Papers by Alexander Turbiner
Journal of physics, Nov 28, 2001
We study maps preserving the Heisenberg commutation relation ab − ba = 1. We find a one-parameter... more We study maps preserving the Heisenberg commutation relation ab − ba = 1. We find a one-parameter deformation of the standard realization of the above algebra in terms of a coordinate and its dual derivative. It involves a non-local "coordinate" operator while the dual "derivative" is just the Jackson finite-difference operator. Substitution of this realization into any differential operator involving x and d dx , results in an isospectral deformation of a continuous differential operator into a finite-difference one. We extend our results to the deformed Heisenberg algebra ab − qba = 1. As an example of potential applications, various deformations of the Hahn polynomials are briefly discussed.
Physics Letters, Mar 1, 2015
The Schroedinger equation for two electrons in the field of a charged fixed center Z is solved wi... more The Schroedinger equation for two electrons in the field of a charged fixed center Z is solved with the Lagrange mesh method for charges close to the critical charge Z cr. We confirm the value of the nuclear critical charge Z cr recently calculated in Estienne et al. Phys. Rev. Lett. 112, 173001 (2014) to 11 decimal digits using an inhomogeneous (non-uniform) three-dimensional lattice of size 70 × 70 × 20. We show that the ground state energy for H − is accurate to 14 decimals on the lattice 50 × 50 × 40 in comparison with the highly accurate result by Nakashima-Nakatsuji, J. Chem.
Journal of Physics A, Jun 23, 2021
It is shown that for the one-dimensional quantum anharmonic oscillator with potential V (x) = x 2... more It is shown that for the one-dimensional quantum anharmonic oscillator with potential V (x) = x 2 +g 2 x 4 the Perturbation Theory (PT) in powers of g 2 (weak coupling regime) and the semiclassical expansion in powers of for energies coincide. It is related to the fact that the dynamics in x-space and in (gx)-space corresponds to the same energy spectrum with effective coupling constant g 2. Two equations, which govern the dynamics in those two spaces, the Riccati-Bloch (RB) and the Generalized Bloch (GB) equations, respectively, are derived. The PT in g 2 for the logarithmic derivative of wave function leads to PT (with polynomial in x coefficients) for the RB equation and to the true semiclassical expansion in powers of for the GB equation, which corresponds to a loop expansion for the density matrix in the path integral formalism. A 2-parametric interpolation of these two expansions leads to a uniform approximation of the wavefunction in x-space with unprecedented accuracy ∼ 10 −6 locally and unprecedented accuracy ∼ 10 −9 − 10 −10 in energy for any g 2 ≥ 0. A generalization to the radial quartic oscillator is briefly discussed.
Astrophysics and Space Science, Mar 21, 2007
Letters in Mathematical Physics, Nov 1, 2005
A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction fo... more A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction for both the quantum-mechanical anharmonic oscillator and the doublewell potential given by V = m 2 x 2 + gx 4 at arbitrary g ≥ 0 for m 2 > 0 and m 2 < 0, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory.
Journal of Mathematical Physics, Jul 1, 2021
As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor... more As a generalization and extension of our previous paper [Turbiner et al., J. Phys. A: Math. Theor. 53, 055302 (2020)], in this work, we study a quantum four-body system in Rd (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, rij ≡ |ri − rj|, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρij≡rij2, the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation, it is shown that the four-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly solvable. We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite and three others are equal), molecular two-center (two masses are infinite and two others are equal), and molecular three-center (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born–Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. In addition, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for the four-body harmonic oscillator case, there exists an infinite family of eigenfunctions that depend on the single variable, which is the moment of inertia of the system.
arXiv (Cornell University), Oct 30, 2021
It is already known that the quantum quartic single-well anharmonic oscillator V ao (x) = x 2 +g ... more 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). 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 uniformlyaccurate 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) ± Ψ ao (u − 1) allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
arXiv (Cornell University), May 20, 2014
The 1/Z-expansion for the ground state energy of the Coulomb system of an infinitely massive cent... more 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, Phys. Rev. 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 ≥ 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 = ∞ 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 (N a 9+) and Z = 12 (M g 10+) is calculated with 12 decimal digits.
arXiv (Cornell University), Mar 2, 2002
The existence of the molecular ion H ++ 3 in a magnetic field in a triangular configuration is re... more The existence of the molecular ion H ++ 3 in a magnetic field in a triangular configuration is revised. A variational method with an optimization of the form of the vector potential (gauge fixing) is used. It is shown that in the range of magnetic fields 10 8 < B < 10 11 G the system (pppe), with the protons forming an equilateral triangle perpendicular to the magnetic line, has a well-pronounced minimum in the total energy. This configuration is unstable under the decays H-atom + p + p and H + 2 + p. The triangular configuration of H ++ 3 complements H ++ 3 in the linear configuration which exists for B 10 10 G.
arXiv (Cornell University), Jun 14, 2006
A trial function is presented for the H 2 molecule which provides the most accurate (the lowest) ... more A trial function is presented for the H 2 molecule which provides the most accurate (the lowest) Bohr-Oppenheimer ground state energy among few-parametric trial functions (with ≤ 14 parameters). It includes the electronic correlation term in the form ∼ exp (γr 12) where γ is a variational parameter.
arXiv (Cornell University), Jul 21, 2017
It is shown that the non-relativistic ground state energy of helium-like and lithium-like ions wi... more 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 ∈ [1, 50] for 2-,3-electron systems and the leading relativistic and QED corrections leave unchanged 3-4 s.d. for Z ∈ [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, two-point Pade approximant (for both small and large Z expansions) with fitting some parameters at intermediate Z.
Journal of Mathematical Physics, Jun 1, 2019
As a generalization and extension of our previous paper J. Phys. A: Math. Theor. 53 055302 [1], i... more As a generalization and extension of our previous paper J. Phys. A: Math. Theor. 53 055302 [1], in this work we study a quantum 4-body system in R d (d ≥ 3) with quadratic and sextic pairwise potentials in the relative distances, r ij ≡ |r i − r j |, between particles. Our study is restricted to solutions in the space of relative motion with zero total angular momentum (S-states). In variables ρ ij ≡ r 2 ij , the corresponding reduced Hamiltonian of the system possesses a hidden sl(7; R) Lie algebra structure. In the ρ-representation it is shown that the 4-body harmonic oscillator with arbitrary masses and unequal spring constants is exactly-solvable (ES). We pay special attention to the case of four equal masses and to atomic-like (where one mass is infinite, three others are equal), molecular two-center (two masses are infinite, two others are equal) and molecular threecenter (three infinite masses) cases. In particular, exact results in the molecular case are compared with those obtained within the Born-Oppenheimer approximation. The first and second order symmetries of non-interacting system are searched. Also, the reduction to the lower dimensional cases d = 1, 2 is discussed. It is shown that for four body harmonic oscillator case there exists an infinite family of eigenfunctions which depend on the single variable which is the moment-of-inertia of the system.
Physical Review A, Dec 26, 2006
A detailed study of the ground state of the Coulomb system (ααee) which corresponds to the He 2+ ... more A detailed study of the ground state of the Coulomb system (ααee) which corresponds to the He 2+ 2 molecular ion in a magnetic field B = 0 − 4.414 × 10 13 G in parallel configuration (infinitely massive α−particles are situated along a magnetic field line) is presented. The variational method is employed using a trial function which includes electronic correlation in the form exp (γr 12) where γ is a variational parameter. It is shown that the quantum numbers of the lowest total energy state depend on the magnetic field strength. It evolves from the spin-singlet 1 Σ g metastable state at 0 ≤ B 0.85 a.u. to a repulsive spin-triplet 3 Σ u state for 0.85 a.u. B 1100 a.u. and, finally, to a strongly bound spin-triplet 3 Π u state at stronger fields B 1100 a.u.
Symmetry, integrability and geometry: methods and applications, Feb 3, 2024
Czechoslovak Journal of Physics, Nov 1, 2003
Journal of Physical Chemistry A, Apr 30, 2013
Journal of Mathematical Physics, Feb 1, 2013
A classical mechanics of two Coulomb charges on a plane (e 1 , m 1) and (e 2 , m 2) subject to a ... more 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), (electronα-particle) are described explicitly.
Journal of physics, Jan 21, 1994
arXiv (Cornell University), Sep 2, 2022
It is evident that the positions of 4 bodies in d > 2 dimensional space can be identified with ve... more It is evident that the positions of 4 bodies in d > 2 dimensional space can be identified with vertices of a tetrahedron. Square of volume of the tetrahedron, weighted sum of squared areas of four facets and weighted sum of squared edges are called the volume variables. A family of translation-invariant potentials which depend on volume variables alone is considered as well as solutions of the Newton equations which solely depend on volume variables. For the case of zero angular momentum L = 0 the corresponding Hamiltonian, which describes these solutions, is derived. Three examples are studied in detail: (I) the (super)integrable 4-body closed chain of harmonic oscillators for d > 2 (the harmonic molecule), (II) a generic, two volume variable dependent potential whose trajectories possess a constant moment of inertia (d > 1), and (III) the 4-body anharmonic oscillator for d ≥ 1. This work is the second of the sequel: the first one [IJMPA 36, No. 18 (2021)] was dedicated to study the 3-body classical problem in volume variables.
arXiv: Atomic Physics, 2017
It is shown that in the Bohr-Oppenheimer approximation for four lowest electronic states $1s\sigm... more It is shown that in the Bohr-Oppenheimer approximation for four lowest electronic states $1s\sigma_g$ and $2p\sigma_u$, $2p \pi_u$ and $3d \pi_g$ of H$_2^+$ and the ground state X$^2\Sigma^+$ of HeH, the potential curves can be well-approximated analytically in full range of internuclear distances $R$ with not less than 4-5-6 figures. Approximation is based on straightforward interpolation of the Taylor-type expansion at small $R$ and a combination of the multipole expansion with one-instanton type expansion at large distances $R$. 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 $(\nu,L)$ associated with $1s\sigma_g$ and $2p\sigma_u$, $2p \pi_u$ and $3d \pi_g$ analytically derived potential curves is calculated. In general, it coincides with spectra found via numerical solution of the Schr\"odinger equation when available. It is s...
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Papers by Alexander Turbiner