Abstract We consider a system of three identical bosons in $$\mathbb {R}^3$$ R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $$ a > 0$$ a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues $$E_n$$ E n accumulating at zero and fulfilling the asymptotic geometrical law $$\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty $$ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where $$s_0\approx 1.00624$$ s 0 ≈ 1.00624 .

arXiv:2306.12157Letters in Mathematical PhysicsNov 3, 20236 citations

Improved bounds on many-body hamiltonians

Jean-Louis Basdevant, André Martin, J. M. Richard

Nuclear Physics BOct 1, 199046 citations

Springer eBooks

Number of eigenvalues of many-body Hamiltonians and Efimov's effect

Israel Michael Sigal

Published Apr 6, 2008• 4 citations

Publisher

DOI: 10.1007/3-540-09964-6_301

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Rigorous derivation of the Efimov effect in a simple model

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Abstract We consider a system of three identical bosons in $\mathbb {R}^3$ R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a > 0$ a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues $E_n$ E n accumulating at zero and fulfilling the asymptotic geometrical law $\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty$ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where $s_0\approx 1.00624$ s 0 ≈ 1.00624 .Abstract We consider a system of three identical bosons in $$\mathbb {R}^3$$ R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $$ a > 0$$ a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues $$E_n$$ E n accumulating at zero and fulfilling the asymptotic geometrical law $$\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty $$ E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where $$s_0\approx 1.00624$$ s 0 ≈ 1.00624 .

arXiv:2306.12157Letters in Mathematical PhysicsNov 3, 20236 citations

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Jean-Louis Basdevant, André Martin, J. M. Richard

Nuclear Physics BOct 1, 199046 citations

Abstract We consider a system of three identical bosons in

\mathbb {R}^3

R 3 with two-body zero-range interactions and a three-body hard-core repulsion of a given radius

a > 0

a > 0 . Using a quadratic form approach, we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of a . In particular, this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, i.e., we show that the Hamiltonian has an infinite sequence of negative eigenvalues

E_n

E n accumulating at zero and fulfilling the asymptotic geometrical law

\;E_{n+1} / E_n \; \rightarrow \; e^{-\frac{2\pi }{s_0}}\,\; \,\text {for} \,\; n\rightarrow +\infty

E n + 1 / E n → e - 2 π s 0 for n → + ∞ holds, where

s_0\approx 1.00624

s 0 ≈ 1.00624 .