Spectral Shift Function in the Large Coupling Constant Limit

Safronov, Oleg

doi:10.1006/jfan.2000.3720

Cited by 17 publications

(14 citation statements)

References 9 publications

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“…Also they allow one to study the strong coupling limit. In particular, in Section 3 we will prove For other results related to the strong coupling limit we refer to [20,21,23]. Most of the results of the present note have appeared previously in [13] in a slightly less general form.…”

Section: Introduction and Main Resultsmentioning

confidence: 81%

Concavity of Eigenvalue Sums and the Spectral Shift Function

Kostrykin

2000

Journal of Functional Analysis

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It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix $V$ is concave (convex) with respect to $V$. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from $-\infty$ to $\lambda$ (from $\lambda$ to $+\infty$) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered

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Section: Introduction and Main Resultsmentioning

confidence: 81%

Concavity of Eigenvalue Sums and the Spectral Shift Function

Kostrykin

2000

Journal of Functional Analysis

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“…Remark 2. The signs ascribed to ITEs in Theorem 1.1 resemble the standard procedure used in the definition of spectral flows (see, for example, [1], [30], [36], [37]). The principal difference is that these signs in the present paper are defined not by the direction of motion of the eigenvalues of the operator under consideration, but by the direction of motion of another object, namely, the eigenvalues of the scattering matrix.…”

Section: Resultsmentioning

confidence: 98%

Sharp Weyl Law for Signed Counting Function of Positive Interior Transmission Eigenvalues

Lakshtanov

Vainberg²

2015

SIAM J. Math. Anal.

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We consider the interior transmission eigenvalue (ITE) problem that arises when scattering by inhomogeneous media is studied. The ITE problem is not self-adjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (as they approach z = 1). We obtain a Weyl-type formula for the counting function of positive ITEs, which are taken together with the ascribed signs. The results are applicable to the case when the medium contains an unpenetrable obstacle. ∂u ∂ν − ∂v ∂ν = 0, x ∈ ∂O,

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“…An alternate approach to proving that a(λ) ≡ 1 implies V ≡ 0, is through recent work on the large-λ asymptotics of the spectral shift function arg[a(λ)]; see [9,11], for example. We would like to thank A. Pushnitski for explaining some of this material to us.…”

Section: Proofsmentioning

confidence: 99%

Absence of reflection as a function of the coupling constant

Killip

Sims

2006

Journal of Mathematical Physics

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Abstract. We consider solutions of the one-dimensional equation −u ′′ +(Q+ λV )u = 0 where Q : R → R is locally integrable, V : R → R is integrable with supp(V ) ⊂ [0, 1], and λ ∈ R is a coupling constant. Given a family of solutions {u λ } λ∈R which satisfy u λ (x) = u 0 (x) for all x < 0, we prove that the zeros of b(λ) := W [u 0 , u λ ], the Wronskian of u 0 and u λ , form a discrete set unless V ≡ 0. Setting Q(x) := −E, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment −u ′′ + λV u = Eu gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then V ≡ 0.

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Spectral Shift Function in the Large Coupling Constant Limit

Cited by 17 publications

References 9 publications

Concavity of Eigenvalue Sums and the Spectral Shift Function

Concavity of Eigenvalue Sums and the Spectral Shift Function

Sharp Weyl Law for Signed Counting Function of Positive Interior Transmission Eigenvalues

Absence of reflection as a function of the coupling constant

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