Self-adjoint Extensions in Quantum Mechanics

Гитман, Д. М.; Tyutin, I. V.; Voronov, B. L.

doi:10.1007/978-0-8176-4662-2

Cited by 156 publications

(236 citation statements)

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“…It should be noted that a choice of a self-adjoint Hamiltonian requires additional physical arguments [5]. We emphasize that the radial Hamiltonian h contains a singular potential (M aδ(r)/r).…”

Section: Physical Regularization Proceduresmentioning

confidence: 99%

“…Such a potential affects the behavior of wave functions at the origin but it not grasped by an initial (symmetric) radial Hamiltonian, whose domain includes functions vanishing at the origin. Mathematically, such a potential is grasped by constructing self-adjoint extensions of Hamiltonian that are parameterized by asymptotic self-adjoint boundary conditions at the origin so, physically, the nonzero extension parameter can be treated as a manifestation of additional singular (∼ δ(r)/r) potentials [5]. For each ξ, we find a possible domain for a self-adjoint h ξ and different choices ξ lead to inequivalent physical cases (see, also, [4] and [15][16][17][18]).…”

Section: Physical Regularization Proceduresmentioning

confidence: 99%

“…To put it more exactly, a domain, including the singular r = 0 region, in the corresponding Hilbert space of square-integrable functions must be indicated for each self-adjoint Hamiltonian. Different choices of selfadjoint extension parameter values lead to inequivalent physical cases [4] so a physical interpretation of self-adjoint extensions is a purely physical problem and each extension can be understood through an appropriate physical regularization [5].…”

Section: Introductionmentioning

confidence: 99%

“…We note that self-adjoint Hamiltonians with the AB potential and ACF have been considered to show the presence of fermion bound states in [4][5][6][7][8][9][10]. Self-adjoint Schrödinger and Dirac Hamiltonians with singular potentials (such as the one-dimensional Calogero potential, the Coulomb potential, a superposition of the Aharonov-Bohm field and the so-called magnetic-solenoid field and the potentials localized at the origin, in particular, delta-like potentials) were considered in many works (see [5] and references there).…”

Section: Introductionmentioning

confidence: 99%

“…Self-adjoint Schrödinger and Dirac Hamiltonians with singular potentials (such as the one-dimensional Calogero potential, the Coulomb potential, a superposition of the Aharonov-Bohm field and the so-called magnetic-solenoid field and the potentials localized at the origin, in particular, delta-like potentials) were considered in many works (see [5] and references there). Also, it should be noted that the quantum dynamics of a neutral fermion in the cosmic string space-time have been analyzed by self-adjoint extension method in Refs.…”

Section: Introductionmentioning

confidence: 99%

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Quantum states of a neutral massive fermion with an anomalous magnetic moment in an Aharonov–Casher field

Khalilov

2017

Int. J. Mod. Phys. A

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The planar non-relativistic quantum dynamics of a neutral massive fermion with an anomalous magnetic moment (AMM) in the presence of the electric field of infinitely long and thin thread with a charge density distributed uniformly along it (an Aharonov-Casher field) is examined. The relevant Hamiltonian is singular and requires additional specification of a one-parameter self-adjoint extension, which can be given in terms of physically acceptable (self-adjoint) boundary conditions. We find all possible self-adjoint Hamiltonians with an Aharonov-Casher field (ACF) by constructing the corresponding Hilbert space of square-integrable functions, including the r = 0 region, for all their Hamiltoniahs. We determine the most relevant physical quantities, such as energy spectrum and wave functions and discuss their correspondence with those obtained by the physical regularization procedure. We show that energy levels of bound states are simple poles of the scattering amplitude. Expressions for the scattering amplitude and cross section depending on the fermion spin are reexamined.

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Section: Physical Regularization Proceduresmentioning

confidence: 99%

Section: Physical Regularization Proceduresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Quantum states of a neutral massive fermion with an anomalous magnetic moment in an Aharonov–Casher field

Khalilov

2017

Int. J. Mod. Phys. A

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Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces

Antoine

Trapani

2015

Non‐Selfadjoint Operators in Quantum Physics

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A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP-space). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP-space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.

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Eigenvalues of Robin Laplacians in infinite sectors

Khalile

Pankrashkin

2017

Mathematische Nachrichten

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Abstract. For α ∈ (0, π), let Uα denote the infinite planar sector of opening 2α, Uα = (x 1 , x 2 ) ∈ R 2 : arg(x 1 + ix 2 ) < α , and T γ α be the Laplacian in L 2 (Uα), T γ α u = −∆u, with the Robin boundary condition ∂ν u = γu, where ∂ν stands for the outer normal derivative and γ > 0. The essential spectrum of T γ α does not depend on the angle α and equals [−γ 2 , +∞), and the discrete spectrum is non-empty iff α < π 2 . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for α ≥ π 6 . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by κ/α with a suitable κ > 0, and the nth eigenvalue En(T γ α ) of T γ α behaves asand admits a full asymptotic expansion in powers of α 2 . The eigenfunctions are exponentially localized near the origin. The results are also applied to δ-interactions on star graphs.

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Self-adjoint Extensions in Quantum Mechanics

Cited by 156 publications

References 0 publications

Quantum states of a neutral massive fermion with an anomalous magnetic moment in an Aharonov–Casher field

Quantum states of a neutral massive fermion with an anomalous magnetic moment in an Aharonov–Casher field

Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces

Eigenvalues of Robin Laplacians in infinite sectors

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