Spin-dependent point potentials in one and three dimensions

Cacciapuoti, Claudio; Carlone, Raffaele; Figari, Rodolfo

doi:10.1088/1751-8113/40/2/004

Cited by 22 publications

(31 citation statements)

References 24 publications

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“…By the unitary isomorphism L 2 (R 3 ) ⊗ f ≃ L 2 (R 3 ; f) given by ψ ⊗ ζ → ψζ, which transforms ∆ ⊗ 1 + 1 ⊗ B into A defined in (5.10), and by taking f = ⊗ n k=1 C 2 ≃ C 2 n , this example reproduces (for a particular choice of B) the self-adjoint extensions given in [6] describing systems made of a spin-less quantum particle and an array of n spin 1/2 (there the parametrisation is given by a couple of n2 n ×n2 n matrices satisfying (4.1) and (4.2)).…”

Section: Example 53 (The Laplacian With N Point Interactions) Letmentioning

confidence: 70%

Self-adjoint extensions of restrictions

Posilicano¹

2008

Operators and Matrices

116

192

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Abstract. We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the self-adjoint operator A : D (A) ⊆ H → H to the dense, closed with respect to the graph norm, subspace N ⊂ D (A) . Neither the knowledge of S * nor of the deficiency spaces of S is required. Typically A is a differential operator and N is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle p : E(h) → P(h) , where P(h) denotes the set of orthogonal projections in the Hilbert space h D (A)/N and p −1 (Π) is the set of self-adjoint operators in the range of Π . The set of self-adjoint operators in h , i.e. p −1 (1) , parametrises the relatively prime extensions. Any (Π, Θ) ∈ E(h) determines a boundary condition in the domain of the corresponding extension A Π,Θ and explicitly appears in the formula for the resolvent (−A Π,Θ + z) −1 . The connection with both von Neumann's and Boundary Triples theories of self-adjoint extensions is explained. Some examples related to quantum graphs, to Schrödinger operators with point interactions and to elliptic boundary value problems are given.Mathematics subject classification (2000): 47B25, 47B38, 35J25.

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Section: Example 53 (The Laplacian With N Point Interactions) Letmentioning

confidence: 70%

Self-adjoint extensions of restrictions

Posilicano¹

2008

Operators and Matrices

116

192

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“…For d = 2, 3, theorem 1 gives all possible self-adjoint extensions of S. For d = 1, Hamiltonians H AB include only extensions of S for which the wave function part of vectors in D(H AB ) is continuous in the origin. To cover the whole family of self-adjoint extensions of S, and include Hamiltonians defined on vectors with discontinuous wave function, we should have taken into account the whole bases of the defect spaces of S. For the analysis of the general case one can refer to [10]. Remark 2.…”

Section: Basic Notation and Preliminary Resultsmentioning

confidence: 99%

Resonances in models of spin-dependent point interactions

Cacciapuoti¹,

Carlone²,

Figari³

2008

J. Phys. A: Math. Theor.

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In dimension d = 1, 2, 3 we define a family of two-channel Hamiltonians obtained as point perturbations of the generator of the free decoupled dynamics. Within the family we choose two Hamiltonians,Ĥ 0 andĤ ε , giving rise respectively to the unperturbed and to the perturbed evolution. The HamiltonianĤ 0 does not couple the channels and has an eigenvalue embedded in the continuous spectrum. The HamiltonianĤ ε is a small perturbation, in resolvent sense, ofĤ 0 and exhibits a small coupling between the channels.We take advantage of the complete solvability of our model to prove with simple arguments that the embedded eigenvalue ofĤ 0 shifts into a resonance forĤ ε . In dimension three we analyze details of the time behavior of the projection onto the region of the spectrum close to the resonance.

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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%

“…In particular, our purpose is to provide a model of a finite volume quantum measurement apparatus whose interaction with the system under measurement is described by a delta-shaped quantum potential. An analogous problem in R 1 and R 3 have been investigated in [4].…”

Section: Introductionmentioning

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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Spin-dependent point potentials in one and three dimensions

Cited by 22 publications

References 24 publications

Self-adjoint extensions of restrictions

Self-adjoint extensions of restrictions

Resonances in models of spin-dependent point interactions

Point interaction Hamiltonians in bounded domains

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