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Spectral averaging and the Krein spectral shift
, 1996
"... We provide a new proof of a theorem of Birman and Solomyak that if A(s) =A0+ sB with B ≥ 0 trace class and dµs(·) =Tr(B1/2EA(s)(·)B1/2), then ∫ 1 0 [dµs(λ)] ds = ξ(λ) dλ where ξ is the Krein spectral shift from A(0) to A(1). Our main point is that this is a simple consequence of the formula: d ds ..."
Abstract

Cited by 25 (0 self)
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We provide a new proof of a theorem of Birman and Solomyak that if A(s) =A0+ sB with B ≥ 0 trace class and dµs(·) =Tr(B1/2EA(s)(·)B1/2), then ∫ 1 0 [dµs(λ)] ds = ξ(λ) dλ where ξ is the Krein spectral shift from A(0) to A(1). Our main point is that this is a simple consequence of the formula: d ds Tr(f(A(s)) = Tr(Bf ′ (A(s))).
WEAK ORDER FOR THE DISCRETIZATION OF THE STOCHASTIC HEAT EQUATION
, 2008
"... Abstract. In this paper we study the approximation of the distribution of Xt Hilbert–valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dXt + AXt dt = Q 1/2 dW(t), X0 = x ∈ H, t ∈ [0,T], driven by a Gaussian space time noi ..."
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Cited by 12 (2 self)
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Abstract. In this paper we study the approximation of the distribution of Xt Hilbert–valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dXt + AXt dt = Q 1/2 dW(t), X0 = x ∈ H, t ∈ [0,T], driven by a Gaussian space time noise whose covariance operator Q is given. We assume that A−α is a finite trace operator for some α>0andthatQis bounded from H into D(Aβ)forsomeβ≥0. It is not required to be nuclear or to commute with A. The discretization is achieved thanks to finite element methods in space (parameter h>0) and a θmethod in time (parameter ∆t = T/N). We define a discrete solution Xn h and for suitable functions ϕ defined on H, we show that E ϕ(X N h) − E ϕ(XT)  = O(h 2γ +∆t γ) where γ<1 − α + β. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations. 1.
Perturbation determinants, the spectral shift function, trace identities, and all that, Funct
 Anal. Appl
, 2007
"... Abstract. We discuss applications of the M. G. Kreĭn theory of the spectral shift function to the multidimensional Schrödinger operator as well as specific properties of this function, for example, its highenergy asymptotics. Trace identities for the Schrödinger operator are derived. 1. ..."
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Cited by 3 (0 self)
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Abstract. We discuss applications of the M. G. Kreĭn theory of the spectral shift function to the multidimensional Schrödinger operator as well as specific properties of this function, for example, its highenergy asymptotics. Trace identities for the Schrödinger operator are derived. 1.
Resonances of the cusp family
, 2001
"... We study a family of chaotic maps with limit cases the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius–Perron operator in different function spaces including spaces of analytic functions. A numerical study of the eigenvalues and eigenfun ..."
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Cited by 1 (1 self)
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We study a family of chaotic maps with limit cases the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius–Perron operator in different function spaces including spaces of analytic functions. A numerical study of the eigenvalues and eigenfunctions is performed. 1
VAST MULTIPLICITY OF VERY SINGULAR SELFSIMILAR SOLUTIONS OF A SEMILINEAR HIGHERORDER DIFFUSION EQUATION WITH TIMEDEPENDENT ABSORPTION
, 901
"... Abstract. As a basic model, the Cauchy problem in R N × R+ for the 2mthorder semilinear parabolic equation of the diffusionabsorption type ut = −(−∆) m u − t α u  p−1 u, with p> 1, α> 0, m ≥ 2, with singular initial data u0 ̸ = 0 such that u0(x) = 0 for any x ̸ = 0 is studied. The additional mu ..."
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Abstract. As a basic model, the Cauchy problem in R N × R+ for the 2mthorder semilinear parabolic equation of the diffusionabsorption type ut = −(−∆) m u − t α u  p−1 u, with p> 1, α> 0, m ≥ 2, with singular initial data u0 ̸ = 0 such that u0(x) = 0 for any x ̸ = 0 is studied. The additional multiplier h(t) = t α → 0 as t → 0 in the absorption term plays a role of timedependent nonhomogeneous potential that affects the strength of the absorption term in the PDE. Existence and nonexistence of the corresponding very singular solutions (VSSs) is studied. For m = 1 and h(t) ≡ 1, first nonexistence result for p ≥ p0 = 1 + 2 N was proved in the celebrated paper by Brezis and Friedman in 1983. Existence of VSSs in the complement interval 1 < p < p0 was established in the middle of the 1980s. The main goal is to justify that, in the subcritical range 1 < p < p0 = 1 + 2m(1+α) N there exists a finite number of different VSSs of the selfsimilar form u∗(x, t) = t −β V (y), y = x/t 1 2m, β = 1+α p−1, where each V is an exponentially decaying as y → ∞ solution of the elliptic equation −(−∆) m V + 1 2m y · ∇V + βV − V p−1 V = 0 in R N. Complicated families of VSSs in 1D and also nonradial VSS patterns in R N are detected. Some of these VSS profiles Vl are shown to bifurcate from 0 at the bifurcation exponents pl = 1 + 2m(1+α) l+N, where l = 0, 1, 2,.... 1. Introduction: VSSs