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One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as $G_2/I_6$-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian $H$ and two integrals ${\cal I}_{1}, {\cal I}_{2}$, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of $g^{(2)}$ or $g^{(3)}$ (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that $(H, {\cal I}_1, {\cal I}_2, {\cal I}_{12} \equiv [{\cal I}_1, {\cal I}_2])$ are the four generating elements of the {\it quartic} polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra $g^{(3)}$. In turn, 3-body/$A_2$-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly., Comment: 15 pages, typos corrected, editing, final version to be published in J Math Phys

As a continuation of Parts I \cite{Part-1:2020}, II \cite{Part-2:2021}, III \cite{Part-3:2022}, where ultra-compact wave functions were constructed for a few low-lying states of He-like and Li-like sequences, the family of spin-singlet $(1s\,ns)$ type excited states $n\,{}^1 S$ of the He-like sequence is studied with an emphasis on the $n=3,4,5$: $3\,{}^1 S, 4\,{}^1 S, 5\,{}^1 S$ states, for nuclear charges $Z \leq 20$. Particular attention is given to finding of critical charges $Z=Z_B$ at which the ultra-compact wave functions lose their square-integrability. For each ${}^1 S$ state an ultra-compact, seven-parametric trial function is constructed, which describes the domain of applicability of the non-relativistic Quantum Mechanics of Coulomb Charges (QMCC) for the total energies (4-5 significant digits (s.d.)) and reproduces 3 decimal digits (d.d.) of the spin-singlet states $n\,{}^1 S$ of He-like ions (in the static approximation with point-like, infinitely heavy nuclei) for $n=1,2,3,\ldots$ and any $Z \leq 20$\,. All energies are well described by second degree polynomials in $Z$ (the Majorana formula). Critical charges $Z=Z_B^{(n)}$, where the ultra-compact trial function for the $n^1 S, n=1,2,3,\ldots$ states loses its square-integrability, are estimated: for all studied states $Z_B^{(n)}$ increases slowly with $n$; it seems they lie in the interval $Z_B(n^1 S) \sim 0.90 - 0.95$, in particular, with $Z_B^{(1)}=Z_B^{(2)}\,=\,0.904$, $Z_B^{(3)}=Z_B^{(4)}\,=\,0.928$, $Z_B^{(5)}\ =\ 0.939$., Comment: 31 pages, 5 figures, 12 Tables; dedicated to the memory of Frank E Harris; extended, typos fixed, accepted for publication at Advances in Quantum Chemistry

As a continuation of Part I, dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z \leq 20$, and Part II, dedicated to two excited states of He-like sequence, two ultra-compact wave functions in the form of generalized Guevara-Harris-Turbiner functions are constructed for Li-like sequence. They describe accurately the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (2-3 significant digits (s.d.)) of the spin-quartet state $1^40^+$ of Li-like ions (in static approximation with point-like, infinitely heavy nuclei). Variational parameters are fitted in $Z$ by 2nd degree polynomials. The most accurate ultra-compact function leads to the absolute accuracy $\sim 10^{-3}$\,a.u. for energy, and $\sim 10^{-4}$ for the normalized electron-nuclear cusp parameter for $Z \leq 20$. Critical charge $Z=Z_B$, where the ultra-compact trial function for the $1^40^+$ state looses its square-integrability, is estimated, $Z_B(1^4\,0^+) \sim 1.26 - 1.30$. As a complement to Part I, square integrability for the compact functions constructed for the {\it ground, spin-doublet state} $1^2\,0^+$ of the Li-like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as $Z_B( 1^2\,0^+) = 1.62 - 1.65$. It implies that at $Z=2$ - the negative helium ion He${}^-$ - both states $1^2\,0^+$ and $1^4\,0^+$ exist as states embedded to continuum., Comment: 29 pages, 5 figures, 2 tables

We describe the emission, detection and structure of multiphoton states of light. We include the effect of frequency filtering, which describes, at a fundamental level, physical detection of a quantum emitter. The case of the spontaneous emission of Fock states is treated fully and analytically. We stress this picture by contrasting it to the numerical simulation of two-photon bundles emitted from a two-level system in a cavity. We show that dynamical factors exist that allow for a more robust multiphoton emission than spontaneous emission. We also describe how this relates to thermal light., Comment: Replacement of the original file, includes the complete supplementary material. 25 pages, 8 figures in total (6 pages, 2 figures for the main text and 19 pages, 6 figures for supplementary material)

As a continuation of Part I \cite{Part-1:2020} (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z \leq 20$, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for the energies (4-5 significant digits (s.d.)) of two excited states of He-like ions: the spin-singlet (first) excited state $2^1 S$ and the lowest spin-triplet $1^3 S$ state. For both states it provides absolute accuracy for energy $\sim 10^{-3}$\,a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy $\sim 10^{-2}$. The Bressanini-Reynolds observation about the special form of the nodal surface of the $2^1 S$ state of Helium is confirmed and extended to He-like ions with $Z > 2$. Critical charges $Z=Z_B$, where ultra-compact trial functions lose their square-integrability, are estimated: $Z_B(1^1 S)\approx Z_B(2^1 S)\sim 0.905$ and $Z_B(1^3 S)\sim 0.902$. For both states the Majorana formula - the energy as a second degree polynomial in $Z$ - provides accurately 4-5 significant digits for $Z \leq 20$., Comment: 27 pages, 6 figures, 4 tables: typos fixed, 8 extra references added, the text essentially extended, to be published at Int Journ Quantum Chemistry

A compact, few-parametric, physically adequate, 3-term variational trial function is used to calculate with high accuracy the energy of the ground state ${}^3\Pi_u$ of the hydrogen molecule ${\rm H}_2$ in strong magnetic field ${\bf B}$ in the range $5\times10^{10}\, {\rm G} \leq B \leq 10^{13}\,$G. The nuclei (protons) are assumed as infinitely massive (BO appproximation of zero order) and situated along the magnetic field line (parallel configuration).

Several ultra-compact accurate wave functions in the form of generalized Hylleraas-Kinoshita functions and Guevara-Harris-Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), or, equivalently, the Non-Relativistic QED (NRQED), for the ground state energies (4-5 significant digits (s.d.)) of He-like and Li-like iso-electronic sequences in the static approximation with point-like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in $Z$ by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He-like two-electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy $\sim 10^{-3}$\,a.u. and the same relative accuracies $\sim 10^{-2}-10^{-3}$ for both the cusp parameters and the six expectation values, is found. For the Li-like three-electron sequence the most accurate ultra-compact function taken as the variational trial function provides absolute accuracy for energy $\sim 10^{-3}$\,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for $Z \leq 20$ and 3 s.d. for the two expectation values for $Z=3$., Comment: post-publication, extended version, 43 pages, 4 figures, 7 tables: perturbation theory view added, Tables dramatically extended, Appendix with interpolated parameters vs Z added, published at IJQC (early view)

We study a prototypical model of two coupled two-level systems, where the competition between coherent and dissipative coupling gives rise to a rich phenomenology. In particular, we analyze the case of asymmetric coupling, as well as the limiting case of chiral (or one-way) coupling. We investigate various quantum optical properties of the system, including its steady state populations, power spectrum, and second-order correlation functions, and outline the characteristic features which emerge in each quantity as one sweeps through the nontrivial landscape of effective complex couplings. Most importantly, we reveal instances of population trapping, unexpected spectral features and strong photon correlations., Comment: 22 pages, 16 figures

Highly accurate variational calculations, based on a few-parameter, physically adequate trial function, are carried out for the hydrogen molecule \hh in inclined configuration, where the molecular axis forms an angle $\theta$ with respect to the direction of a uniform constant magnetic field ${\bf B}$, for $B=0,\, 0.1,\, 0.175$ and $0.2\,$a.u. Three inclinations $\theta=0^\circ,\,45^\circ,\,90^\circ$ are studied in detail with emphasis to the ground state $1_g$. Diamagnetic and paramagnetic susceptibilities are calculated (for $\theta=45^\circ$ for the first time), they are in agreement with the experimental data and with other calculations. For $B=0,\, 0.1$ and $0.2\,$a.u. potential energy curves $E$ vs $R$ are built for each inclination, they are interpolated by simple, two-point Pad\'e approximant $Pade[2/6](R)$ with accuracy of not less than 4 significant digits. Spectra of rovibrational states are calculated for the first time. It was found that the optimal configuration of the ground state for $B \leq B_{cr}=0.178\,$a.u. corresponds always to the parallel configuration, $\theta=0$, thus, it is a $^1\Sigma_g$ state. The state $1_g$ remains bound for any magnetic field, becoming metastable for $B > B_{cr}$, while for $B_{cr} < B < 12$\,a.u. the ground state corresponds to two isolated hydrogen atoms with parallel spins., Comment: 31 pages, 11 Tables, 7 Figures (2 new), following referee's suggestions parts 4,5,6 essentially rewritten, to be published at Journal of Quantitative Spectroscopy and Radiative Transfer

The existence and stability of the linear hydrogenic chain H$_3$ and H${}_2^-$ in a strong magnetic field is established. Variational calculations for H$_3$ and H${}_2^-$ are carried out in magnetic fields in the range $10^{11}\leq B \leq 10^{13}\,$G with 17-parametric (13-parametric for H${}_2^-$), physically adequate trial function. Protons are assumed infinitely massive, fixed along the magnetic line. States with total spin projection $S_z=-3/2$ and magnetic quantum numbers $M=-3,-4,-5$ are studied. It is shown that for both H$_3$ and H${}_2^-$ the lowest energy state corresponds to $M=-3$ in the whole range of magnetic fields studied. As for a magnetic field $B \gtrsim 10^{11}\,$G both H$_3$ and H${}_2^-$ exist as metastable states, becoming stable for $B \geq 1.9 \times 10^{11}\,$G and for $B \geq 2.7 \times 10^{11}\,$G, respectively. The excited states $^4(-4)^+$, $^4(-5)^+$ of ${\rm H}_3$ and H${}_2^-$ appear at magnetic fields $B > 7 \times 10^{11}$ and $10^{12}$\,G, respectively., Comment: 4 pages, 1 Figure, 4 Tables, the second half of the paper rewritten, Table split into four Tables, results on first two excited states added, estimate on domain of stability H$_4$ molecule given

We show how the photon statistics emitted by a large variety of light-matter systems under weak coherent driving can be understood, to lowest order in the driving, in the framework of an admixture of (or interference between) a squeezed state and a coherent state, with the resulting state accounting for all bunching and antibunching features. One can further identify two mechanisms that produce resonances for the photon correlations: i) conventional statistics describes cases that involve a particular quantum level or set of levels in the excitation/emission processes with interferences occurring to all orders in the photon numbers, while unconventional statistics describes cases where the driving laser is far from resonance with any level and the interference occurs for a particular number of photons only, yielding stronger correlations but only for a definite number of photons. Such an understanding and classification allows for a comprehensive and transparent description of the photon statistics from a wide range of disparate systems, where optimum conditions for various types of photon correlations can be found and realized., Comment: Comments and links to further relevant literature are welcome

We present a combined classical and quantum electrodynamics description of the coupling between two circularly-polarized quantum emitters held above a metal surface supporting surface plasmons. Depending on their position and their natural frequency, the emitter-emitter interactions evolve from being reciprocal to nonreciprocal, which makes the system a highly tunable platform for chiral coupling at the nanoscale. By relaxing the stringent material and geometrical constraints for chirality, we explore the interplay between coherent and dissipative coupling mechanisms in the system. Thus, we reveal a quasi-chiral regime in which its quantum optical properties are governed by its subradiant state, giving rise to extremely sharp spectral features and strong photon correlations., Comment: 6 pages, 4 figures

Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are considered. We present all real standard potentials that allow separation of variables in polar coordinates and admit an independent fourth-order integral of motion. The general form of the potentials satisfies a linear ODE. In the classical case, the standard potentials coincide with the Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models. In the quantum case new superintegrable systems are obtained, in addition to the TTW and PW ones. Their classical limit is free motion., Comment: 30 pages

We present all real quantum mechanical potentials in a two-dimensional Euclidean space that have the following properties: 1. They allow separation of variables of the Schr\"odinger equation in polar coordinates, 2. They allow an independent fourth order integral of motion, 3. It turns out that their angular dependent part $S(\theta)$ does not satisfy any linear differential equation. In this case it satisfies a nonlinear ODE that has the Painlev\'e property and its solutions can be expressed in terms of the Painlev\'e transcendent $P_6$. We also study the corresponding classical analogs of these potentials. The polynomial algebra of the integrals of motion is constructed in the classical case., Comment: 43 pages

We study the question about existence i.e. stability with respect to dissociation of the spin-quartet, permutation- and reflection-symmetric ${}^4(-3)^+_g$ ($S_z=-3/2, M=-3$) state of the $(\alpha\alpha e e e)$ Coulomb system: the ${\rm He}_2^+$ molecular ion, placed in a magnetic field $0 \le B \le 10000$ a.u. We assume that the $\alpha$-particles are infinitely massive (Born-Oppenheimer approximation of zero order) and adopt the parallel configuration, when the molecular axis and the magnetic field direction coincide, as the optimal configuration. The study of the stability is performed variationally with a physically adequate trial function. To achieve this goal, we explore several Helium-contained compounds in strong magnetic fields, in particular, we study the spin-quartet ground state of ${\rm He}^-$ ion, and the ground (spin-triplet) state of the Helium atom, both for a magnetic field in $100 \leq B\leq 10000$ a.u. The main result is that the ${\rm He}_2^+$ molecular ion in the state ${}^4(-3)^+_g$ is stable towards all possible decay modes for magnetic fields $B \gtrsim 120$ a.u. and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At $B=1000$ a.u., the dissociation energy of ${\rm He}_2^+$ into ${\rm He}^- + \alpha$ is $\sim 701.8$ eV and the dissociation energy for the decay channel to ${\rm He} + \alpha + e $ is $\sim 729.1$ eV, latter both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209., Comment: LaTeX revtex4 BibTeX, 34 pages, 7 Tables and 5 Postscript Figures. Typos corrected. Several clarifying sentences were added. List of References was updated. 3 references were removed, 9 new references were added. Some sections were renamed and Section III was added

We study the excitation of a two-level system (2LS) by quantum light, thereby bringing our previous studies~(see part~I. of this series) to a target that is quantum itself. While there is no gain for the quantum state of the target as compared to driving it with classical light, its dynamical features, such as antibunching, can be improved. We propose a chain of two-level systems, i.e., setting the emission of each 2LS as the driving source of the following one, as an arrangement to provide better single-photon sources. At a fundamental level, we discuss the notion of strong-coupling between quantum light from a source and its target, and the several versions of the Mollow triplet that follow from various types of driving light. We discuss the Heitler effect of antibunched photons from the scattered light off a laser., Comment: 21 pages, 12 figures

We map the Hilbert space of the quantum Harmonic oscillator to the space of Glauber's $n$th-order intensity correlators, in effect showing "the correlations between the correlators" for a random sampling of the quantum states. In particular, we show how the popular $g^{(2)}$ function is correlated to the mean population and how a recurrent criterion to identify single-particle states or emitters, namely $g^{(2)}<1/2$, is incorrect as states exist that satisfy this condition with average population larger than one. Our charting of the Hilbert space allows to capture its structure in a simpler and physically more intuitive way that can be used to classify quantum sources by surveying which territory they can access., Comment: 12 pages, 5 figures

We start a series of studies of the excitation of an optical target by quantum light. In this first part, we introduce the problematic and address the first case of interest, that of exciting the quantum harmonic oscillator, corresponding to, e.g., a single-mode passive cavity or a non-interacting bosonic field. We introduce a mapping of the Hilbert space that allows to chart usefully the accessible regions. We then consider the quantum excitation from single photon sources in the form of a two-level system under various regimes of (classical) pumping: incoherent, coherent and in the Mollow triplet regime. We close this first opus with an overview of the material to be covered in the subsequent papers., Comment: 15 pages, 10 figures

The second (unphysical) critical charge in the 3-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by F~Stillinger has been calculated to be equal to $Z_{B}^{\infty}\ =\ 0.904854$ and $Z_{B}^{m_p}\ =\ 0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $Re Z < Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^-$ is predicted to be at -0.51554 a.u. (-0.51531 a.u. for the finite proton mass $m_p$). The first critical charge $Z_c$ is found accurately for a finite proton mass $m_p$ in the Lagrange mesh method, $Z^{m_p}_{c}\ =\ 0.911\, 069\, 724\, 655$., Comment: 12 pages, 1 figure, 3 tables: title changed and Figure modified, several explanatory sentences added, text improved for better understanding, some typos fixed, to be published at Mod Phys Lett A

We discuss the excitation of polaritons---strongly-coupled states of light and matter---by quantum light, instead of the usual laser or thermal excitation. As one illustration of the new horizons thus opened, we introduce Mollow spectroscopy, a theoretical concept for a spectroscopic technique that consists in scanning the output of resonance fluorescence onto an optical target, from which weak nonlinearities can be read with high precision even in strongly dissipative environments., Comment: 5 pages, 3 figures

We report the experimental observation and control of space and time-resolved light-matter Rabi oscillations in a microcavity. Our setup precision and the system coherence are so high that coherent control can be implemented with amplification or switching off of the oscillations and even erasing of the polariton density by optical pulses. The data is reproduced by a fundamental quantum optical model with excellent accuracy, providing new insights on the key components that rule the polariton dynamics., Comment: 5 pages, 3 figures, supplementary 7 pages, 4 figures. Supplementary videos: https://drive.google.com/folderview?id=0B0QCllnLqdyBNjlMLTdjZlNhbTQ&usp=sharing

Recently, the singular value decomposition (SVD) was applied to standard Gaussian ensembles of Random Matrix Theory (RMT) to determine the scale invariance in the spectral fluctuations without performing any unfolding procedure. Here, SVD is applied directly to the $\nu$-Hermite ensemble and to a sparse matrix ensemble, decomposing the corresponding spectra in trend and fluctuation modes. In correspondence with known results, we obtain that fluctuation modes exhibit a cross-over between soft and rigid behavior. By using the trend modes we performed a data-adaptive unfolding, and we calculate traditional spectral fluctuation measures. Additionally, ensemble-averaged and individual-spectrum averaged statistics are calculated consistently within the same basis of normal modes., Comment: 62 figures

Algebraic-rational nature of the four-dimensional, $F_4$-invariant integrable quantum Hamiltonians, both rational and trigonometric, is revealed and reviewed. It was shown that being written in $F_4$ Weyl invariants, polynomial and exponential, respectively, both similarity-transformed Hamiltonians are in algebraic form, they are quite similar the second order differential operators with polynomial coefficients; the flat metric in the Laplace-Beltrami operator has polynomial (in invariants) matrix elements. Their potentials are calculated for the first time: they are meromorphic (rational) functions with singularities at the boundaries of the configuration space. Ground state eigenfunctions are algebraic functions in a form of polynomials in some degrees. Both Hamiltonians preserve the same infinite flag of polynomial spaces with characteristic vector $(1, 2, 2, 3)$, it manifests exact solvability. A particular integral common for both models is derived. The first polynomial eigenfunctions are presented explicitly., Comment: 19 pages, Sec.2 rewritten, typos corrected, to be published in Journal of Physics (conference series), talk presented at QTS-8, Mexico-City, August 2014

The $1/Z$-expansion for the Coulomb system of infinitely massive center of charge Z and two electrons is discussed. Numerical deficiency in Baker et al, {\em Phys. Rev. \bf A41}, 1247 (1990) is indicated which continue to raise doubts in correctness of their calculations of the higher order coefficients in $1/Z$-expansion expressed in Refs.[4-5]. It is shown that a minor modification of a few first coefficients found in Ref.[3] allows to calculate the ground state energies at $Z\ =\ 1, 2,\ldots \ 10$ (as well as at $Z > 10$) with a portion of 15th decimal digit in comparison with highly accurate calculations by C. Schwartz and by Nakashima-Nakatsuji. Ground state energies of two-electron ions $Z=11\ (Na^{9+})$ and $Z=12\ (Mg^{10+})$ are found with 14 decimal digits., Comment: 7 pages, 2 Tables, two extra references added, Conclusions extended, two "Note added.I-II." added

The existence of bound states of the system $(\al,e,e,e)$ in a magnetic field $B$ is studied using the variational method. It is shown that for $B \gtrsim 0.13\,{\rm a.u.}$ this system gets bound with total energy below the one of the $(\al,e,e)$ system. It manifests the existence of the stable He$^-$ atomic ion. Its ground state is a spin-doublet $^2(-1)^{+}$ at $0.74\, {\rm a.u.} \gtrsim B \gtrsim 0.13\, {\rm a.u.}$ and it becomes a spin-quartet $^4(-3)^{+}$ for larger magnetic fields. For $0.8\, {\rm a.u.} \gtrsim B \gtrsim 0.7\, {\rm a.u.}$ the He$^-$ ion has two (stable) bound states $^2(-1)^{+}$ and $^4(-3)^{+}$., Comment: 4 pages, 1 figure, 2 tables, one reference added, typos corrected, small modifications done, to be published at Phys Rev Letters

Five physics mechanisms of interaction leading to the binding of the ${\rm H}_3^+$ molecular ion are identified. They are realized in a form of variational trial functions and their respective total energies are calculated. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer (BO) ground state energy among (two-three-seven)-parametric trial functions being correspondingly, H$_2$-molecule plus proton (two variational parameters), H$_2^+$-ion plus H-atom (three variational parameters) and generalized Guillemin-Zener (seven variational parameters). These trial functions are chosen following a criterion of physical adequacy. They include the electronic correlation in the exponential form $\sim\exp{(\gamma r_{12})}$, where $\gamma$ is a variational parameter. Superpositions of two different mechanisms of binding are investigated and a particular one, which is a generalized Guillemin-Zener plus H$_2$-molecule plus proton (ten variational parameters), provides the total energy at the equilibrium of $E=-1.3432$\ a.u. The superposition of three mechanisms: generalized Guillemin-Zener plus (H$_2$ -molecule plus proton) plus (H$_2^+$ -ion plus H) (fourteen parameters) leads to the total energy which deviates from the best known BO energy to $\sim 0.0004$\ a.u., {\it it reproduces two-three significant digits in exact, non-BO total energy}. In general, our variational energy agrees in two-three-four significant digits with the most accurate results available at present as well as major expectation values., Comment: 26 pages, 2 figures, 4 tables, invited contribution to Takeshi Oka Festschrift: Celebrating 45 Years of Astrochemistry, accepted to Jour Phys Chem typos corrected, Introduction and Conclusions are extended

Three simple $7-, (7+3)-, 10-$parametric trial functions for the ${\rm H}_3^+$ molecular ion are presented. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer ground state energy among several-parametric trial functions. These trial functions are chosen following a criterion of physical adequacy and includes the electronic correlation in the exponential form $\sim\exp{(\gamma r_{12})}$, where $\gamma$ is a variational parameter. The Born-Oppenheimer energy is found to be $E=-1.340 34, -1.340 73, -1.341 59$\,a.u., respectively, for optimal equilateral triangular configuration of protons with the equilibrium interproton distance $R=1.65$\,a.u. The variational energy agrees in three significant digits (s.d.) with most accurate results available at present as well as for major expectation values., Comment: 12 pages, 1 figure, 3 tables

It is shown that the $E_8$ trigonometric Olshanetsky-Perelomov Hamiltonian, when written in terms of the Fundamental Trigonometric Invariants (FTI), is in algebraic form, i.e., has polynomial coefficients, and preserves two infinite flags of polynomial spaces marked by the Weyl (co)-vector and $E_8$ highest root (both in the basis of simple roots) as characteristic vectors. The explicit form of the Hamiltonian in new variables has been obtained both by direct calculation and by means of the orbit function technique. It is shown a triangularity of the Hamiltonian in the bases of orbit functions and of algebraic monomials ordered through Weyl heights. Examples of first eigenfunctions are presented., Comment: 39 pages, 3 Tables, 3 Appendices, no figures

The one-electron Li-containing Coulomb systems of atomic type $(li, e)$ and molecular type $(li, li, e)$, $(li, \alpha, e)$ and $(li, p, e)$ are studied in the presence of a strong magnetic field $B \leq 10^{7}$ a.u. in the non-relativistic framework. They are considered at the Born-Oppenheimer approximation of zero order (infinitely massive centers) within the parallel configuration (molecular axis parallel to the magnetic field). The variational and Lagrange-mesh methods are employed in complement to each other. It is demonstrated that the molecular systems ${\rm LiH}^{3+}$, ${\rm LiHe}^{4+}$ and ${\rm Li}_{2}^{5+}$ can exist for sufficiently strong magnetic fields $B \gtrsim 10^{4}$ a.u. and that ${\rm Li}_{2}^{5+}$ can even be stable at magnetic fields typical of magnetars., Comment: 22 pages, 9 figures, 4 tables

A non-relativistic classification of charged molecular hydrogenic, helium and mixed helium-hydrogenic chains with one or two electrons which can exist in a strong magnetic field $B \lesssim 10^{16} $G is given. It is shown that for both $1e-2e$ cases at the strongest studied magnetic fields the longest hydrogenic chain contains at most five protons indicating to the existence of the $\rm{H}_5^{4+}$ and $\rm{H}_5^{3+}$ ions, respectively. In the case of the helium chains the longest chains can exist at the strongest studied magnetic fields with three and four $\al-$particles for $1e-2e$ cases, respectively. For mixed helium-hydrogenic chains the number of heavy centers can reach five for highest magnetic fields studied. In general, for a fixed magnetic field two-electron chains are more bound than one-electron ones., Comment: 32 pages, 2 figures, 9 tables

The Garvey-Kelson relations are used in an iterative process to predict nuclear masses in the neighborhood of nuclei with measured masses. Average errors in the predicted masses for the first three iteration shells are smaller than those obtained with the best nuclear mass models. Their quality is comparable with the Audi-Wapstra extrapolations, offering a simple and reproducible procedure for short range mass predictions. A systematic study of the way the error grows as a function of the iteration and the distance to the known masses region, shows that a correlation exists between the error and the residual neutron-proton interaction, produced mainly by the implicit assumption that $V_{np}$ varies smoothly along the nuclear landscape., Comment: 10 pages, 18 figures

It is shown that the $E_7$ trigonometric Olshanetsky-Perelomov Hamiltonian, when written in terms of the Fundamental Trigonometric Invariants (FTI), is in algebraic form, i.e., has polynomial coefficients, and preserves the infinite flag of polynomial spaces with the characteristic vector $\vec \alpha = (1,2,2,2,3,3,4)$. Its flag coincides with one of the minimal characteristic vector for the $E_7$ rational model., Comment: 11 pages, LaTeX

It is conjectured that any trigonometric Olshanetsky-Perelomov Hamiltonian written in Fundamental Trigonometric Invariants (FTI) as coordinates takes an algebraic form and preserves an infinite flag of spaces of polynomials. It is shown that try-and-guess variables which led to the algebraic form of trigonometric Olshanetsky-Perelomov Hamiltonians related to root spaces of the classical $A_N, B_N, C_N, D_N, BC_N$ and exceptional $G_2, F_4$ Lie algebras are FTI. This conjecture is also confirmed for the trigonometric $E_6$ Olshanetsky-Perelomov Hamiltonian whose algebraic form is found with the use of FTI., Comment: 17 pages, to appear in Contemporary Mathematics

A detailed study of the lowest states $1s_0, 2p_{-1}, 2p_0$ of the hydrogen atom placed in a magnetic field $B\in(0-4.414\times 10^{13} {\rm G})$ and their electromagnetic transitions ($1s_{0} \leftrightarrow 2p_{-1}$ and $ 1s_{0} \leftrightarrow 2p_{0}$) is carried out in the Born Oppenheimer approximation. The variational method is used with a physically motivated recipe to design simple trial functions applicable to the whole domain of magnetic fields. We show that the proposed functions yield very accurate results for the ionization (binding) energies. Dipole and oscillator strengths are in good agreement with results by Ruder {\em et al.} \cite{Ruderbook} although we observe deviations up to $\sim 30%$ for the oscillator strength of the (linearly polarized) electromagnetic transition $1s_{0} \leftrightarrow 2p_{0}$ at strong magnetic fields $B\gtrsim 1000$ a.u., Comment: RevTeX4, 18 pages, 5 Tables, 5 Figures

A first detailed study of the ground state of the H$_3^+$ molecular ion in linear configuration, parallel to a magnetic field direction, and its low-lying $\Si,\Pi,\De$ states is carried out for magnetic fields $B=0-4.414 \times 10^{13} $G in the Born-Oppenheimer approximation. The variational method is employed with a single trial function which includes electronic correlation in the form $\exp{(\ga r_{12})}$, where $\ga$ is a variational parameter. It is shown that the quantum numbers of the state of the lowest total energy (ground state) depend on the magnetic field strength. The ground state evolves from the spin-singlet ${}^1\Si_g$ state for weak magnetic fields $B \lesssim 5 \times 10^{8} $G to a weakly-bound spin-triplet ${}^3\Si_u$ state for intermediate fields and, eventually, to a spin-triplet $^3\Pi_u$ state for $5 \times 10^{10} \lesssim B \lesssim 4.414 \times 10^{13} $G. Local stability of the linear parallel configuration with respect to possible small deviations is checked., Comment: 27 pages, 6 figures, 12 tables (the text expanded, several new references added, typos corrected)

Easy physics-inspired approximations of the total and binding energies for the ${\rm H}$ atom and for the molecular ions $${\rm H}_2^{(+)} ({\rm ppe}), {\rm H}_3^{(2+)} ({\rm pppe}), ({\rm HeH})^{++} (\al {\rm p e}), {\rm He}_2^{(3+)} (\al \al {\rm e})$$ as well as quadrupole moment for the ${\rm H}$ atom and the equilibrium distances of the molecular ions in strong magnetic fields $> 10^{9}$ G are proposed. The idea of approximation is based on the assumption that the dynamics of the one-electron Coulomb system in a strong magnetic field is governed by the ratio of transverse to longitudinal sizes of the electronic cloud., Comment: 22 Pages, 22 Figures, 4 Tables, Invited Contribution to appear in Collection of Czechoslovak Chemical Communications, Special Issue in honor of Professor Josef Paldus

The Coulombic systems $(\al p e)$ and $(\al\al e)$, $(\al p p e)$, $(\al \al p e)$ and $(Li^{3+} Li^{3+} e)$ placed in a magnetic field $B \gtrsim 10^{11} {G}$ are studied. It is demonstrated a theoretical existence of the exotic ion $(He H)^{2+}$ for $B\gtrsim 5\times 10^{12} {G}$ in parallel configuration (the magnetic field is directed along internuclear axis) as optimal as well as its excited states $1\pi, 1\delta$. As for the exotic ion ${He}_{2}^{3+}$ it is shown that in spite of strong electrostatic repulsion of $\al-$particles this ion can also exist for $B \gtrsim 100 {a.u.} (= 2.35\times 10^{11} {G})$ in parallel configuration as optimal in the states $1\si_g$ (ground state), $1\pi_u, 1\delta_g$. Upon appearance both ions are unstable towards dissociation with $He^+$ in the final state but with very large lifetime. However, at $B\gtrsim 10000$ a.u. the ion $(He H)^{2+}$ becomes stable, while at $B\gtrsim 1000$ a.u. the ion ${He}_{2}^{3+}$ becomes stable. With a magnetic field growth, both exotic ions become more and more tightly bound and compact, their lowest rotational and vibrational energies grow. At the edge of applicability of non-relativistic approximation, $B \sim 4.414 \times 10^{13}$ G, there are indications that three more exotic linear ions $(H-He-H)^{3+}$, $(He-H-He)^{4+}$ and even $Li_2^{5+}$ in parallel configuration may also occur., Comment: 24 pages, 11 Figures, 7 Tables, revtex4 (preprint single column format). The text is slightly modified. To Table 4 a complete set of the lowest rotational energies for the ${He}_{2}^{3+}$ is added. Also a behavior of the rotational and vibrational energies viz magnetic field is briefly discussed

An accurate study of the lowest $1\sigma_g$ and the low-lying excited $1\sigma_u$, $1\pi_{u,g}$, $1\delta_{g,u}$ electronic states of the exotic molecular ion $H_3^{2+}$ in linear configuration parallel to a magnetic field is carried out. The magnetic field ranges from $10^{10}$ G up to $4.414 \times 10^{13}$ G where non-relativistic considerations are justified. The variational method is exploited and the {\it same} trial function is used for different magnetic fields. It is shown that the states of positive $z$-parity $1\sigma_g, 1\pi_u, 1\delta_{g}$ are bound states of the $H_3^{2+}$ exotic ion for all magnetic fields studied. We also demonstrate that for magnetic fields $B\gtrsim 2.35\times 10^{12}$ G the potential energy surface well corresponding to the $1\sigma_g$ state contains at least one longitudinal vibrational state. It is also shown that the negative $z$-parity states $1\sigma_u, 1\pi_g, 1\delta_{u}$, are purely repulsive in the whole range of magnetic fields studied, $B=10^{10}- 4.414 \times 10^{13}$ G., Comment: 26 pages, 6 figures, 4 tables

A model of a hydrogenic content of atmosphere of the isolated neutron star 1E1207.4-5209 is proposed. It is based on the assumption that the main component in the atmosphere is the exotic molecular ion $H_3^{2+}$ and that there exists a magnetic field in the range of $(4 \pm 2) \times 10^{14}$ G. Photoionization $H_3^{2+} \rar e + 3p$ and photodissociation $H_3^{2+} \rar H + 2p$ correspond to two absorption features at 0.7 KeV and 1.4 KeV, respectively, discovered by {\it Chandra} observatory (Sanwal et al, 2002). The model predicts one more absorption feature at 80-150 eV corresponding to photodissociation $H_3^{2+} \rar H_2^+ + p$., Comment: 8 pages, 1 figure

As a continuation of our previous work ({\it Phys. Rev. A68, 012504 (2003)}) an accurate study of the lowest $1\si_g$ and the low-lying excited $1\si_u$, $2\si_g$, $1\pi_{u,g}$, $1\de_{g,u}$ electronic states of the molecular ion $H_2^+$ is made. Since the parallel configuration where the molecular axis coincides with the magnetic field direction is optimal, this is the only configuration which is considered. The variational method is applied and the {\it same} trial function is used for different magnetic fields. The magnetic field ranges from $10^9 G$ to $4.414 \times 10^{13} G$ where non-relativistic considerations are justified. Particular attention is paid to the $1\si_u$ state which was studied for an arbitrary inclination. For this state a one-parameter vector potential is used which is then variationally optimized., Comment: 25 pages, 2 figures

A detailed quantitative analysis of the system $(ppe)$ placed in magnetic field ranging from $0 - 4.414 \times 10^{13} G$ is presented. The present study is focused on the question of the existence of the molecular ion $H_2^{+}$ in a magnetic field. As a tool, a variational method with an optimization of the form of the vector potential (optimal gauge fixing) is used. It is shown that in the domain of applicability of the non-relativistic approximation the system $(ppe)$ in the Born-Oppenheimer approximation has a well-pronounced minimum in the total energy at a finite interproton distance for $B \lesssim 10^{11} G$, thus manifesting the existence of $H_2^{+}$. For $B \gtrsim 10^{11} G$ and large inclinations (of the molecular axis with respect to the magnetic line) the minimum disappears and hence the molecular ion $H_2^{+}$ does not exist. It is shown that the most stable configuration of $H_2^{+}$ always corresponds to protons situated along the magnetic line. With magnetic field growth the ion $H_2^{+}$ becomes more and more tightly bound and compact, and the electronic distribution evolves from a two-peak to a one-peak pattern. The domain of inclinations where the $H_2^{+}$ ion exists reduces with magnetic field increase and finally becomes $0^o - 25^o$ at $B = 4.414 \times 10^{13} G$. Phase transition type behavior of variational parameters for some interproton distances related to the beginning of the chemical reaction $H_2^{+} \leftrightarrow H + p$ is found., Comment: RevTeX 4, 36 pages, 19 Figs. (figures 14-18 should be requested from the authors)