I was born in Moscow, Russia at 1950 in the family of two mechanical engineers. My father, who was the outstanding engineer, taught me personally the concepts of scientific method. At 1967 I was admitted to Moscow Aviation Institute from there I entered to the Moscow Institute for Physics and Technology. At 1974 I received the diploma with honors (summa cum laude) in nuclear engineering. My first job was in the Institute for Theoretical and Experimental Physics as an engineer, where I defended PhD thesis on high-energy physics phenomenology at 1978 and got DSc degree on non-perturbative methods in quantum mechanics and field theory at 1989. This year I was promoted to a leading scientific researcher. From 1990 I began to travel visiting 20 different countries in the Europe, US, Canada and Mexico. I am a full professor in Nuclear Science Institute at UNAM (Mexico), a foreign associate in CRM, Canada (from 2012) and a scholar in Stony Brook University (from 2015). Supervisors: Karen A Ter-Martirosyan (official, from 1970) and Ya B Zeldovich (inofficial, from 1977)
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...
The quantum H3 integrable system is a three-dimensional system with rational potential related to... more The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattic...
We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable... more We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure linear algebra means. A Lie-algebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. The approach also allows to calculate the ratios of a certain generalized Dyson-Mehta integrals algebraically, which are interested by themselves.
It is already known that for the one-dimensional quantum quartic single-well anharmonic oscillato... more It is already known that for the one-dimensional quantum quartic single-well anharmonic oscillator V = x2 + g2x4 and double-well anharmonic oscillator with potential V (x) = x2(1 − gx)2 the (trans)series in g (Perturbation Theory (PT) in powers of g plus exponentially-small terms in g, weak coupling regime) and the semiclassical expansion in ~ (including the exponentially small terms in ~) for energies coincide. It implies that both problems are essentially one-parametric, they depend on a combination (g2~). Hence, these problems are reduced to study the potentials Vao = u 2 + u4 and Vdw = u 2(1 − u)2, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction Ψao(u) obtained in [1-2] and defining the function Ψdw(u) = Ψao(u)±Ψao(u− 1) this leads to highly accurate approximation for the eigenfunctions of the double-well potential and its eigenvalues.
It is already known that the quantum quartic single-well anharmonic oscillator Vao(x) = x 2+g2x4 ... more It is already known that the quantum quartic single-well anharmonic oscillator Vao(x) = x 2+g2x4 and double-well anharmonic oscillator Vdw(x) = x 2(1 − gx)2 are essentially one-parametric, their eigenstates depend on a combination (g2~). Hence, these problems are reduced to study the potentials Vao = u 2 + u4 and Vdw = u 2(1− u)2, respectively. It is shown that by taking uniformlyaccurate approximation for anharmonic oscillator eigenfunction Ψao(u), obtained recently, see JPA 54 (2021) 295204 [1] and Arxiv 2102.04623 [2], and then forming the function Ψdw(u) = Ψao(u)± Ψao(u − 1) allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
Following the first principles the analytic Born-Oppenheimer (BO) potential curve for the ground ... more Following the first principles the analytic Born-Oppenheimer (BO) potential curve for the ground state X1Σ+ of the molecule ClF is proposed for whole range of internuclear distances R ∈ [0,∞). It is based on matching the perturbation theory at small internuclear distances R and multipole expansion at large distances R, it has the form of two-point Pade approximant and provides 3-4 figures in rovibrational energies. It supports 5719 rovibrational states with maximal vibrational number νmax = 47 and maximal angular momentum Lmax = 210 including 36 weakly-bound states close to threshold (to dissociation limit) with the energies . 10−4 Hartree. The van der Waals constant C (ClF ) 6 ∼ 29.3 a.u. is predicted.
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...
The quantum H3 integrable system is a three-dimensional system with rational potential related to... more The quantum H3 integrable system is a three-dimensional system with rational potential related to the noncrystallographic root system H3. It is shown that the gauge-rotated H3 Hamiltonian as well as one of the integrals, when written in terms of the invariants of the Coxeter group H3, is in algebraic form: it has polynomial coefficients in front of derivatives. The Hamiltonian has infinitely-many finite-dimensional invariant subspaces in polynomials, they form the infinite flag with the characteristic vector [Formula: see text]. One among possible integrals is found (of the second order) as well as its algebraic form. A hidden algebra of the H3 Hamiltonian is determined. It is an infinite-dimensional, finitely-generated algebra of differential operators possessing finite-dimensional representations characterized by a generalized Gauss decomposition property. A quasi-exactly-solvable integrable generalization of the model is obtained. A discrete integrable model on the uniform lattic...
We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable... more We show that the existence of algebraic forms of quantum, exactly-solvable, completely-integrable $A-B-C-D$ and $G_2, F_4, E_{6,7,8}$ Olshanetsky-Perelomov Hamiltonians allow to develop the {\it algebraic} perturbation theory, where corrections are computed by pure linear algebra means. A Lie-algebraic classification of such perturbations is given. In particular, this scheme admits an explicit study of anharmonic many-body problems. The approach also allows to calculate the ratios of a certain generalized Dyson-Mehta integrals algebraically, which are interested by themselves.
It is already known that for the one-dimensional quantum quartic single-well anharmonic oscillato... more It is already known that for the one-dimensional quantum quartic single-well anharmonic oscillator V = x2 + g2x4 and double-well anharmonic oscillator with potential V (x) = x2(1 − gx)2 the (trans)series in g (Perturbation Theory (PT) in powers of g plus exponentially-small terms in g, weak coupling regime) and the semiclassical expansion in ~ (including the exponentially small terms in ~) for energies coincide. It implies that both problems are essentially one-parametric, they depend on a combination (g2~). Hence, these problems are reduced to study the potentials Vao = u 2 + u4 and Vdw = u 2(1 − u)2, respectively. It is shown that by taking uniformly-accurate approximation for anharmonic oscillator eigenfunction Ψao(u) obtained in [1-2] and defining the function Ψdw(u) = Ψao(u)±Ψao(u− 1) this leads to highly accurate approximation for the eigenfunctions of the double-well potential and its eigenvalues.
It is already known that the quantum quartic single-well anharmonic oscillator Vao(x) = x 2+g2x4 ... more It is already known that the quantum quartic single-well anharmonic oscillator Vao(x) = x 2+g2x4 and double-well anharmonic oscillator Vdw(x) = x 2(1 − gx)2 are essentially one-parametric, their eigenstates depend on a combination (g2~). Hence, these problems are reduced to study the potentials Vao = u 2 + u4 and Vdw = u 2(1− u)2, respectively. It is shown that by taking uniformlyaccurate approximation for anharmonic oscillator eigenfunction Ψao(u), obtained recently, see JPA 54 (2021) 295204 [1] and Arxiv 2102.04623 [2], and then forming the function Ψdw(u) = Ψao(u)± Ψao(u − 1) allows to get the highly accurate approximation for both the eigenfunctions of the double-well potential and its eigenvalues.
Following the first principles the analytic Born-Oppenheimer (BO) potential curve for the ground ... more Following the first principles the analytic Born-Oppenheimer (BO) potential curve for the ground state X1Σ+ of the molecule ClF is proposed for whole range of internuclear distances R ∈ [0,∞). It is based on matching the perturbation theory at small internuclear distances R and multipole expansion at large distances R, it has the form of two-point Pade approximant and provides 3-4 figures in rovibrational energies. It supports 5719 rovibrational states with maximal vibrational number νmax = 47 and maximal angular momentum Lmax = 210 including 36 weakly-bound states close to threshold (to dissociation limit) with the energies . 10−4 Hartree. The van der Waals constant C (ClF ) 6 ∼ 29.3 a.u. is predicted.
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