Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

Gesztesy, Fritz; Weikard, Rudi; Zinchenko, Maxim

doi:10.7153/oam-07-15

Cited by 21 publications

(36 citation statements)

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“…Our proofs are essentially based on a translation of the electromagnetic assumptions of passive linear media and transparency window into the distributional approach of Zemanian on the theory of passive linear systems [16,Chap. 8] and connecting all of this to the theory of Herglotz functions (see, for instance, [17,19,20] and [37,Appendix A], and references within).…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

“…In fact, a necessary and sufficient condition for h(ω) = ω χ (ω) to be a Herglotz function is that dχ dt * is a passive convolution operator, i.e, that the passivity condition (5) holds. Herglotz functions have been heavily studied (see, for instance, [17,19,20,37], and references within) and their properties, along with a transparency window, are the key to the positivity of the energy density and the speed-of-light limitation.…”

Section: B Passivitymentioning

confidence: 99%

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Speed-of-light limitations in passive linear media

2014

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We prove that well-known speed-of-light restrictions on electromagnetic energy velocity can be extended to a new level of generality, encompassing even nonlocal chiral media in periodic geometries, while at the same time weakening the underlying assumptions to only passivity and linearity of the medium (either with a transparency window or with dissipation). As was also shown by other authors under more limiting assumptions, passivity alone is sufficient to guarantee causality and positivity of the energy density (with no thermodynamic assumptions). Our proof is general enough to include a very broad range of material properties, including anisotropy, bianisotropy (chirality), nonlocality, dispersion, periodicity, and even delta functions or similar generalized functions. We also show that the "dynamical energy density" used by some previous authors in dissipative media reduces to the standard Brillouin formula for dispersive energy density in a transparency window. The results in this paper are proved by exploiting deep results from linear-response theory, harmonic analysis, and functional analysis that had previously not been brought together in the context of electrodynamics.

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Section: Proofs Of Main Resultsmentioning

confidence: 99%

Section: B Passivitymentioning

confidence: 99%

Speed-of-light limitations in passive linear media

2014

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“…The inner product in L 2 ((a, b); H), in obvious notation, then reads (f, g) L 2 ((a,b);H) =ˆb a (f (x), g(x)) H dx, f, g ∈ L 2 ((a, b); H). For applications of these concepts to Schrödinger operators with operator-valued potentials we refer to [17]; applications to scattering theory for multi-dimensional Schrödinger operators are studied in great detail in [28, Chs. IV, V].…”

Section: The Vector-valued Casementioning

confidence: 99%

“…17) suggests equality in(7.8) holds only for functions of the formg 0 (x) = d(x − a) µ (7.18)for some d ∈ C, µ ∈ R. To prove d = 0, we will argue as follows. First, one notes that g (n) 0 ∈ L 2 ((a, c)) implies 2(µ − n) > −1 or, µ > n − 1/2.…”

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confidence: 99%

On Birman's sequence of Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

Michael

et al. 2018

Journal of Differential Equations

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Dedicated with great pleasure to Eduard Tsekanovskii on the occasion of his 80th birthday.Abstract. In 1961, Birman proved a sequence of inequalities {In}, for n ∈ N, valid for functions in C n 0 ((0, ∞)) ⊂ L 2 ((0, ∞)). In particular, I 1 is the classical (integral) Hardy inequality and I 2 is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite b > 0, these inequalities hold on the standard Sobolev space H n 0 ((0, b)). Furthermore, in all cases, the Birman constants [(2n − 1)!!] 2 /2 2n in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in L 2 ((0, ∞)) (resp., L 2 ((0, b))). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail.

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“…We will return to this circle of ideas elsewhere. ⋄ At this point we turn to the content of each section: Section 2 recalls our basic results in [50] on the initial value problem associated with Schrödinger operators with bounded operator-valued potentials. We use this section to introduce some of the basic notation employed subsequently and note that our conditions on V (·) (cf.…”

Section: Introductionmentioning

confidence: 99%

Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

Gesztesy

Naboko

Weikard

et al. 2019

JAMA

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Given a complex, separable Hilbert space H, we consider differential expressions of the type τ = −(d 2 /dx 2 )I H + V (x), with x ∈ (x 0 , ∞) for some x 0 ∈ R, or x ∈ R (assuming the limit-point property of τ at ±∞). Here V denotes a bounded operator-valued potential V (·) ∈ B(H) such that V (·) is weakly measurable, the operator norm V (·) B(H) is locally integrable, andWe focus on two major cases. First, on m-function theory for self-adjoint half-line L 2 -realizations H +,α in L 2 ((x 0 , ∞); dx; H) (with x 0 a regular endpoint for τ , associated with the selfadjoint boundary condition sin(α)u ′ (x 0 ) + cos(α)u(x 0 ) = 0, indexed by the self-adjoint operator α = α * ∈ B(H)), and second, on m-function theory for self-adjoint full-line L 2 -realizations H of τ in L 2 (R; dx; H).In a nutshell, a Donoghue-type m-function M Do A,N i (·) associated with selfadjoint extensions A of a closed, symmetric operatorȦ in H with deficiency spaces Nz = ker Ȧ * − zI H and corresponding orthogonal projections P Nz onto Nz is given byIn the concrete case of half-line and full-line Schrödinger operators, the role ofȦ is played by a suitably defined minimal Schrödinger operator H +,min in L 2 ((x 0 , ∞); dx; H) and H min in L 2 (R; dx; H), both of which will be proven to be completely non-self-adjoint. The latter property is used to prove that if H +,α in L 2 ((x 0 , ∞); dx; H), respectively, H in L 2 (R; dx; H), are self-adjoint extensions of H +,min , respectively, H min , then the corresponding operator-valued measures in the Herglotz-Nevanlinna representations of the Donoghue-type mfunctions M Do H +,α ,N +,i (·) and M Do H,N i (·) encode the entire spectral information of H +,α , respectively, H.Next, we briefly turn to Donoghue-type m-functions which abstractly can be introduced as follows (cf. [47], [48]). Given a self-adjoint extension A of a densely defined, closed, symmetric operatorȦ in K (a complex, separable Hilbert space) and the deficiency subspace N i ofȦ in K, withthe Donoghue-type m-operator M Do A,Ni (z) ∈ B(N i ) associated with the pair (A, N i ) is given bywith I Ni the identity operator in N i , and P Ni the orthogonal projection in K onto N i . Then M Do A,Ni (·) is a B(N i )-valued Nevanlinna-Herglotz function that admits the representation M DoA,Ni (z) =ˆR dΩ Do A,Ni (λ)

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Initial value problems and Weyl-Titchmarsh theory for Schrödinger operators with operator-valued potentials

Cited by 21 publications

References 79 publications

Speed-of-light limitations in passive linear media

Speed-of-light limitations in passive linear media

On Birman's sequence of Hardy–Rellich-type inequalities

Donoghue-type m-functions for Schrödinger operators with operator-valued potentials

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