arXiv subjects

M. A. Escobar-Ruiz

Publications and source records attributed to M. A. Escobar-Ruiz.

15 recordsLinked to original sources

arXiv2018

Four-body problem in d-dimensional space: ground state, (quasi)-exact-solvability. IV

Due to its great importance for applications, we generalize and extend the approach of our previous papers to study aspects of the quantum and classical dynamics of a $4$-body system with equal masses in {\it $d$}-dimensional space with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. The ground state (and some other states) in the quantum case and some trajectories in the classical case are of this type. We construct the quantum Hamiltonian for which these states are eigenstates. For $d \geq 3$, this describes a six-dimensional quantum particle moving in a curved space with special $d$-independent metric in a certain $d$-dependent singular potential, while for $d=1$ it corresponds to a three-dimensional particle and coincides with the $A_3$ (4-body) rational Calogero model; the case $d=2$ is exceptional and is discussed separately. The kinetic energy of the system has a hidden $sl(7,{\bf R})$ Lie (Poisson) algebra structure, but for the special case $d=1$ it becomes degenerate with hidden algebra $sl(4,R)$. We find an exactly-solvable four-body $S_4$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable four-body sextic polynomial type potential with singular terms. Naturally, the tetrahedron whose vertices correspond to the positions of the particles provides pure geometrical variables, volume variables, that lead to exactly solvable models. Their generalization to the $n$-body system as well as the case of non-equal masses is briefly discussed.

math-ph↗

arXiv2018

Three charges on a plane in a magnetic field: Special trajectories

As a generalization and extension of JMP 54 (2013) 022901, the classical dynamics of three non-relativistic Coulomb charges $(e_1, m_1)$, $(e_2, m_2)$ and $(e_3, m_3)$ on the plane placed in a constant magnetic field perpendicular to the plane is considered. Special trajectories for which the distances between the charges remain unchanged are presented and their corresponding integrals of motion are indicated. For these special trajectories the number of integrals of motion is larger than the dimension of the configuration space and hence they can be called \emph{particularly superintegrable}. Three physically relevant cases are analyzed in detail, namely that of three electrons, a neutral system and a Helium-like system. The $n$-body case is discussed as well.

math-ph↗

arXiv2018

Three-body problem in $d$-dimensional space: ground state, (quasi)-exact-solvability

As a straightforward generalization and extension of our previous paper, J. Phys. A50 (2017) 215201 we study aspects of the quantum and classical dynamics of a $3$-body system with equal masses, each body with $d$ degrees of freedom, with interaction depending only on mutual (relative) distances. The study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only. It is shown that the ground state (and some other states) in the quantum case and the planar trajectories (which are in the interaction plane) in the classical case are of this type. It corresponds to a three-dimensional quantum particle moving in a curved space with special $d$-dimension-independent metric in a certain $d$-dependent singular potential, while at $d=1$ it elegantly degenerates to a two-dimensional particle moving in flat space. It admits a description in terms of pure geometrical characteristics of the interaction triangle which is defined by the three relative distances. The kinetic energy of the system is $d$-independent, it has a hidden $sl(4,R)$ Lie (Poisson) algebra structure, alternatively, the hidden algebra $h^{(3)}$ typical for the $H_3$ Calogero model as in the $d=3$ case. We find an exactly-solvable three-body $S^3$-permutationally invariant, generalized harmonic oscillator-type potential as well as a quasi-exactly-solvable three-body sextic polynomial type potential with singular terms. For both models an extra first order integral exists. It is shown that a straightforward generalization of the 3-body (rational) Calogero model to $d>1$ leads to two primitive quasi-exactly-solvable problems. The extension to the case of non-equal masses is straightforward and is briefly discussed.

math-ph↗

arXiv2017

Separation equations for 2D superintegrable systems on constant curvature spaces

Second-order conformal quantum superintegrable systems in 2 dimensions are Laplace equations on a manifold with an added scalar potential and $3$ independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: Each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are 8 pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra so(4,C) to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality.

math-ph↗

arXiv2017

Fluctuations in quantum mechanics and field theories from a new version of semiclassical theory. II

This is the second paper on semiclassical approach based on the density matrix given by the Euclidean time path integral with fixed coinciding endpoints. The classical path, interpolating between this point and the classical vacuum, called "flucton", plus systematic one- and two-loop corrections, has been calculated in the first paper \cite{Escobar-Ruiz:2016aqv} for double-well potential and now extended for a number of quantum-mechanical problems (anharmonic oscillator, sine-Gordon potential). The method is based on systematic expansion in Feynman diagrams and thus can be extended to QFTs. We show that the loop expansion in QM reminds the leading log-approximations in QFT. In this sequel we present complete set of results obtained using this method in unified way. Alternatively, starting from the Schrödinger equation we derive a {\it generalized} Bloch equation which semiclassical-like, iterative solution generates the loop expansion. We re-derive two loop expansions for all three above potentials and now extend it to three loops, which has not yet been done via Feynman diagrams. All results for both methods are fully consistent with each other. Asymmetric (tilted) double-well potential (non-degenerate minima) is also studied using the second method.

hep-th↗

arXiv2016

Toward a classification of semidegenerate 3D superintegrable systems

Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter potentials are intriguing objects. Next to the nondegenerate 4-parameter potential systems they admit the maximum number of symmetry operators but their symmetry algebras don't close under commutation and not enough is known about their structure to give a complete classification. Some examples are known for which the 3-parameter system can be extended to a 4th order superintegrable system with a 4-parameter potential and 6 linearly independent symmetry generators. In this paper we use Bôcher contractions of the conformal Lie algebra $so(5,C)$ to itself to generate a large family of 3-parameter systems with 4th order extensions, on a variety of manifolds, and all from Bôcher contractions of a single "generic" system on the 3-sphere. We give a contraction scheme relating these systems. The results have myriad applications for finding explicit solutions for both quantum and classical systems.

math-ph↗

arXiv2016

Quantum and thermal fluctuations in quantum mechanics and field theories from a new version of semiclassical theory

We develop a new semiclassical approach, which starts with the density matrix given by the Euclidean time path integral with fixed coinciding endpoints, and proceed by identifying classical (minimal Euclidean action) path, to be referred to as {\it flucton}, which passes through this endpoint. Fluctuations around flucton path are included, by standard Feynman diagrams, previously developed for instantons. We calculate the Green function and evaluate the one loop determinant both by direct diagonalization of the fluctuation equation, and also via the trick with the Green functions. The two-loop corrections are evaluated by explicit Feynman diagrams, and some curious cancellation of logarithmic and polylog terms is observed. The results are fully consistent with large-distance asymptotics obtained in quantum mechanics. Two classic examples -- quartic double-well and sine-Gordon potentials -- are discussed in detail, while power-like potential and quartic anharmonic oscillator are discussed in brief. Unlike other semiclassical methods, like WKB, we do not use the Schrödinger equation, and all the steps generalize to multi-dimensional or quantum fields cases straightforwardly.

hep-th↗

arXiv2016

Conformal Laplace superintegrable systems in 2D: polynomial invariant subspaces

2nd-order conformal superintegrable systems in $n$ dimensions are Laplace equations on a manifold with an added scalar potential and $2n - 1$ independent 2nd order conformal symmetry operators. They encode all the information about Helmholtz (eigenvalue) superintegrable systems in an efficient manner: there is a 1-1 correspondence between Laplace superintegrable systems and Stackel equivalence classes of Helmholtz superintegrable systems. In this paper we focus on superintegrable systems in two dimensions, $n = 2$, where there are 44 Helmholtz systems, corresponding to 12 Laplace systems. For each Laplace equation we determine the possible 2-variate polynomial subspaces that are invariant under the action of the Laplace operator, thus leading to families of polynomial eigenfunctions. We also study the behavior of the polynomial invariant subspaces under a Stackel transform. The principal new results are the details of the polynomial variables and the conditions on parameters of the potential corresponding to polynomial solutions. The hidden gl_3-algebraic structure is exhibited for the exact and quasi-exact systems. For physically meaningful solutions, the orthogonality properties and normalizability of the polynomials are presented as well. Finally, for all Helmholtz superintegrable solvable systems we give a unified construction of 1D and 2D quasi-exactly solvable potentials possessing polynomial solutions, and a construction of new 2D PT-symmetric potentials is established.

math-ph↗

arXiv2015

Three-loop Correction to the Instanton Density. I. The Quartic Double Well Potential

This paper deals with quantum fluctuations near the classical instanton configuration. Feynman diagrams in the instanton background are used for the calculation of the tunneling amplitude (the instanton density) in the three-loop order for quartic double-well potential. The result for the three-loop contribution coincides in six significant figures with one given long ago by J.~Zinn-Justin. Unlike the two-loop contribution where all involved Feynman integrals are rational numbers, in the three-loop case Feynman diagrams can contain irrational contributions.

hep-th↗

arXiv2015

Three-loop Correction to the Instanton Density. II. The Sine-Gordon potential

In this second paper on quantum fluctuations near the classical instanton configurations, see {\em Phys. Rev. D \bf 92}, 025046 (2015) and arXiv:1501.03993, we focus on another well studied quantum-mechanical problem, the one-dimensional Sine-Gordon potential (the Mathieu potential). Using only the tools from quantum field theory, the Feynman diagrams in the instanton background, we calculate the tunneling amplitude (the instanton density) to the three-loop order. The result confirms (to seven significant figures) the one recently recalculated by G. V. Dunne and M. Ünsal, {\it Phys. Rev. \bf D 89}, 105009 (2014) from the resurgence perspective. As in the double well potential case, we found that the largest contribution is given by the diagrams originating from the Jacobian. We again observe that in the three-loop case individual Feynman diagrams contain irrational contributions, while their sum does not.

hep-th↗

arXiv2014

Two charges on plane in a magnetic field: III. $He^+$ ion

The $He^+$ ion on a plane subject to a constant magnetic field $B$ perpendicular to the plane is considered taking into account the finite nuclear mass. Factorization of eigenfunctions permits to reduce the four-dimensional problem to three-dimensional one. The ground state energy of the composite system is calculated in a wide range of magnetic fields from $B=0.01$ up to $B=100$ a.u. and center-of-mass Pseudomomentum $K$ from $0$ to $1000$ a.u. using a variational approach. The accuracy of calculations for $B = 0.1 $ a.u. is cross-checked in Lagrange-mesh method and not less than five significant figures are reproduced in energy. Similarly to the case of moving neutral system on the plane a phenomenon of a sharp change of energy behavior as a function of $K$ for a certain critical $K_c$ but a fixed magnetic field occurs.

physics.atom-ph↗

arXiv2014

Two charges on plane in a magnetic field: II. Moving neutral quantum system across a magnetic field

The moving neutral system of two Coulomb charges on a plane subject to a constant magnetic field $B$ perpendicular to the plane is considered. It is shown that the composite system of finite total mass is bound for any center-of-mass momentum $P$ and magnetic field strength; the energy of the ground state is calculated accurately using a variational approach. Their accuracy is cross-checked in a Lagrange-mesh method for $B=1$ a.u. and in a perturbation theory at small $B$ and $P$. The constructed trial function has the property of being a uniform approximation of the exact eigenfunction. For a Hydrogen atom and a Positronium a double perturbation theory in $B$ and $P$ is developed and the first corrections are found algebraically. A phenomenon of a sharp change of energy behavior for a certain center-of-mass momentum and a fixed magnetic field is indicated.

quant-ph↗

arXiv2013

Two charges on plane in a magnetic field I. "Quasi-equal" charges and neutral quantum system at rest cases

Low-lying bound states for the problem of two Coulomb charges of finite masses on a plane subject to a constant magnetic field $B$ perpendicular to the plane are considered. Major emphasis is given to two systems: two charges with the equal charge-to-mass ratio (quasi-equal charges) and neutral systems with concrete results for the Hydrogen atom and two electrons (quantum dot). It is shown that for these two cases, but when a neutral system is at rest (the center-of-mass momentum is zero), some outstanding properties occur: in double polar coordinates in CMS $(R, ϕ)$ and relative $(ρ, φ)$ coordinate systems (i) the eigenfunctions are factorizable, all factors except for $ρ$-dependent are found analytically, they have definite relative angular momentum, (ii) dynamics in $ρ$-direction is the same for both systems being described by a funnel-type potential; (iii) at some discrete values of dimensionless magnetic fields $b \leq 1$ the system becomes {\it quasi-exactly-solvable} and a finite number of eigenfunctions in $ρ$ are polynomials. The variational method is employed. Trial functions are based on combining for the phase of a wavefunction (a) the WKB expansion at large distances, (b) the perturbation theory at small distances (c) with a form of the known analytically (quasi-exactly-solvable) eigenfunctions. For the lowest states with relative magnetic quantum numbers $s=0,1,2$ this approximation gives not less than 7 s.d., 8 s.d., 9 s.d., respectively, for the total energy $E(B)$ for magnetic fields $0.049\, \text{a.u.} < B < 2000\, \text{a.u.} $ (Hydrogen atom) and $0.025\, \text{a.u.}\eqslantless B \eqslantless 1000\, \text{a.u.} $ (two electrons).

physics.atom-ph↗

arXiv2013

Two charges on a plane in a magnetic field: hidden algebra, (particular) integrability, polynomial eigenfunctions

The quantum mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and $(e_2, m_2)$ subject to a constant magnetic field $B$ perpendicular to the plane is considered. Four integrals of motion are explicitly indicated. It is shown that for two physically-important particular cases, namely that of two particles of equal Larmor frequencies, ${e_c} \propto \frac{e_1}{m_1}-\frac{e_2}{m_2}=0$ (e.g. two electrons) and one of a neutral system (e.g. the electron - positron pair, Hydrogen atom) at rest (the center-of-mass momentum is zero) some outstanding properties occur. They are the most visible in double polar coordinates in CMS $(R, ϕ)$ and relative $(ρ, φ)$ coordinate systems: (i) eigenfunctions are factorizable, all factors except one with the explicit $ρ$-dependence are found analytically, they have definite relative angular momentum, (ii) dynamics in $ρ$-direction is the same for both systems, it corresponds to a funnel-type potential and it has hidden $sl(2)$ algebra; at some discrete values of dimensionless magnetic fields $b \leq 1$, (iii) particular integral(s) occur, (iv) the hidden $sl(2)$ algebra emerges in finite-dimensional representation, thus, the system becomes {\it quasi-exactly-solvable} and (v) a finite number of polynomial eigenfunctions in $ρ$ appear. Nine families of eigenfunctions are presented explicitly.

math-ph↗

arXiv2012

Two charges on plane in a magnetic field: special trajectories

A classical mechanics of two Coulomb charges on a plane $(e_1, m_1)$ and $(e_2, m_2)$ subject to a constant magnetic field perpendicular to a plane is considered. Special "superintegrable" trajectories (circular and linear) for which the distance between charges remains unchanged are indicated as well as their respectful constants of motion. The number of the independent constants of motion for special trajectories is larger for generic ones. A classification of pairs of charges for which special trajectories occur is given. The special trajectories for three particular cases of two electrons, (electron - positron), (electron - $α$-particle) are described explicitly.

physics.class-ph↗