Point interactions in the problem of three quantum particles with internal structure

For quantum systems of zero-range interaction we discuss the mathematical scheme within which modelling the two-body interaction by means of the physically relevant ultra-violet asymptotics known as the "Ter-Martirosyan-Skornyakov condition" gives rise to a self-adjoint realisation of the corresponding Hamiltonian. This is done within the self-adjoint extension scheme of Kreȋn, Višik, and Birman. We show that the Ter-Martirosyan-Skornyakov asymptotics is a condition of selfadjointness only when is imposed in suitable functional spaces, and not just as a point-wise asymptotics, and we discuss the consequences of this fact on a model of two identical fermions and a third particle of different nature.

show abstract

“…for arbitrary f ∈ D(H) and ξ ∈ C, and comparing it with (25), one recognises that (26) follows from (14). This concludes the proof of part (i).…”

Section: Point Interaction Hamiltonian Through the Kreȋn-višik-birmanmentioning

confidence: 99%

On point interactions realised as Ter-Martirosyan–Skornyakov Hamiltonians

Michelangeli

Ottolini

2017

Reports on Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…• a more operator-theoretic line in the Faddeev-Minlos spirit, developed from the mid 1980's to the recent years by Minlos (also in collaboration with Menlikov, Mogilner, and Shermatov) [176,185,186,169,170,177,215,178,179,180,181,182], with also recent contributions by Yoshitomi [253], by Michelangeli and Ottolini [172,173], and by Becker, Michelangeli, and Ottolini [30]; • a line exploiting quadratic forms methods, initiated at the end of the 1980's by Dell'Antonio, Figari, and Teta and mainly developed in the following decades by an Italian community [230,74,75,101,64,175,66,65,29,174,27], with also recent contributions by Moser and Seiringer [187,188]; • a side line by Pavlov and his school [154,168], retaining the same ideas, but aimed at rigorously constructing variants of the formal Hamiltonian (6.5) for particles with spin, and a spin-spin contact interaction; • a further line where (three-dimensional) three-body Hamiltonians with zerorange interactions are constructed as rigorous limits, in the resolvent sense, of ordinary Schrödinger operators with potentials that scale up to a delta-like profile -an idea discussed first by Albeverio, Høegh-Krohn, and Wu [10] in the early 1980's, and subsequently by Dimock and Rajeev [83] in the planar analogue (more recently, one-and two-dimensional counterpart results have been established in [26,126,125]).…”

Section: Introductionmentioning

confidence: 99%

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

Gallone¹,

Michelangeli²

2022

Preprint

View full text Add to dashboard Cite

In particular, we are indebted to S. Albeverio, for his overall support and scientific advice, being indeed one of the world's leading figures, among many other fields, in the area of self-adjoint solvable models in quantum mechanics.Last, we gratefully acknowledge the support of the Italian National Institute for Higher Mathematics (INdAM), the Hausdorff Center for Mathematics, and the Alexander von Humboldt Foundation.

show abstract

“…Sections 4, 5, and 6 contain our main analysis of three-particle equations with some details placed in Appendix B. One can trace the long history of the development of singular two-and three-particle problems in the recent articles [6] (and references therein). We would like to notice here that our consideration follows the idea of refs.…”

Section: Introductionmentioning

confidence: 99%

“…As shown in refs. [11] and [6], the asymptotic behavior of our separable off-shell T-matrix (64) provides that we deal with a self-adjoint three-particle Hamiltonian semi-bounded from below in both cases. However, the Hamiltonians related to more slowly vanishing T-matrices for other two-particle extensions (41) are unbounded, manifesting the "collapse" in the threeparticle system under consideration.…”

mentioning

confidence: 99%

Two- and Three-Particle States in a Nonrelativistic Four-Fermion Model in the Fine-Tuning Renormalization Scheme: Goldstone Mode Versus Extension Theory

et al. 2001

View full text Add to dashboard Cite

In a nonrelativistic contact four-fermion model we show that simple regularisation prescriptions together with a definite fine-tuning of the cut-off-parameter dependence of "bare" quantities give the exact solutions for the two-particle sector and Goldstone modes. Their correspondence with the self-adjoint extension into Pontryagin space is established leading to self-adjoint semi-bounded Hamiltonians in three-particle sectors as well. Renormalized Faddeev equations for the bound states with Fredholm properties are obtained and analysed.The interaction between all particles in the systems B and C is the same, as in the AA and A A-channels of system A. So it is enough to consider the last one. Hereafter BB means BB, B B, B B, and the same for CC. Let us introduce the two-particle interaction kernels occurring in Eq. (11) and the two-particle energies as: K (QQ ′ ) (s, k) = K P {±} (s, k), for QQ ′ = A A, AA, BB, CC A A respectively,

show abstract

Point interactions in the problem of three quantum particles with internal structure

Cited by 6 publications

References 9 publications

On point interactions realised as Ter-Martirosyan–Skornyakov Hamiltonians

On point interactions realised as Ter-Martirosyan–Skornyakov Hamiltonians

Self-adjoint extension schemes and modern applications to quantum Hamiltonians

Two- and Three-Particle States in a Nonrelativistic Four-Fermion Model in the Fine-Tuning Renormalization Scheme: Goldstone Mode Versus Extension Theory

Contact Info

Product

Resources

About