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Bose-Einstein condensates in a double-well potential contain the essential ingredients to study many-body systems within a rich classical phase-space that includes an unstable point and a separatrix. Employing a selfconsistent finite difference method, we study some of their quantum properties and their dependency on the strength of the boson-boson interaction. We observe a deviation in the critical parameters associated with a behavior change in both the energy distribution and the eigenstates of the system. We also examine the trends of the nonclassicality via the Wigner function, the tunneling transmission coefficient, and the nonorthogonality of eigenstates associated with the nonlinearity aspects of the Gross-Pitaevskii equation.

Gaussian potentials serve as a valuable tool for the comprehensive modeling of short-range interactions, spanning applications from nuclear physics to the artificial confinement of electrons within quantum dots. This study focuses on examining the lowest states within Gaussian wells, with particular emphasis on the weakly bound regime. The analysis delves into the asymptotic behavior of the exact wave function at both small and large distances, motivating the development of a few-parametric Ansatz which is locally accurate and yields to a fast convergent basis set. To validate its efficacy, we assess its convergence rate using a toy model of Nuclear Physics, specifically for Deuteron. Furthermore, we employ the expansion of the energy close to the threshold to derive an analytical formula for the binding energy of the deuteron whose accuracy improves as the effective parameter approaches the critical. In concluding our investigation, we evaluate the performance of our Ansatz as an orbital in the exploration of the electronic structure of a two-electron quantum dot.

A compact yet accurate approach for representing the wave functions of members of the He and Li isoelectronic series is using explicitly correlated wave functions. These wave functions, however, often have nonlinear forms, which make them challenging to optimize. In this work, we show how Variational Monte Carlo (VMC) can efficiently optimize explicitly correlated wave functions that accurately describe the quartet 1s2s2p state of the Li isoelectronic sequence with ten or fewer parameters. We find that our compact wave functions correctly describe cusp conditions and reproduce at least 99.9% percent of the exact energy.

The quasi-energy spectrum recently measured in experiments with a squeeze-driven superconducting Kerr oscillator showed good agreement with the energy spectrum of its corresponding static effective Hamiltonian. The experiments also demonstrated that the dynamics of low-energy states can be explained with the same emergent static effective model. The spectrum exhibits real (avoided) level crossings for specific values of the Hamiltonian parameters, which can then be chosen to suppress (enhance) quantum tunneling. Here, we analyze the spectrum and the dynamics of the effective model up to high energies, which should soon be within experimental reach. We show that the parameters values for the crossings, which can be obtained from a semiclassical approach, can also be identified directly from the dynamics. Our analysis of quantum tunneling is done with the effective flux of the Husimi volume of the evolved states between different regions of the phase space. Both initial coherent states and quench dynamics are considered. We argue that the level crossings and their consequences to the dynamics are typical to any quantum system with one degree of freedom, whose density of states presents a local logarithmic divergence and a local step discontinuity., Comment: 13 pages, 8 figures

Within the Bohr's correspondence principle, the quantum theory should reproduce the classical world when $\hbar\to 0$. In practice, the discrete energies come close to each other before passing to continuum, causing that highly excited states become inaccessible. However it is possible to identify signatures that the spectrum is approaching the continuum limit. In this work, for $\hbar \to 0$, we focus our attention on the spectrum of the 1D quartic double well and find signatures of the classical instability in the quantum spectrum near the critical energy, and explore how the tunneling phenomenon remains only at energies close to this critical energy.

Assuming a logarithmic interaction between constituent particles, compact and locally accurate wave functions that describe bound states of the two-particle neutral and three-particle charged complexes in two dimensions are designed. Prime examples of these complexes are excitons and trions that appear in monolayers of Transition-Metal DichalCogenides (TMDCs). In the case of excitons, these wave functions led to 5-6 correct decimal digits in the energy and the diamagnetic shifts. In addition, it is demonstrated that they can be used as zero-order approximations to study magnetoexcitons via perturbation theory in powers of the magnetic field strength. For the trion, making a comparison with experimental data for concrete TMDCs, we established that the logarithmic potential leads to binding energies $\lesssim 25\%$ greater than experimental ones. Finally, the structure of the wave function at small distances is established for excitons whose carriers interact via the Rytova-Keldysh potential., Comment: 21 pages, 4 figures

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

In this paper we study the effects of a topological Weyl semimetal (WSM) upon the ground state and polarization of an hydrogen-like atom near its surface. The WSM is assumed to be in the equilibrium state and at the neutrality point, such that the interaction between the atomic charges and the material is fully described (in the non retarded regime) by axion electrodynamics, which is an experimentally observable signature of the anomalous Hall effect in the bulk of the WSM. The atom-WSM interaction provides additional contributions to the Casimir-Polder potential thus modifying the energy spectra and wave function, which now became distance dependent. Using variational methods, we solve the corresponding Schr\"{o}dinger equation for the atomic electron. The ground state and the polarization are analyzed as a function of the atom-surface distance, and we directly observe the effects of the nontrivial topology of the material by comparing our results with that of a topologically trivial sample. We also study the impact of the medium's permittivity by assuming a hydrogen atom in vacuum, and a donor impurity in the semiconductors gallium arsenide (GaAs) and gallium phosphide (GaP). We found that the topological interaction behaves as an effective-attractive charge so that the electronic cloud tends to be polarized to the interface of materials. Moreover, the loss of wave-function normalization is interpreted as a critical location from below which the bound state is broken., Comment: 17 pages, 14 figures. Material systems and parameter changed. Discussion improved. Data repository at https://github.com/djuliannader/WSMs-H

Using the Wherl entropy, we study the delocalization in phase-space of energy eigenstates in the vicinity of avoided crossing in the Lipkin-Meshkov-Glick model. These avoided crossing, appearing at intermediate energies in a certain parameter region of the model, originate classically from pairs of trajectories lying in different phase space regions, which contrary to the low energy regime, are not connected by the discrete parity symmetry of the model. As coupling parameters are varied, a sudden increase of the Wherl entropy is observed for eigenstates close to the critical energy of the excited-state quantum phase transition (ESQPT). This allows to detect when an avoided crossing is accompanied by a superposition of the pair of classical trajectories in the Husimi functions of eigenstates. This superposition yields an enhancement of dynamical tunneling, which is observed by considering initial Bloch states that evolve partially into the partner region of the paired classical trajectories, thus breaking the quantum-classical correspondence in the evolution of observables.

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)

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

Using three different approaches, Perturbation Theory (PT), the Lagrange Mesh Method (Lag-Mesh) and the Variational Method (VM), we study the low-lying states of the Yukawa potential $V(r)=-(\lambda/r)e^{-\alpha r}\,$. First orders in PT in powers of $\alpha$ are calculated in the framework of the Non-Linerization Procedure. It is found that the Pad\'e approximants to PT series together with the Lag-Mesh provide highly accurate values of the energy and the positions of the radial nodes of the wave function. The most accurate results, at present, of the critical screening parameters ($\alpha_c$) for some low-lying states and the first coefficients in the expansion of the energy at $\alpha_c$ are presented. A locally-accurate and compact approximation for the eigenfunctions of the low-lying states for any $r\in [ 0,\infty)$ is discovered. This approximation used as a trial function in VM eventually leads to energies as precise as those of PT and Lag-Mesh. Finally, a compact analytical expression for the energy as a function of $\alpha$, that reproduce at least $6$ decimal digits in the entire physical range of $\alpha$, is found., Comment: 28 pages, 4 figures

This manuscript is in progress and the first version was removed by decision of the autors., Comment: This paper has been withdrawn by the author due to a crucial sign error in equation 17