On point interactions realised as Ter-Martirosyan–Skornyakov Hamiltonians

Michelangeli, Alessandro; Ottolini, Andrea

doi:10.1016/s0034-4877(17)30036-8

Cited by 34 publications

(46 citation statements)

References 26 publications

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“…While (26) appears to give an explicit expression for Hϕ, it does not because it depends on the vector w ϕ whose dependence on ϕ is not explicit. (25) can be seen as an abstract, operator theoretic version of the so called TMS boundary condition [9,13,20]. In Section 8 we show how this condition reduces to the usual TMS condition in the case of the Fermi-polaron in R 2 .…”

Section: From This Equation and From The Strong Convergencementioning

confidence: 99%

Spectral Theory of the Fermi Polaron

Griesemer

Linden

2019

Ann. Henri Poincaré

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The Fermi polaron refers to a system of free fermions interacting with an impurity particle by means of two-body contact forces. Motivated by the physicists' approach to this system, the present article describes a general mathematical framework for defining many-body Hamiltonians with two-body contact interactions by means of a renormalization procedure. In the case of the Fermi polaron the well-known TMS Hamiltonians are shown to emerge. For the Fermi polaron in a box [0, L] 2 ⊂ R 2 a novel variational principle, established within the general framework, links the low-lying eigenvalues of the system to the zero-modes of a Birman-Schwinger type operator. It allows us to show, e.g., that the polaron-and molecule energies, computed in the physical literature, are indeed upper bounds to the ground state energy of the system.

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Section: From This Equation and From The Strong Convergencementioning

confidence: 99%

Spectral Theory of the Fermi Polaron

Griesemer

Linden

2019

Ann. Henri Poincaré

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“…Models of this kind have been studied extensively in the literature (see, e.g., [6][7][8][9][10][11]13,15,[18][19][20][21][22][23]28,37]) and can be defined via a suitable regularization procedure. More precisely, the formal expression (1.1) can be given a meaning in terms of a suitable quadratic form [7,10,15], which will be introduced in the next section.…”

Section: Introductionmentioning

confidence: 99%

Stability of a Fermionic N + 1 Particle System with Point Interactions

Moser

Seiringer

2017

Commun. Math. Phys.

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Abstract:We prove that a system of N fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical m * . The value of m * is independent of N and turns out to be less than 1. This fact has important implications for the stability of the unitary Fermi gas. We also characterize the domain of the Hamiltonian of this model, and establish the validity of the Tan relations for all wave functions in the domain.

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“…This is to overcome the difficulty of defining W −1 λ T λ when ℓ = 0, since owing to Proposition 3 the map W −1 λ cannot pull arbitrary functions T λ ξ back to H −1/2 (R 3 ). To our understanding, this simple fact had been overlooked in all the previous operator-theoretic approaches to the 2+1 fermionic model of TMS type [27,15,16,30,20,21,22,24,23] until when we pointed it out in [18]. We shall characterise A λ,α on the sector ℓ = 0 in Section 5.…”

Section: With Respect To the Decomposition (See Remark 2 Below)mentioning

confidence: 98%

“…More in detail, here are the results of our investigation. Our first main result is to make the auxiliary operator A λ,α acting on the space of charges completely explicit, as compared to the somewhat implicit characterisation we gave in [18].…”

Section: Introductionmentioning

confidence: 99%

Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan—Skornyakov type

Michelangeli

Ottolini

2018

Reports on Mathematical Physics

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We reconstruct the whole family of self-adjoint Hamiltonians of Ter-Martirosyan-Skornyakov type for a system of two identical fermions coupled with a third particle of different nature through an interaction of zero range. We proceed through an operator-theoretic approach based on the self-adjoint extension theory of Kreȋn, Višik, and Birman. We identify the explicit 'Kreȋn-Višik-Birman extension parameter' as an operator on the 'space of charges' for this model (the 'Kreȋn space') and we come to formulate a sharp conjecture on the dimensionality of its kernel. Based on our conjecture, for which we also discuss an amount of evidence, we explain the emergence of a multiplicity of extensions in a suitable regime of masses and we reproduce for the first time the previous partial constructions obtained by means of an alternative quadratic form approach.

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On point interactions realised as Ter-Martirosyan–Skornyakov Hamiltonians

Cited by 34 publications

References 26 publications

Spectral Theory of the Fermi Polaron

Spectral Theory of the Fermi Polaron

Stability of a Fermionic N + 1 Particle System with Point Interactions

Multiplicity of self-adjoint realisations of the (2+1)-fermionic model of Ter-Martirosyan—Skornyakov type

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