A remark on Krein's resolvent formula and boundary conditions

Albeverio, Sergio; Pankrashkin, Konstantin

doi:10.1088/0305-4470/38/22/010

Cited by 38 publications

(57 citation statements)

References 14 publications

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“…3) which involves the gamma field γ and the Weyl function M corresponding to this change of boundary conditions [12]. The latter is a 4 × 4 matrix-valued Herglotz function z → M (z) of the spectral parameter z ∈ C + .…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Edge Switching Transformations of Quantum Graphs

et al. 2017

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Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by {En} ∞ n=1 and { En} ∞ n=1 correspondingly, are level-2 interlaced, so that En−2 ≤ En ≤ En+2. The proofs are guided by considerations of the quantum graphs' discrete analogs. [2,3], and they are defined and discussed at great length and detail in review articles and books (see e.g., [4][5][6][7]). In this note, we consider a general compact quantum graph, G = (V, E, L), whose edges e ∈ E are metrized and of finite lengths belonging to the list, may include external potential V and possibly also a magnetic potential A:The operator definition is incomplete without specifying also the boundary conditions (bc) at vertices. They are assumed here to be local, i.e., expressed in terms of linear relations on the limiting values of the function and of its derivatives along the deg(v) edges which meet at each vertex v ∈ V. The possible choices which ensure self-adjointness are reviewed in detail in, e.g., [5,6]. Under mild conditions on V and A the spectrum of the Schrödinger operator {E n } ∞ n=1 is discrete and bounded below, but not above. It suffices to assume V, A are integrable, but for a more transparent presentation we focus on the case these are piecewise continuous [6].This spectrum of H is unaffected by the operation of edge splitting, through the insertion of a vertex of degree 2 with Kirchhoff boundary conditions. These require Ψ to be continuous at v ∈ V having there directional derivatives satisfying:(To avoid confusion: these boundary conditions are not assumed here for the other quantum graph vertices.) Our purpose here is to discuss the effects on the spectrum of another basic graph transformation, that of edge switching. It is defined in the following extension of a notion which is used in combinatorial graph theory [8]. Definition 1.1 For a quantum graph G with a selfadjoint Schrödinger operator H, an edge switch is a transformation in which a pair of edges in E exchange the graph designations of one of their end points. Preserved under this exchange are the edge lengths, the local action of H along the corresponding edges, and the vertex boundary conditions -up to the corresponding transposition of the functions whose limiting behavior at the two vertices enter the local boundary conditions.Under an edge switch the collection of the edge lengths remains unchanged. However the spectra will in the generic case be affected. For example, for quantum graph Laplacians it is shown in [9] that the topology of the quantum graph with rationally independent edge length L "can be heard" in the sense of Marc Kac [10]. For clarity let us add that the graph topology may be affected by a switch, but it does not have to. However, regardless of that, and even in case the swi...

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

Edge Switching Transformations of Quantum Graphs

et al. 2017

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“…Moreover, in [6] (see also [21]) it was shown, in a very general setting, that a generalized Krein's formula for the resolvent exists. Such a formula explicitly gives the resolvent of a selfadjoint extension of a given symmetric operator in terms of the parameters characterizing the boundary conditions satisfied by the vectors in its domain.…”

Section: Point Perturbations Of Hmentioning

confidence: 99%

“…This proves that the operators H AB are self-adjoint. We use the proposition proved in [6] (see also Theorem 10 in [21]) to write down the resolvent of H AB . Define γ z : C m → K z in the following way: γ z = (Λ|K z ) −1 .…”

Section: Proof Define Two Linear Applications λ : D(hmentioning

confidence: 99%

“…Such a formula explicitly gives the resolvent of a selfadjoint extension of a given symmetric operator in terms of the parameters characterizing the boundary conditions satisfied by the vectors in its domain. Moreover the generalized formula for the resolvent given in [6] and [21] avoids the problem of finding the maximal common part of two extensions. In this paper we use the results of [6] and [21] …”

Section: Point Perturbations Of Hmentioning

confidence: 99%

See 1 more Smart Citation

Spin-dependent point potentials in one and three dimensions

Cacciapuoti¹,

Carlone²,

Figari³

2006

J. Phys. A: Math. Theor.

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We consider a system realized with one spinless quantum particle and an array of N spins 1/2 in dimension one and three. We characterize all the Hamiltonians obtained as point perturbations of an assigned free dynamics in terms of some generalized boundary conditions. For every boundary condition we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin dependent point potentials that could be of interest as multi-component solvable models.

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“…To this aim, and following the analogous definitions given for the case Ω = R 3 (see for instance [8]- [10], and [4]), we define…”

Section: Parametrization Of Selfadjoint Extensionsmentioning

confidence: 99%

Point interaction Hamiltonians in bounded domains

Blanchard

Figari

Mantile

2007

Journal of Mathematical Physics

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Making use of recent techniques in the theory of selfadjoint extensions of symmetric operators, we characterize the class of point interaction Hamiltonians in a 3-D bounded domain with regular boundary. In the particular case of one point interaction acting in the center of a ball, we obtain an explicit representation of the point spectrum of the operator togheter with the corresponding related eigenfunctions. These operators are used to build up a model-system where the dynamics of a quantum particle depends on the state of a quantum bit.

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A remark on Krein's resolvent formula and boundary conditions

Cited by 38 publications

References 14 publications

Edge Switching Transformations of Quantum Graphs

Edge Switching Transformations of Quantum Graphs

Spin-dependent point potentials in one and three dimensions

Point interaction Hamiltonians in bounded domains

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