arXiv subjects

Alexander V. Turbiner

Publications and source records attributed to Alexander V. Turbiner.

At least 19 recordsLinked to original sources

arXiv2025

Power-like potentials: from the Bohr-Sommerfeld energies to exact ones

For one-dimensional power-like potentials $|x|^m, m > 0$ the Bohr-Sommerfeld Energies (BSE) extracted explicitly from the Bohr-Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones contrary to the negative parity states where BSE remain above the exact ones for $m>2$ but they are below them for $m < 2$. The ground state BSE as the function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but relative deviation grows with $m$ reaching the value 4 at $m=\infty$. For physically important cases $m=1,4,6$ for the $100$th excited state BSE coincide with exact ones in 5-6 figures. It is demonstrated that modifying the right-hand-side of the Bohr-Sommerfeld quantization condition by introducing the so-called {\it WKB correction} $γ$ (coming from the sum of higher order WKB terms taken at the exact energies) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is small, bounded function $|γ| < 1/2$ for all $m \geq 1$, it is slow growing with increase in $m$ for fixed quantum number, while it decays with quantum number growth at fixed $m$. For the first time for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are written explicitly in closed analytic form with high relative accuracy $10^{-9 \ -11}$ (and $10^{-6}$).

quant-ph↗

arXiv2024

Towards the "puzzle" of Chromium dimer Cr$_2$: predicting the Born-Oppenheimer rovibrational spectrum

The experimentally-observed non-trivial electronic structure of the Cr$_2$ dimer has made the calculation of its potential energy curve a theoretical challenge in the last decades. By matching the perturbation theory at small internuclear distances $R$ and the multipole expansion at large distances $R$ (supposedly both of asymptotic nature), and by adding a few Rydberg-Klein-Rees (RKR) turning points, extracted from experimental data by Casey-Leopold (1993), the analytic form of the potential energy curve for the ground state $X^1Σ^+$ of the Cr$_2$ dimer is found for the first time for the whole range of internuclear distances $R$. This has the form of a two-point Padé approximant and provides an accuracy of 3-4 decimal digits in 29 experimental vibrational energies. The resulting ground state $X^1Σ^+$ potential curve supports 19694 rovibrational states with a maximal vibrational number $ν_\text{max}=104$ at zero angular momentum and with a maximal angular momentum $L_\text{max}=312$ with energies $> 10^{-4}$ { hartree}, and additionally 218 weakly-bound states (close to the dissociation limit) with energies $< 10^{-4}$ { hartree}.

physics.chem-ph↗

arXiv2024

$\mathfrak{gl}(3)$ Polynomial Integrable System: Different Faces of the 3-Body/${\mathcal A}_2$ Elliptic Calogero Model

It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system, introduced by Sokolov-Turbiner in [arXiv:1409.7439], is equivalent to the $\mathfrak{gl}(3)$ quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian as well as their third-order integral can be rewritten in terms of $\mathfrak{gl}(3)$ algebra generators. In turn, all these $\mathfrak{gl}(3)$ generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)$, thus, the Hamiltonian and integral are two elements of the universal enveloping algebra $U_{\mathfrak{h}_5}$. In this paper, four different representations of the $\mathfrak{h}_5$ Heisenberg algebra are used: (I) by differential operators in two real (complex) variables, (II) by finite-difference operators on uniform or exponential lattices. We discovered the existence of two 2-parametric bilinear and trilinear elements (denoted $H$ and $I$, respectively) of the universal enveloping algebra $U(\mathfrak{gl}(3))$ such that their Lie bracket (commutator) can be written as a linear superposition of nine so-called artifacts - the special bilinear elements of $U(\mathfrak{gl}(3))$, which vanish once the representation of the $\mathfrak{gl}(3)$-algebra generators is written in terms of the $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2},I)$-algebra generators. In this representation all nine artifacts vanish, two of the above-mentioned elements of $U(\mathfrak{gl}(3))$ (called the Hamiltonian $H$ and the integral $I$) commute(!); in particular, they become the Hamiltonian and the integral of the 3-body elliptic Calogero model, if $(\hat{p},\hat{q})$ are written in the standard coordinate-momentum representation.

math-ph↗

arXiv2023

Two-body Coulomb problem and hidden $g^{(2)}$ algebra: superintegrability and cubic polynomial algebra

It is shown that the two-body Coulomb problem in the Sturm representation leads to a new two-dimensional, exactly-solvable, superintegrable quantum system in curved space with a $g^{(2)}$ hidden algebra and a cubic polynomial algebra of integrals. The two integrals are of orders two and four, they are made from two components of the angular momentum and from the modified Laplace-Runge-Lenz vector, respectively. It is demonstrated that the cubic polynomial algebra is an infinite-dimensional subalgebra of the universal enveloping algebra $U_{g^{(2)}}$.

math-ph↗

arXiv2023

HCl, DCl and TCl diatomic molecules in their ground state: predicting Born-Oppenheimer rovibrational spectra

The analytic Born-Oppenheimer (B-O) potential curve for the ground state $X^1Σ^+$ of the molecule (H,D,T)Cl is constructed for the whole range of internuclear distances $R \in [0,\infty)$ with an accuracy of 3-5 figures in comparison with the RKR-style potential curve derived from available experimental data on vibrational energies. With an accuracy of 3-4 significant figures in the energies, it is predicted for HCl (DCl, TCl) the 836 (1625, 2366) B-O rovibrational bound states with maximal vibrational number $ν_{max} = 20\, (29, 35)$ and maximal angular momentum $L_{max} = 64\, (90, 109)$ including 24 (46, 63) weakly-bound states (close to the dissociation limit) with energies $\lesssim 10^{-4}$ Hartree. Insufficiency of existing experimental data is indicated and a prediction of the bulk of missing rovibrational states is made for all HCl, DCl, TCl molecules.

physics.chem-ph↗

arXiv2022

Towards the analytic theory of Potential Energy Curves for diatomic molecules. Studying He${}_2^+$ and LiH diatomics as illustration

Following the first principles the elements of the analytic theory of potential curves for diatomic molecules (diatomics) are presented. It is based on matching the perturbation theory at small internuclear distances $R$ and multipole expansion at large distances, modified by instanton-induced trans-series for homonuclear case, with addition of the phenomenologically described equilibrium configuration, if exists. It leads to a new class of (generalized) meromorphic potentials (modified by exponential terms) with difference in degrees of polynomials in numerator and denominator equal to 4 (6) for positively charged (neutral) diatomics. As examples the He$_2^+$ and LiH diatomics in Born-Oppenheimer (adiabatic) approximation are considered. For ${}^{4}$He$_2^+$ (${}^{3}$He$_2^+$) diatomics it is found the approximate analytic expression for the potential energy curves (analytic PEC) $V(R)$ for the ground state $X^2 Σ_u^+$ and the first excited state $A^2 Σ_g^+$. It provides 3-4 s.d. correctly for distances $R \in [1, 10]$\,a.u. with some irregularities for $A^2 Σ_g^+$ PEC at small distances (much smaller than equilibrium distances) probably related to level crossings which may occur there. The analytic PEC for the ground state $X^2 Σ_u^+$ supports 829 (626) rovibrational states with 3-4 s.d. of accuracy in energy, which is only by 1 state less (more) than 830 (625) reported in the literature. In turn, the analytic PEC for the excited state $A^2 Σ_g^+$ supports all 9 reported weakly-bound rovibrational states. Entire rovibrational spectra is found in a single calculation using the code based on the Lagrange mesh method.

physics.atom-ph↗

arXiv2021

From quartic anharmonic oscillator to double well potential

It is already known that the quantum quartic single-well anharmonic oscillator $V_{ao}(x)=x^2+g^2 x^4$ and double-well anharmonic oscillator $V_{dw}(x)= x^2(1 - gx)^2$ are essentially one-parametric, their eigenstates depend on a combination $(g^2 \hbar)$. Hence, these problems are reduced to study the potentials $V_{ao}=u^2+u^4$ and $V_{dw}=u^2(1-u)^2$, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction $Ψ_{ao}(u)$, obtained recently, see JPA 54 (2021) 295204 [1] and Arxiv 2102.04623 [2], and then forming the function $Ψ_{dw}(u)=Ψ_{ao}(u) \pm Ψ_{ao}(u-1)$ allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.

quant-ph↗

arXiv2021

Superintegrability of $(2n+1)$-body choreographies, $n=1,2,3,\ldots, \infty$ on the algebraic Lemniscate by Bernoulli (inverse problem of classical mechanics)

For one 3-body and two 5-body planar choreographies on the same algebraic Lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies $(2n+1)$ moving choreographically (without collisions) along given algebraic Lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit $n \rar \infty$ is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic Lemniscate and it is characterized by infinitely-many constants of motion.

physics.class-ph↗

arXiv2019

Few-electron atomic ions in non-relativistic QED: the Ground state energy

Following detailed analysis of relativistic, QED and mass corrections for helium-like and lithium-like ions with static nuclei for $Z \leq 20$ 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 $Z \leq 20$ 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 not less than 13 s.d. for $Z \leq 12$, and not less than 12 s.d. for $Z \leq 50$ for helium-like as well as for $Z \leq 20$ for lithium-like ions by a compact meromorphic function in $λ=\sqrt{Z-{Z_B}}$ ($Z_B$ is the 2nd critical charge, see {TLO:2016}), $P_9(λ)/Q_5(λ)$. It is found that 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 $Z \leq 20$ in the domain of applicability of NRQED, thus, at least, 3 s.d. It is noted that $\gtrsim 99.9\%$ 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 $Z \leq 20$, respectively. It may imply that these trial functions are, in fact, {\it 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 $Z$ for $Z \leq 20$.

physics.atom-ph↗

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↗

arXiv2017

{H$_2^+$, HeH and H$_2$}: approximating potential curves, calculating rovibrational states

Analytic consideration of the Bohr-Oppenheimer (BO) approximation for diatomic molecules is proposed: accurate analytic interpolation for potential curve consistent with its rovibrational spectra is found. It is shown that in the Bohr-Oppenheimer approximation for four lowest electronic states $1sσ_g$ and $2pσ_u$, $2p π_u$ and $3d π_g$ of H$_2^+$, the ground state X$^2Σ^+$ of HeH and the two lowest states $^1Σ^+_g$ and $^3Σ^+_u$ of H$_2$, the potential curves can be analytically interpolated in full range of internuclear distances $R$ with not less than {4-5-6} figures. Approximation based on matching the Taylor-type expansion at small $R$ and a combination of the multipole expansion with one-instanton type contribution at large distances $R$ is given by two-point Padé approximant. The position of minimum, when exists, is predicted within 1$\%$ or better. For the molecular ion H$_2^+$ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states $(ν,L)$ associated with $1sσ_g$ and $2pσ_u$, $2p π_u$ and $3d π_g$ potential curves is calculated. In general, $1sσ_g$ electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state $2pσ_u$ the total number of rovibrational states (all with $ν=0$) is equal to 3, within or beyond Bohr-Oppenheimer approximation. As for the state $2pπ_u$ within the Bohr-Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state $3dπ_g$ is repulsive, no rovibrational state is found. The ground state potential curve of the heteronuclear molecule HeH does not support rovibrational states. Accurate analytical expression for the potential curves of the hydrogen molecule H$_2$ for the states $^1Σ^+_g$ and $^3Σ^+_u$ is presented.

physics.atom-ph↗

arXiv2017

The quantum n-body problem in dimension $d\ge n-1$: ground state

We employ generalized Euler coordinates for the $n$ body system in $d \geq n-1$ dimensional space, which consists of the centre-of-mass vector, relative (mutual), mass-independent distances $r_{ij}$ and angles as remaining coordinates. We prove that the kinetic energy of the quantum $n$-body problem for $d \geq n-1$ can be written as the sum of three terms: (i) kinetic energy of centre-of-mass, (ii) the second order differential operator $Δ_{rad}$ which depends on relative distances alone and (iii) the differential operator $Ω$ which annihilates any angle-independent function. The operator $Δ_{rad}$ has a large reflection symmetry group $Z_2^{\oplus \frac{n(n-1)}{2}}$ and in $ρ_{ij}=r_{ij}^2$ variables is an algebraic operator, which can be written in terms of generators of their {\it hidden} algebra $sl(\frac{n(n-1)}{2}+1, R)$. Thus, $Δ_{rad}$ makes sense of the Hamiltonian of a quantum Euler-Arnold $sl(\frac{n(n-1)}{2}+1, R)$ top in a constant magnetic field. It is conjectured that for any $n$, the similarity-transformed $Δ_{rad}$ is the Laplace-Beltrami operator plus (effective) potential; thus, it describes a $\frac{n(n-1)}{2}$-dimensional quantum particle in curved space. This was verified for $n=2,3,4$. After de-quantization the similarity-transformed $Δ_{rad}$ becomes the Hamiltonian of the classical top with variable tensor of inertia in an external potential. This approach allows a reduction of the $dn$-dimensional spectral problem to a $\frac{n(n-1)}{2}$ -dimensional spectral problem if the eigenfunctions depend only on relative distances. We prove that the ground state function of the $n$ body problem depends on relative distances alone.

math-ph↗

arXiv2017

Helium-like and Lithium-like ions: Ground state energy

It is shown that the non-relativistic ground state energy of helium-like and lithium-like ions with static nuclei can be interpolated in full physics range of nuclear charges $Z$ with accuracy of not less than 6 decimal digits (d.d.) or 7-8 significant digits (s.d.) using a meromorphic function in appropriate variable with a few free parameters. It is demonstrated that finite nuclear mass effects do not change 4-5 s.d. for $Z \in [1,50]$ for 2-,3-electron systems and the leading relativistic and QED corrections leave unchanged 3-4 s.d. for $Z \in [1,12]$ in the ground state energy for 2-electron system, thus, the interpolation reproduces definitely those figures. A meaning of proposed interpolation is in a construction of unified, {\it two-point} Pade approximant (for both small and large $Z$ expansions) with fitting some parameters at intermediate $Z$.

physics.atom-ph↗

arXiv2016

The H$_2^+$ molecular ion: low-lying states

Matching for a wavefunction the WKB expansion at large distances and Taylor expansion at small distances leads to a compact, few-parametric uniform approximation found in {\it J. Phys. B44, 101002 (2011)}. The ten low-lying eigenstates of H$_2^+$ of the quantum numbers $(n,m,\La,\pm)$\, with $n=m=0$ at $\La=0,1,2$, with $n=1$, $m=0$ and $n=0$, $m=1$ at $\La=0$ of both parities are explored for all interproton distances $R$. For all these states this approximation provides the relative accuracy $\lesssim 10^{-5}$ (not less than 5 s.d.) locally, for any real coordinate $x$ in eigenfunctions, when for total energy $E(R)$ it gives 10-11 s.d. for $R \in [0,50]$~a.u. Corrections to the approximation are evaluated in the specially-designed, convergent perturbation theory. Separation constants are found with not less than 8 s.d. The oscillator strength for the electric dipole transitions $E1$ is calculated with not less than 6~s.d. A dramatic dip in the $E1$ oscillator strength $f_{1s\si_g-3p\si_u}$ at $R \sim R_{eq}$ is observed. The magnetic dipole and electric quadrupole transitions are calculated for the first time with not less than 6~s.d. in oscillator strength. For two lowest states $(0,0,0,\pm)$ (or, equivalently, $1s\si_g$ and $2p\si_u$ states) the potential curves are checked and confirmed in the Lagrange mesh method within 12~s.d. Based on them the Energy Gap between $1s\si_g$ and $2p\si_u$ potential curves is approximated with modified Pade $R e^{-R} [Pade(8/7)](R)$ with not less than 4-5 figures at $R \in [0, 40]$\,a.u. Sum of potential curves $E_{1s\si_g} + E_{2p\si_u}$ is approximated by Pade $1/R [Pade(5/8)](R)$ in $R \in [0, 40]$\,a.u. with not less than {3-4} figures.

quant-ph↗

arXiv2016

The Heun operator as a Hamiltonian

IIt is shown that the celebrated Heun operator $H_e=-(a_0 x^3 + a_1 x^2 + a_2 x) \frac{d^2}{dx^2} + (b_0 x^2 + b_1 x + b_2)\frac{d}{dx} + c_0 x$ is the Hamiltonian of the $sl(2,R)$-quantum Euler-Arnold top of spin $ν$ in a constant magnetic field. For $a_0 \neq 0$ it is canonically-equivalent to $BC_1(A_1)-$ Calogero-Moser-Sutherland quantum models, if $a_0=0$, ten known one-dimensional quasi-exactly-solvable problems are reproduced, and if, in addition, $b_0=c_0=0$, then four well-known one-dimensional quantal exactly-solvable problems are reproduced. If spin $ν$ of the top takes (half)-integer value the Hamiltonian possesses a finite-dimensional invariant subspace and a number of polynomial eigenfunctions occurs. Discrete systems on uniform and exponential lattices are introduced which are canonically-equivalent to one described by the Heun operator.

math-ph↗

arXiv2015

On $1/Z$ expansion, the critical charge for two-electron system and the Kato theorem

The $1/Z$-expansion for the ground state energy of the Coulomb system of an infinitely massive center of charge Z and two electrons (two electron ionic sequence) is studied. A critical analysis of the $1/Z$ coefficients presented in Baker et al, {\em Phys. Rev. \bf A41}, 1247 (1990) is performed and its numerical deficiency is indicated, leading, in particular, to unreliable decimal digits beyond digits 11-12 of the first coefficients. We made a consistency check of the $1/Z$-expansion with accurate energies for $Z = 1 - 10$: the weighted partial sums of the $1/Z$-expansion with Baker et al. coefficients, reproduce systematically the ground state energies of two-electron ions with $Z \geq 2$ up to 12 decimal digits and for $Z=1$ up to 10 decimal digits. This rules out the presence of non-analytic terms at $Z=\infty$ contributing into the first 10-12 decimal digits in the ground state energy; it agrees with the Kato theorem about convergence of the $1/Z$-expansion within that accuracy. The ground state energy of two-electron ions $Z=11\ (Na^{9+})$ and $Z=12\ (Mg^{10+})$ is calculated with 12 decimal digits.

quant-ph↗

arXiv2015

Quasi-exact-solvability of the $A_{2}/G_2$ Elliptic model: algebraic forms, $sl(3)/g^{(2)}$ hidden algebra, polynomial eigenfunctions

The potential of the $A_2$ quantum elliptic model (3-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through Weierstrass $\wp$-function and has a single coupling constant. A change of variables has been found, which are $A_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown that the model possesses the hidden $sl(3)$ algebra - the Hamiltonian is an element of the universal enveloping algebra $U_{sl(3)}$ for arbitrary coupling constant - thus, it is equivalent to $sl(3)$-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial, is constructed explicitly, being also an element of $U_{sl(3)}$. It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the $G_2$ quantum elliptic model (3-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through Weierstrass $\wp$-function and has two coupling constants. A change of variables has been found, which are $G_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown the model possesses the hidden $g^{(2)}$ algebra. It is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a (non-singular) gauge factor occur.

math-ph↗

arXiv2014

The $BC_{1}$ Elliptic model: algebraic forms, hidden algebra $sl(2)$, polynomial eigenfunctions

The potential of the $BC_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the Lamé Hamiltonian. It is shown that in the space of $BC_1$ elliptic invariant, the potential becomes a rational function, while the flat space metric becomes a polynomial. The model possesses the hidden $sl(2)$ algebra for arbitrary coupling constants: it is equivalent to $sl(2)$-quantum top in three different magnetic fields. It is shown that there exist three one-parametric families of coupling constants for which a finite number of polynomial eigenfunctions (up to a factor) occur.

math-ph↗