Three-body problem in 3D space: ground state, (quasi)-exact-solvability

Turbiner, A. V.; Miller, Willard; Escobar-Ruiz, A. M.

doi:10.1088/1751-8121/aa6cc2

Cited by 18 publications

(62 citation statements)

References 17 publications

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“…In three recent papers, [15,20,21], Escobar-Ruiz, Miller, and Turbiner made the following remarkable observation. Once the center of mass coordinates have been separated out, the quantum n body Schrödinger operator decomposes into a "radial" component that depends only upon the distances between the masses plus an "angular" component that involves the remaining coordinates and annihilates all functions of the interpoint distances.…”

Section: March 5 2019mentioning

confidence: 95%

“…since each entry of B (n) is linear in the α i 's. For example when n = 3, we write (6) as and recognize the nonzero entries of its three matrix summands as forming order 3 reduced Euclidean distance matrices (20).…”

Section: March 5 2019mentioning

confidence: 99%

“…Let us now specialize the identity in Theorem 12 to the n body case, so that C S , C T reduce, respectively, to C A , C R . We further set x = y and then use (21), (22), (23), (25), (26), and (44) to obtain the following intriguing result, reminiscent of Schoenberg's formula (23) for the quadratic forms based on the (reduced) Euclidean distance matrices.…”

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confidence: 99%

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The $n$ Body Matrix and Its Determinant

Grinberg¹,

Olver

2019

SIAM J. Appl. Algebra Geometry

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The primary purpose of this note is to prove two recent conjectures concerning the n body matrix that arose in recent papers of Escobar-Ruiz, Miller, and Turbiner on the classical and quantum n body problem in ddimensional space. First, whenever the positions of the masses are in a nonsingular configuration, meaning that they do not lie on an affine subspace of dimension ≤ n − 2, the n body matrix is positive definite and, hence, defines a Riemannian metric on the space coordinatized by their interpoint distances. Second, its determinant can be factored into the product of the order n Cayley-Menger determinant and a mass-dependent factor that is also of one sign on all nonsingular mass configurations. The factorization of the n body determinant is shown to be a special case of an intriguing general result proving the factorization of determinants of a certain form.

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Section: March 5 2019mentioning

confidence: 95%

Section: March 5 2019mentioning

confidence: 99%

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The $n$ Body Matrix and Its Determinant

Grinberg¹,

Olver

2019

SIAM J. Appl. Algebra Geometry

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“…We end up with six-dimensional integrals over the relative distances (r 1 , r 2 , r 3 , r 12 , r 13 , r 23 ) (for the general discussion see [28]). It was shown a long ago by Fromm and Hill [32] that these integrals can be reduced to one-dimensional ones (!)…”

Section: Two-electron Case: Effective Potentialmentioning

confidence: 99%

Few-electron atomic ions in non-relativistic QED: The ground state

2019

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Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithiumlike ions with static nuclei for Z ≤ 20 the domain of applicability of Non-Relativistic QED (NRQED) is localized for ground state energy. It is demonstrated that for both helium-like and lithium-like ions with Z ≤ 20 the finite nuclear mass effects do not change 4-5 significant digits (s.d.) and the leading relativistic and QED effects leave unchanged 3-4 s.d. in the ground state energy. It is shown that the non-relativistic ground state energy can be interpolated with accuracy of not less than 6 decimal digits (d.d.) (or 7-8 s.d.) for Z ≤ 50 for helium-like and for Z ≤ 20 for lithium-like ions by a meromorphic function in, which is well inside the domain of applicability of NRQED. It is found that both the Majorana formula -a second degree polynomial in Z with two free parameters -and a fourth degree polynomial in λ (a generalization of the Majorana formula) reproduce the ground state energy of the helium-like and lithium-like ions for Z ≤ 20 in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that 99.9% of the ground state energy is given by the variational energy for properly optimized trial function of the form of (anti)-symmetrized product of three (six) screened Coulomb orbitals for two-(three) electron system with 3 (7) free parameters for Z ≤ 20, respectively. It may imply that these trial functions are, in fact, exact wavefunctions in non-relativistic QED, thus, the NRQED effective potential can be derived. It is shown that the sum of relativistic and QED effects in leading approximation -3 s.d. -for both 2 and 3 electron systems is interpolated by 4th degree polynomial in Z

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“…Replacing the individual quadratic potential A j ω 2 (r (J) j ) 2 by the quasi-exactly-solvable (QES) sextic potential, we arrive at the QES anharmonic Jacobi oscillator. It is easy to check that in the (n − 1)-dimensional space of relative radial motion of modules of Jacobi coordinates, or, saying differently, of the Jacobi distances r Since the spectrum of the Jacobi oscillators (6), (7) is known explicitly, their eigenfunctions can be used as the basis to study many-body problems, as was proposed in [1].…”

Section: Introductionmentioning

confidence: 99%

Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$

Turbiner

Miller

Escobar-Ruiz

2020

J. Phys. A: Math. Theor.

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In this work we study 2-and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative distances, |r i − r j |, between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator, while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only (S-states). We pay special attention to the cases where the masses of the particles and spring constants are unequal as well as to the atomic, where one mass is infinite, and molecular, where two masses are infinite, limits. In general, three-body harmonic oscillator is 7-parametric depending on 3 masses and 3 spring constants, and frequency. In particular, the first and second order integrals of the 3-body oscillator for unequal masses are searched: it is shown that for certain relations involving masses and spring constants the system becomes maximally (minimally) superintegrable in the case of two (one) relations.(J) 1 is always proportional to the vector of relative distance between particles 1 and 2. Evidently, the variables in T r are separated and the kinetic energy of relative motion is the sum of kinetic energies in the Jacobi coordinate directions.By adding to T r the harmonic oscillator potentials in each Jacobi coordinate direction we

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Three-body problem in 3D space: ground state, (quasi)-exact-solvability

Cited by 18 publications

References 17 publications

The $n$ Body Matrix and Its Determinant

The $n$ Body Matrix and Its Determinant

Few-electron atomic ions in non-relativistic QED: The ground state

Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$

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