and 0 ! q 1 on IR n cover many well-known classical spaces such as (fractional) Sobolev spaces, H older-Zygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. In [1...

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In this paper we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and NPIV models under two basic regularity conditions: the approximation number and the link condition. We show that both a simple projection estimator for the NPIR model and a sieve minimum distance estimator for the NPIV model can achieve the minimax risk lower bounds and are rate optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases. 1.

This paper gives theoretical results on spinodal decomposition for the CahnHillard equation. We prove a mechanism which explains why most solutions for the Cahn-Hilliard equation which start near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R. Namely, most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected. The Cahn-Hilliard equation depends on a small parameter ", modeling the (atomic scale) interaction length; we quantify the behavior of solutions as " ! 0. Specifically, we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance O(" 2\Gamman=2 ) up to a ball of radius R, where R is proportional to " \Gamma1+%+n=4 as " ! 0. Here, n 2 f1; 2; 3g denotes the dimension of the considered domain, and % ? 0 can be chosen ...

this paper to seal the gap by providing intrinsic atomic characterizations for all these spaces under very mild and natural restrictions on the (non-- smooth) domain\Omega\Gamma A second problem connected with the approach (i) is the question whether there exists a (linear bounded) extension operator from the spaces on\Omega into the corresponding spaces on R

We consider a magnetic Schrödinger operator H in R n or on a Riemannian manifold M of bounded geometry. Sufficient conditions for the spectrum of H to be discrete are given in terms of behavior at infinity for some effective potentials Veff which are expressed through electric and magnetic fields. These conditions can be formulated in the form Veff (x) → + ∞ as x → ∞. They generalize the classical result by K.Friedrichs (1934), and include earlier results of J. Avron, I. Herbst and B. Simon (1978), A. Dufresnoy (1983) and A. Iwatsuka (1990) which were obtained in the absence of an electric field. More precise sufficient conditions can be formulated in terms of the Wiener capacity and extend earlier work by A.M. Molchanov (1953) and V. Kondrat’ev and M. Shubin (1999) who considered the case of the operator without a magnetic field. These conditions become necessary and sufficient in case there is no magnetic field and the electric potential is semi-bounded below.

. Let B be a generator of positivity preserving (shortly, positive) semigroup in L 2 on a space with oe-finite measure. It is supposed that the semigroup e \GammatB ; t ? 0 acts continuously from L 2 to L1 . For a measurable function V 0 an estimate for the number of negative eigenvalues of the operator B \Gamma V is obtained. The well-known CLR-estimate for the number of negative eigenvalues of the Schrodinger operator on R d ; d 3 is a particular case of the general result, and the best known constant here is reproduced. A more general theorem replaces the positivity property of the semigroup by domination by a positive semigroup. 1. Introduction 1.1. The CLR inequality, in its original form, reads (1.1) N \Gamma (\Gamma\Delta \Gamma V ) C(d) Z R d V d=2 dx; d 3: Here \Delta is the Laplacian on R d and V 0 is a measurable function (potential). By N \Gamma we denote the number of negative eigenvalues of a selfadjoint operator, provided its negative spectrum ...

The paper deals with weighted function spaces of type B s p;q (R n ; w(x)) and F s p;q (R n ; w(x)), where w(x) is a weight function of at most polynomial growth. Of special interest are weight functions of type w(x) = (1 + jxj 2 ) ff=2 (log(2 + jxj)) with ff 0 and 2 R. Our main result deals with estimates for the entropy numbers of compact embeddings between spaces of this type; more precisely, we may extend and tighten some of our previous results in [12]. AMS Subject Classification: 46E 35 Key Words: weighted function spaces, compact embeddings, entropy numbers Introduction 1 Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Weighted embeddings -- the non-limiting case 3 2 Limiting embeddings, entropy numbers 7 2.1 Estimates from above, an approach via duality arguments . . . . . . . . . . . . . . 8 2.2 Estimates from above, an approach via approximation numbers . . . . . . . . . . . 15 2.3 Estimates...

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