Box approximation and related techniques in spectral theory

Borovyk, Vita

doi:10.32469/10355/5566

Cited by 2 publications

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“…In the one-dimensional half-line context, Borovyk and Makarov [14] (see also Borovyk [13]) proved in 2009 that for potentials V ∈ L 1 ((0, ∞); (1 + |x|)dx) realvalued, and denoting by H R the self-adjoint Schrödinger operator in L 2 ((0, R); dx) and H the corresponding self-adjoint Schrödinger operator in L 2 ((0, ∞); dx), both with Dirichlet boundary conditions (and otherwise maximally defined or defined in terms of quadratic forms), and analogously for H 0,R and H 0 in the unperturbed case V = 0, the following vague limit holds:…”

Section: Introductionmentioning

confidence: 99%

An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators

Gesztesy¹,

Nichols²

2012

J. Spectr. Theory

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Abstract. We study the manner in which a sequence of spectral shift functions ξ(·; H j , H 0,j ) associated with abstract pairs of self-adjoint operators (H j , H 0,j ) in Hilbert spaces H j , j ∈ N, converge to a limiting spectral shift function ξ(·; H, H 0 ) associated with a pair (H, H 0 ) in the limiting Hilbert space H as j → ∞ (mimicking the infinite volume limit in concrete applications to multi-dimensional Schrödinger operators).Our techniques rely on a Fredholm determinant approach combined with certain measure theoretic facts. In particular, we show that prior vague convergence results for spectral shift functions in the literature actually extend to the notion of weak convergence. More precisely, in the concrete case of multidimensional Schrödinger operators on a sequence of domains Ω j exhausting R n as j → ∞, we extend the convergence of associated spectral shift functions from vague to weak convergence and also from Dirichlet boundary conditions to more general self-adjoint boundary conditions on ∂Ω j .

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Section: Introductionmentioning

confidence: 99%

An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators

Gesztesy¹,

Nichols²

2012

J. Spectr. Theory

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“…In the one-dimensional half-line context, Borovyk and Makarov [8] (see also Borovyk [7]) proved in 2009 that for potentials V ∈ L 1 ((0, ∞); (1 + |x|)dx) realvalued, and denoting by H R the self-adjoint Schrödinger operator in L 2 ((0, R); dx) and H the corresponding self-adjoint Schrödinger operator in L 2 ((0, ∞); dx), both with Dirichlet boundary conditions (and otherwise maximally defined or defined in terms of quadratic forms), and analogously for H 0,R and H 0 in the unperturbed case V = 0, the following vague limit holds: For any g ∈ C 0 (R), lim R→∞ R ξ λ; H R , H 0,R dλ g(λ) = R ξ λ; H, H 0 dλ g(λ).…”

Section: Introductionmentioning

confidence: 99%

Weak convergence of spectral shift functions for one‐dimensional Schrödinger operators

Gesztesy

Nichols

2012

Mathematische Nachrichten

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We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on the finite interval (0, R) converge in the infinite volume limit R → ∞ to the half-line spectral shift function.Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at x = 0 and x = R.In Section 2 we introduce basic facts on one-dimensional Schrödinger operators, describe associated Green's functions, Krein-type resolvent formulas, compute the integral kernel for the square root of the resolvent of the Dirichlet Laplacian on a finite interval and on a half-line. In addition, we derive Jost-Pais-type [30] (cf. also [18] and the extensive literature therein) reductions of Fredholm determinants corresponding to Birman-Schwinger-type kernels to Wronski determinants of appropriate solutions of the associated Schrödinger equation. Basic convergence results of resolvents and closely related operators (including Birman-Schwinger-type kernels) are discussed in Section 3. Finally, our principal Section 4 provides a new approach to vague, and especially, weak convergence of spectral shift functions in the infinite volume limit based on the convergence of underlying Fredholm determinants combined with certain measure theoretic facts. We refer to Theorem 4.12 for the main result of this paper. The latter considerably extends (1.3) in a variety of ways: First, g(·) in (1.3) can now be replaced by f (·)/(1 + λ 2 ) with f a bounded continuous function (this can further be improved). Second, we can permit any separated self-adjoint boundary conditions (not just Dirichlet boundary conditions) at x = 0 and x = R. Third, we can permit real-valued (singular) potentials V satisfying V ∈ L 1 ((0, ∞); dx) (in the case of Dirichlet boundary conditions at x = 0 one can even permit V ∈ L 1 ((0, ∞); [x/(1 + x)]dx), cf. Remark 2.8). Appendix A collects basic properties of spectral shift functions used in Section 4, and Appendix B derives a particular decomposition formula relating spectral shift functions for the interval (0, R 1 ) and (0, R 2 ), 0 < R 1 < R 2 , associated with Dirichlet boundary conditions at x = 0, R 1 , R 2 .Finally, we briefly summarize some of the notation used in this paper: Let H be a separable complex Hilbert space, (·, ·) H the scalar product in H (linear in the second argument), and I H the identity operator in H. Next, let T be a linear operator mapping (a subspace of) a Hilbert space into another, with dom(T ) and ker(T ) denoting the domain and kernel (i.e., null space) of T . The resolvent set of a closed linear operator in H will be denoted by ρ(·). The Banach spaces of bounded (resp., compact) linear operators on H is denoted by B(H) (resp., B ∞ (H)). The corresponding p -based trace ideals will be denoted by B p (H), p > 0.The form sum (res...

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Box approximation and related techniques in spectral theory

Abstract: This dissertation is concerned with various aspects of the spectral theory of differential and pseudodifferential operators. It consists of two chapters.

Cited by 2 publications

References 0 publications

An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators

An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators

Weak convergence of spectral shift functions for one‐dimensional Schrödinger operators

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