Szego limit theorem for operators with discontinuous symbols and applications to entanglement entropy

Abstract. We use the singular value decomposition of the array response matrix, frequency by frequency, to image selectively the edges of extended reflectors in a homogeneous medium. We show with numerical simulations in an ultrasound regime, and analytically in the Fraunhofer diffraction regime, that information about the edges is contained in the singular vectors for singular values that are intermediate between the large ones and zero. These transition singular vectors beamform selectively from the array onto the edges of the reflector cross-section facing the array, so that these edges are enhanced in imaging with travel-time migration. Moreover, the illumination with the transition singular vectors is done from the sources at the edges of the array. The theoretical analysis in the Fraunhofer regime shows that the singular values transition to zero at the index N (ω) = |A||B|/(λL) 2 . Here |A| and |B| are the areas of the array and the reflector cross-section, respectively, ω is the frequency, λ is the wavelength, and L is the range. Since (λL) 2 /|A| is the area of the focal spot size at range L, we see that N (ω) is the number of focal spots contained in the reflector cross-section. The ultrasound simulations are in an extended Fraunhofer regime. The simulation results are, however, qualitatively similar to those obtained theoretically in the Fraunhofer regime. The numerical simulations indicate, in addition, that the subspaces spanned by the transition singular vectors are robust with respect to additive noise when the array has a large number of elements.

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“…Gioev [19] has recently confirmed that the second term in the asymptotics of tr f (T r ) is of the order predicted by…”

Section: Second Order Asymptotics Of the Eigenvalue Distribution Formentioning

confidence: 73%

Edge Illumination and Imaging of Extended Reflectors

Borcea¹,

Papanicolaou²,

Vasquez³

2008

SIAM J. Imaging Sci.

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“…Once this theorem is proved, the asymptotics can be closed with the help of the sharp bounds (12.11) and (12.17), which were derived in [11], [12] using the abstract version of the Szegő formula with a remainder estimate obtained in [21] (see also [22]).…”

Section: Resultsmentioning

confidence: 99%

Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom’s conjecture

Sobolev

2010

Funct Anal Its Appl

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In 1982 H. Widom conjectured a multi-dimensional generalization of a well-known twoterm quasi-classical asymptotic formula for the trace of the function f (A) of a Wiener-Hopf-type operator A in dimension 1 for a pseudodifferential operator A with symbol a(x, ξ) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs in a hyperplane.This note announces a proof of Widom's conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces. For a (complex-valued1, and an arbitrary Schwartz function u, consider the pseudodifferential operator, which depends on a large parameter α > 0. Let Λ and Ω be two domains in R d , and let χ Λ (x) and χ Ω (ξ) be their characteristic functions. We use the notation P Ω,α = Op α (χ Ω ). In this note we study the operator T α (a) = T α (a; Λ, Ω) = χ Λ P Ω,α Op α (a)P Ω,α χ Λand its symmetrized variant S α (a) = S α (a; Λ, Ω) = χ Λ P Ω,α Re Op α (a)P Ω,α χ Λ .We are interested in the two-term asymptotics of the traces tr g(T α ) and tr g(S α ) as α → ∞, where g is an appropriate smooth function such that g(0) = 0. We introduce the asymptotic coefficientswhere n ∂Λ (x) and n ∂Ω (ξ) denote the outer unit normals to the boundaries ∂Λ and ∂Ω at the points x and ξ, respectively, andThe coefficient A(g; b) is well defined for any complex (real) number b if g is a C 1 -function of a complex (respectively, real) variable. The objective of this paper is to present the two-term asymptotic formulas given in the following theorem.

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“…Before discussing behaviour of the variance of the number of particles in a disordered metal, we remind the reader its behaviour in the absence of disorder. As is well known, for k F L 1 the particle-number variance becomes [4,19,20]:…”

Section: Formalismmentioning

confidence: 89%

Entanglement entropy and particle number cumulants of disordered fermions

et al. 2017

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We study the entanglement entropy and particle number cumulants for a system of disordered noninteracting fermions in d dimensions. We show, both analytically and numerically, that for a weak disorder the entanglement entropy and the second cumulant (particle number variance) are proportional to each other with a universal coefficient. The corresponding expressions are analogous to those in the clean case but with a logarithmic factor regularized by the mean free path rather than by the system size. We also determine the scaling of higher cumulants by analytical (weak disorder) and numerical means. Finally, we predict that the particle number variance and the entanglement entropy are nonanalytic functions of disorder at the Anderson transition.

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Szego limit theorem for operators with discontinuous symbols and applications to entanglement entropy

Cited by 16 publications

References 42 publications

Edge Illumination and Imaging of Extended Reflectors

Edge Illumination and Imaging of Extended Reflectors

Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom’s conjecture

Entanglement entropy and particle number cumulants of disordered fermions

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