Self-adjoint extension schemes and modern applications to quantum Hamiltonians

Gallone, Matteo; Michelangeli, Alessandro

doi:10.48550/arxiv.2201.10205

Cited by 3 publications

(3 citation statements)

References 110 publications

(219 reference statements)

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“…Looking for self-adjoint extensions of H min is equivalent to looking for selfadjoint extensions of the massless problem (as their difference is a bounded operator). The latter can be conveniently expressed in the unitary equivalent form of a direct sum (Proposition 2.1) which can be exploited for the computation of deficiency indices (see [28,Section 1.6]). From Proposition 2.3 we know that the deficiency indices of h ν,k are Using the characterisation of Proposition 2.4 one completes the proof of (i).…”

Section: Proof Of Proposition 23 Let Us Denote By H *mentioning

confidence: 99%

Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

Cassano,

Gallone,

Pizzichillo

2022

Preprint

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We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac-Hardy inequality and exploit the Kato-Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate our results giving a description of the spectrum of these operators.

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Section: Proof Of Proposition 23 Let Us Denote By H *mentioning

confidence: 99%

Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

Cassano,

Gallone,

Pizzichillo

2022

Preprint

Self Cite

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“…This means that a further threebody boundary condition is required, corresponding to a sort of three-body force acting between the particles. The three-body boundary condition introduced in [18] (see also the recent papers [13,17] and the references therein) leads to a Hamiltonian unbounded from below which is unsatisfactory from the physical point of view. Such instability property is known as Thomas effect and it is due to the fact that the interaction becomes too singular when all the three particles are close to each other (we just recall that the situation is rather different for systems made of two species of fermions, see, e.g., [8,9], [20][21][22]).…”

Section: Introductionmentioning

confidence: 99%

Zero-Range Hamiltonian for a Bose Gas with an Impurity

Ferretti

Teta

2023

Complex Anal. Oper. Theory

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We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a three-body force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.

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“…On the other hand, in dimension three for n = 3 it is known ( [15], see also [16]) that the same procedure leads to a symmetric but non s.a. operator and that all its s.a. extensions are unbounded from below. Such instability property is known as Thomas effect and it is due to the fact that the interaction becomes too singular when all the three particles are close to each other (see also the recent papers [14], [9] and the references therein). The Hamiltonian defined in [15] is therefore unsatisfactory from the physical point of view and the construction of other physically reasonable n-body Hamiltonians…”

Section: Introductionmentioning

confidence: 99%

Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity

Ferretti¹,

Teta²

2022

Preprint

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We study the Hamiltonian for a system of N identical bosons interacting with an impurity, i.e., a different particle, via zero-range forces in dimension three. It is well known that, following the standard approach, one obtains the Ter-Martirosyan Skornyakov Hamiltonian which is unbounded from below. In order to avoid such instability problem, we introduce a threebody force acting at short distances. The effect of this force is to reduce to zero the strength of the zero-range interaction between two particles, i.e., the impurity and a boson, when another boson approaches the common position of the first two particles. We show that the Hamiltonian defined with such regularized interaction is self-adjoint and bounded from below if the strength of the three-body force is sufficiently large. The method of the proof is based on a careful analysis of the corresponding quadratic form.

show abstract

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

Cited by 3 publications

References 110 publications

Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

Dirac-Coulomb Operators with Infinite Mass Boundary Conditions in Sectors

Zero-Range Hamiltonian for a Bose Gas with an Impurity

Regularized Zero-Range Hamiltonian for a Bose Gas with an Impurity

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