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97
B: Semiclassical states for nonselfadjoint Schrödinger operators
 Commun. Math. Phys
, 1999
"... We prove that the spectrum of certain nonselfadjoint Schrödinger operators is unstable in the semiclassical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semiclassical modes of the operator by the JWKB method ..."
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Cited by 36 (10 self)
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We prove that the spectrum of certain nonselfadjoint Schrödinger operators is unstable in the semiclassical limit h → 0. Similar results hold for a fixed operator in the high energy limit. The method involves the construction of approximate semiclassical modes of the operator by the JWKB method for energies far from the spectrum.
Unexpectedly linear behavior for the CahnHilliard equation
 SIAM Journal on Applied Mathematics
"... Abstract. This paper gives theoretical results on spinodal decomposition for the Cahn–Hillard equation. We prove a mechanism which explains why most solutions for the Cahn–Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a character ..."
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Cited by 16 (7 self)
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Abstract. This paper gives theoretical results on spinodal decomposition for the Cahn–Hillard equation. We prove a mechanism which explains why most solutions for the Cahn–Hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball ofradius R. Namely, most solutions are driven into a region ofphase space in which linear behavior dominates for much longer than expected. The Cahn–Hilliard equation depends on a small parameter ε, modeling the (atomic scale) interaction length; we quantify the behavior of solutions as ε → 0. Specifically, we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution ofthe linearized equation with relative distance O(ε 2−n/2) up to a ball ofradius R in the H 2 (Ω)norm, where R is proportional to ε −1+ϱ+n/4 as ε → 0. Here, n ∈{1, 2, 3} denotes the dimension ofthe considered domain, and ϱ>0 can be chosen arbitrarily small. Not only does this approach significantly increase the radius ofexplanation for spinodal decomposition, but it also gives a clear picture ofhow the phenomenon occurs. While these results hold for the standard cubic nonlinearity, we also show that considerably better results can be obtained for similar higher order nonlinearities. In particular, we obtain R ∼ ε −2+ϱ+n/2 for every ϱ>0 by choosing a suitable nonlinearity. Key words. Cahn–Hilliard equation, spinodal decomposition, phase separation, pattern formation, linear behavior
Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators II
, 1994
"... 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 distribu ..."
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Cited by 14 (7 self)
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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 BirmanSchwinger principle, we deal with the "negative spectrum" (bound states) of related symmetric operators in L2 .
On Rate Optimality for IllPosed Inverse Problems in Econometrics
, 2007
"... 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 ..."
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Cited by 13 (1 self)
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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 the NPIV models under two basic regularity conditions that allow for both mildly illposed and severely illposed cases. 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 rateoptimal uniformly over a large class of structure functions, allowing for mildly illposed and severely illposed cases. KEY WORDS: Nonparametric instrumental regression; Nonparametric indirect regression; Statistical illposed inverse problems; Minimax risk lower bound; Optimal rate.
Kreins resolvent formula for selfadjoint extensions of symmetric secondorder elliptic differential operators
 J. Phys. A: Math. Theor
, 2009
"... Abstract. Given a symmetric, semibounded, second order elliptic differential operator A on a bounded domain with C 1,1 boundary, we provide a Kreĭntype formula for the resolvent difference between its Friedrichs extension and an arbitrary selfadjoint one. 1. Introduction. Given a bounded open set ..."
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Cited by 12 (1 self)
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Abstract. Given a symmetric, semibounded, second order elliptic differential operator A on a bounded domain with C 1,1 boundary, we provide a Kreĭntype formula for the resolvent difference between its Friedrichs extension and an arbitrary selfadjoint one. 1. Introduction. Given a bounded open set Ω ⊂ Rn, n> 1, let us consider a second order elliptic differential operator A: C ∞ c (Ω) ⊂ L 2 (Ω) → L 2 n∑ n∑
Discreteness of spectrum for the magnetic Schrödinger operators. I.
, 2001
"... 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. T ..."
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Cited by 12 (2 self)
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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
CLREstimate For The Generators Of Positivity Preserving And Positively Dominated Semigroups
, 1997
"... . Let B be a generator of positivity preserving (shortly, positive) semigroup in L 2 on a space with oefinite 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 th ..."
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Cited by 11 (2 self)
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. Let B be a generator of positivity preserving (shortly, positive) semigroup in L 2 on a space with oefinite 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 wellknown CLRestimate 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 ...
Spectral problems with mixed DirichletNeumann boundary conditions: isospectrality and
, 2006
"... Dépt. de mathématiques et de statistique ..."
Intrinsic Atomic Characterizations of Function Spaces on Domains
, 1996
"... 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 operato ..."
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Cited by 10 (2 self)
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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