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In a previous paper [B. Poirier, J. Chem. Phys. 121, 4501 (2004)] a unique bipolar decomposition Ψ=Ψ1+Ψ2 was presented for stationary bound states Ψ of the one-dimensional Schrödinger equation, such that the components Ψ1 and Ψ2 approach their semiclassical WKB analogs in the large-action limit. The corresponding bipolar quantum trajectories, as defined in the usual Bohmian mechanical formulation, are classical-like and well behaved, even when Ψ has many nodes or is wildly oscillatory. A modification for discontinuous potential stationary scattering states was presented in a second, companion paper [C. Trahan and B. Poirier, J. Chem. Phys.124, 034115 (2006), previous paper], whose generalization for continuous potentials is given here. The result is an exact quantum scattering methodology using classical trajectories. For additional convenience in handling the tunneling case, a constant-velocity-trajectory version is also developed. [ABSTRACT FROM AUTHOR]

The excitation energy transfer (EET) process for photosynthetic antenna complexes consisting of subunits, each comprised of multiple chromophores, remains challenging to describe. The multichromophoric Förster resonance energy transfer theory is a popular method to describe the EET process in such systems. This paper presents a new time-domain method for calculating energy transfer based on the combination of multichromophoric Förster resonance energy transfer theory and the Numerical Integration of the Schrödinger Equation method. After validating the method on simple model systems, we apply it to the Light-Harvesting antenna 2 (LH2) complex, a light harvesting antenna found in purple bacteria. We use a simple model combining the overdamped Brownian oscillators to describe the dynamic disorder originating from the environmental fluctuations and the transition charge from the electrostatic potential coupling model to determine the interactions between chromophores. We demonstrate that with this model, both the calculated spectra and the EET rates between the two rings within the LH2 complex agree well with experimental results. We further find that the transfer between the strongly coupled rings of neighboring LH2 complexes can also be well described with our method. We conclude that our new method accurately describes the EET rate for biologically relevant multichromophoric systems, which are similar to the LH2 complex. Computationally, the new method is very tractable, especially for slow processes. We foresee that the method can be applied to efficiently calculate transfer in artificial systems as well and may pave the way for calculating multidimensional spectra of extensive multichromophoric systems in the future. [ABSTRACT FROM AUTHOR]

Quantum mechanics has a deterministic Schrödinger equation for the wave function. The Göttingen–Copenhagen statistical interpretation is based on the Born Rule that interprets the wave function as a "probability amplitude." A precept of this interpretation is the lack of determinism in quantum mechanics. The Bohm interpretation is that the wave function is a source of a field experienced by the electrons, thereby attributing determinism to quantum theory. In this paper, we present a new perspective on such determinism. The ideas are based on the equations of motion or "Quantal Newtonian" Laws obeyed by each electron. These Laws, derived from the temporal and stationary-state Schrödinger equation, are interpreted in terms of "classical" fields whose sources are quantal expectations of Hermitian operators taken with respect to the wave function. According to the Second Law, each electron experiences an external field—the quantal Coulomb-Lorentz law. It also experiences an internal field representative of properties of the system: correlations due to Coulomb repulsion and Pauli principle; the density; kinetic effects; and an internal magnetic field component. There is a response field. The First Law states that the sum of the external and internal fields experienced by each electron vanishes. These fields are akin to those of classical physics: They pervade all space; their structure is descriptive of the quantum system; the energy of the system is stored in these fields. It is in the classical behavior of these fields, which arise from quantal sources that one may then speak of determinism in quantum mechanics. [ABSTRACT FROM AUTHOR]

This paper investigates dynamical behaviors and controllability of some nonautonomous localized waves based on the Gross–Pitaevskii equation with attractive interatomic interactions. Our approach is a relation constructed between the Gross–Pitaevskii equation and the standard nonlinear Schrödinger equation through a new self-similarity transformation which is to convert the exact solutions of the latter to the former's. Subsequently, one can obtain the nonautonomous breather solutions and higher-order rogue wave solutions of the Gross–Pitaevskii equation. It has been shown that the nonautonomous localized waves can be controlled by the parameters within the self-similarity transformation, rather than relying solely on the nonlinear intensity, spectral parameters, and external potential. The control mechanism can induce an unusual number of loosely bound higher-order rogue waves. The asymptotic analysis of unusual loosely bound rogue waves shows that their essence is energy transfer among rogue waves. Numerical simulations test the dynamical stability of obtained localized wave solutions, which indicate that modifying the parameters in the self-similarity transformation can improve the stability of unstable localized waves and prolong their lifespan. We numerically confirm that the rogue wave controlled by the self-similarity transformation can be reproduced from a chaotic initial background field, hence anticipating the feasibility of its experimental observation, and propose an experimental method for observing these phenomena in Bose–Einstein condensates. The method presented in this paper can help to induce and observe new stable localized waves in some physical systems. [ABSTRACT FROM AUTHOR]

Chemical and photochemical reactivity, as well as supramolecular organization and several other molecular properties, can be modified by strong interactions between light and matter. Theoretical studies of these phenomena require the separation of the Schrödinger equation into different degrees of freedom as in the Born–Oppenheimer approximation. In this paper, we analyze the electron–photon Hamiltonian within the cavity Born–Oppenheimer approximation (CBOA), where the electronic problem is solved for fixed nuclear positions and photonic parameters. In particular, we focus on intermolecular interactions in representative dimer complexes. The CBOA potential energy surfaces are compared with those obtained using a polaritonic approach, where the photonic and electronic degrees of freedom are treated at the same level. This allows us to assess the role of electron–photon correlation and the accuracy of CBOA. [ABSTRACT FROM AUTHOR]

In this paper, we present the higher-order nonlinear Schrödinger equation (NLSE) with third order dispersion (3OD), fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms that define the propagation of ultrashort pulses. Two analytical methods, which are the new Kudryashov's method and the unified Riccati equation expansion method, are implemented to extract the analytical soliton solutions of the presented equation for the first time. Thus, bright, dark, and singular soliton solutions are acquired. To illustrate the physical behavior of some of the obtained solutions, 3D, 2D, and contour graphs are depicted. In particular, to understand the effects of the group velocity dispersion, 3OD, 4OD, CQNLs, self-steepening coefficient terms, and group velocity term of the traveling wave transformation on the soliton dynamics of the proposed equation, 2D plots for different values of coefficients are represented. The obtained results provide us with the knowledge that the presented model can be examined from a physical perspective. It can be concluded that the used methods are effective approaches to derive the analytical solutions for the NLSE. [ABSTRACT FROM AUTHOR]

The paper interprets the cubic nonlinear Schrödinger equation as a Hamiltonian system with infinite dimensional phase space. There exists a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The logarithmic Sobolev and concentration of measure inequalities hold for the Gibbs measures, and here are extended to the k-point correlation function and distributions of related empirical measures. By Hasimoto's theorem, the nonlinear Schrödinger equation gives a Lax pair of coupled ordinary differential equations for which the solutions give a system of moving frames. The paper studies the evolution of the measure induced on the moving frames by the Gibbs measure; the results are illustrated by numerical simulations. The paper contains quantitative estimates with well-controlled constants on the rate of convergence of the empirical distribution in Wasserstein metric. [ABSTRACT FROM AUTHOR]

In our research paper, we explore the application of mathematical techniques, both analytical and numerical, to solve the coupled nonlinear Schrödinger equation. To obtain accurate solutions, we use the improved, modified, extended tanh-function method. By breaking down the Schrödinger equation into real and imaginary components, we derive four interconnected equations. We analyze these equations using the generalized tanh method to find precise solutions. This set of equations is of great importance in quantum mechanics and helps us understand the behavior of quantum systems. We provide an analytical and numerical solution using the implicit finite difference. Our method is second-order in both space and time, and we have verified its stability through von Neumann's stability analysis. [ABSTRACT FROM AUTHOR]

Solving the Liouville–von-Neumann equation using a density operator provides a more complete picture of dynamical quantum phenomena than by using a wavepacket and solving the Schrödinger equation. As density operators are not restricted to the description of pure states, they can treat both thermalized and open systems. In practice, however, they are rarely used to study molecular systems as the computational resources required are even more prohibitive than those needed for wavepacket dynamics. In this paper, we demonstrate the potential utility of a scheme based on the powerful multi-layer multi-configurational time-dependent Hartree algorithm for propagating multi-dimensional density operators. Studies of two systems using this method are presented at a range of temperatures and including up to 13 degrees of freedom. The first case is single proton transfer in salicylaldimine, while the second is double proton transfer in porphycene. A comparison is also made with the approach of using stochastic wavepackets. [ABSTRACT FROM AUTHOR]

In order to simulate the exact universal density functional, approximations are nowadays constructed by permitting more flexibility in its ansatz. In view of the difficulty of defining a systematically improvable form for it, this paper argues that an alternative way could be considered. It falls within the class of hybrid functionals with multi-determinant wave functions. The parameter controlling the hybridization is considered as variable. The invariance of the exact result with respect to changes in this variable is used to introduce information about the system under consideration, and to correct the density functional result. The construction considered in this paper accelerates convergence from the model system to the physical one, in the vicinity of the latter. The method, at the present level of implementation, should be seen as a starting point for further development, and not necessarily as a computationally advantageous tool. [ABSTRACT FROM AUTHOR]

This paper presents an analytical quantum model for photoemission from metal surfaces coated with an ultrathin dielectric, by solving the 1D time-dependent Schrödinger equation subject to an oscillating double-triangular potential barrier. The model is valid for an arbitrary combination of metal (of any work function and Fermi level), dielectric (of any thickness, relative permittivity, and electron affinity), laser field (strength and wavelength), and dc field. The effects of dielectric properties on photoemission are systematically investigated. It is found that a flat metal surface with dielectric coating can photoemit a larger current density than the uncoated case when the dielectric has smaller relative permittivity and larger electron affinity. Resonant peaks in the photoemission probability and emission current are observed as a function of dielectric thickness or electron affinity due to the quantum interference of electron waves inside the dielectric. Our model is compared with the effective single-barrier quantum model and modified Fowler–Nordheim equation, for both 1D flat cathodes and pyramid-shaped nanoemitters. While the three models show quantitatively good agreement in the optical field tunneling regime, the present model may be used to give a more accurate evaluation of photoemission from coated emitters in the multiphoton absorption regime. [ABSTRACT FROM AUTHOR]

The chemical formula theory (CFT) proposed in Paper I of this series [H. Nakatsuji et al., J. Chem. Phys. 149, 114105 (2018)] is a simple variational electronic structure theory for atoms and molecules. The CFT constructs simple, conceptually useful wave functions for the ground and excited states, simultaneously, from the ground and excited states of the constituent atoms, reflecting the spirits of the chemical formulas. The CFT wave functions are also designed to be used as the initial wave functions of the free complement (FC) theory, that is, the exact theory producing the exact wave functions of the Schrödinger accuracy. This combined theory is referred to as the FC-CFT. We aim to construct an exact wave function theory that is useful not only quantitatively but also conceptually. This paper shows the atomic applications of the CFT and the FC-CFT. For simplicity, we choose the small atoms, Be and Li, and perform variational calculations to essentially exact levels. For these elements, a simple Hylleraas CI type formulation is known to be potentially highly accurate: we realize it with the CFT and the FC-CFT. Even from the CFT levels, the excitation energies to the Rydberg excited states were calculated satisfactorily. Then, with increasing the order of the FC theory in the FC-CFT, all the absolute energies and the excitation energies of the Be and Li atoms were improved uniformly and reached rapidly to the essentially exact levels in order 3 or 4 with moderately small calculational labors. [ABSTRACT FROM AUTHOR]

General transformation expressions of the second-order non-adiabatic Hamiltonian of the atomic nuclei, including the kinetic-energy correction terms, are derived upon the change from laboratory-fixed Cartesian coordinates to general curvilinear coordinate systems commonly used in rovibrational computations. The kinetic-energy or so-called "mass-correction" tensor elements are computed with the stochastic variational method and floating explicitly correlated Gaussian functions for the H 2 + molecular ion in its ground electronic state. {Further numerical applications for the 4 He 2 + molecular ion are presented in the forthcoming paper, Paper II [E. Mátyus, J. Chem. Phys. 149, 194112 (2018)]}. The general, curvilinear non-adiabatic kinetic energy operator expressions are used in the examples, and non-adiabatic rovibrational energies and corrections are determined by solving the rovibrational Schrödinger equation including the diagonal Born–Oppenheimer as well as the mass-tensor corrections. [ABSTRACT FROM AUTHOR]

Within the well-established optical response function formalism, a new strategy with the central idea of employing the forward-backward stochastic Schrödinger equations in a segmented way to accurately obtain the two-dimensional (2D) electronic spectrum is presented in this paper. Based on the simple excitonically coupled dimer model system, the validity and efficiency of the proposed schemes are demonstrated in detail, along with the comparison against the deterministic hierarchy equations of motion and perturbative second-order time-convolutionless quantum master equations. In addition, an important insight is provided in this paper that the characteristic frequency of the overdamped environment is an extremely crucial factor to regulate the lifetimes of the oscillating signals in 2D electronic spectra and of quantum coherence features of system dynamics. It is worth noting that the proposed scheme benefiting from its stochastic nature and wavefunction framework and many other advantages of substantially reducing the numerical cost has a great potential to systematically investigate various quantum effects observed in realistic large-scale natural and artificial photosynthetic systems. [ABSTRACT FROM AUTHOR]

For molecules with more than three atoms, it is difficult to fit or interpolate a potential energy surface (PES) from a small number of (usually ab initio) energies at points. Many methods have been proposed in recent decades, each claiming a set of advantages. Unfortunately, there are few comparative studies. In this paper, we compare neural networks (NNs) with Gaussian process (GP) regression. We re-fit an accurate PES of formaldehyde and compare PES errors on the entire point set used to solve the vibrational Schrödinger equation, i.e., the only error that matters in quantum dynamics calculations. We also compare the vibrational spectra computed on the underlying reference PES and the NN and GP potential surfaces. The NN and GP surfaces are constructed with exactly the same points, and the corresponding spectra are computed with the same points and the same basis. The GP fitting error is lower, and the GP spectrum is more accurate. The best NN fits to 625/1250/2500 symmetry unique potential energy points have global PES root mean square errors (RMSEs) of 6.53/2.54/0.86 cm−1, whereas the best GP surfaces have RMSE values of 3.87/1.13/0.62 cm−1, respectively. When fitting 625 symmetry unique points, the error in the first 100 vibrational levels is only 0.06 cm−1 with the best GP fit, whereas the spectrum on the best NN PES has an error of 0.22 cm−1, with respect to the spectrum computed on the reference PES. This error is reduced to about 0.01 cm−1 when fitting 2500 points with either the NN or GP. We also find that the GP surface produces a relatively accurate spectrum when obtained based on as few as 313 points. [ABSTRACT FROM AUTHOR]

In this paper we report a new method for resummation of operator expansions of the evolution operator. These resummation techniques are applied to the Feynman propagator and then to the derivation of an effective Schrödinger equation for general system–bath problems. This paper presents a significant advance over work previously reported by us [J. Chem. Phys. 96, 5952 (1992)], and it provides a highly accurate way to calculate quantum mechanical data for many dimensional systems. We then apply this new analytic resummation technique to the calculation of flux–flux correlation functions for two, three, and four dimensional problems in order to study the range of applicability of the approach. For moderate frequencies and coupling strengths, not only is the resummed operator calculational method essentially exact, it also requires a fraction of the computer time that the exact calculation consumes. For truly many dimensional problems this approach should provide the first accurate quantum mechanics of rate processes. [ABSTRACT FROM AUTHOR]

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking. [ABSTRACT FROM AUTHOR]

In this paper, we extend a recently proposed approach for inverse scattering with Neumann boundary conditions [Druskin et al., Inverse Probl. 37, 075003 (2021)] to the 1D Schrödinger equation with impedance (Robin) boundary conditions. This method approaches inverse scattering in two steps: first, to extract a reduced order model (ROM) directly from the data and, subsequently, to extract the scattering potential from the ROM. We also propose a novel data-assimilation (DA) inversion method based on the ROM approach, thereby avoiding the need for a Lanczos-orthogonalization (LO) step. Furthermore, we present a detailed numerical study and A comparison of the accuracy and stability of the DA and LO methods. [ABSTRACT FROM AUTHOR]

Single-reference methods such as Hartree–Fock-based coupled cluster theory are well known for their accuracy and efficiency for weakly correlated systems. For strongly correlated systems, more sophisticated methods are needed. Recent studies have revealed the potential of the antisymmetrized geminal power (AGP) as an excellent initial reference for the strong correlation problem. While these studies improved on AGP by linear correlators, we explore some non-linear exponential Ansätze in this paper. We investigate two approaches in particular. Similar to Wahlen-Strothman et al. [Phys. Rev. B 91, 041114(R) (2015)], we show that the similarity transformed Hamiltonian with a Hilbert-space Jastrow operator is summable to all orders and can be solved over AGP by projecting the Schrödinger equation. The second approach is based on approximating the unitary pair-hopper Ansatz recently proposed for application on a quantum computer. We report benchmark numerical calculations against the ground state of the pairing Hamiltonian for both of these approaches. [ABSTRACT FROM AUTHOR]

GaN-based heterostructures have been used in high-power radio frequency applications for a number of years due to the wide bandgap of GaN and high values of spontaneous and piezoelectric polarization, resulting in high breakdown voltages and high 2D carrier concentrations. However, the accurate modeling of low-field electron mobility within such structures remains a topic of interest. This paper presents a comprehensive numerical model for calculating the carrier mobility within a 2D electron gas in Al x Ga 1 − x N/GaN high electron mobility transistors. The model is based on solving the Schrödinger and Poisson equations self-consistently, taking into account the polarization charges at material interfaces and performing semi-classical numerical calculations of low-field electron mobility within the momentum relaxation time approximation, taking into account all relevant scattering mechanisms. Both intra- and inter-subband transitions are considered, and the differences in intra- and inter-subband scattering rates are analyzed for some of the key scattering mechanisms. The importance of including inter-subband transitions in the calculations is demonstrated by comparing the calculated results with experimentally measured mobilities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the effects of external gaussian white noise on the full counting statistics (FCS) of non-interacting coherent electrons in mesoscopic systems. We first, derive the wave packet formalism of FCS in the presence of time dependent random potential. We then determine the steady state probability distributions of all the moments and cumulants of particle currents, specially, the shot-noise, the Fano factor, and the distribution in the noise-charge plane by numerically solving the stochastic time dependent Schrödinger equation (STDSE) for one and two-electron systems with transparent boundary conditions, using Numerov extension of Crank-Nicolson method. [ABSTRACT FROM AUTHOR]

In this paper, we have analyzed the time series associated with the iterative scheme of a double similarity transformed coupled cluster theory. The coupled iterative scheme to solve the ground state Schrödinger equation is cast as a multivariate time-discrete map, and the solutions show the universal Feigenbaum dynamics. Using recurrence analysis, it is shown that the dynamics of the iterative process is dictated by a small subgroup of cluster operators, mostly those involving chemically active orbitals, whereas all other cluster operators with smaller amplitudes are enslaved. Using synergetics, we will indicate how the master-slave dynamics can suitably be exploited to develop a novel coupled-cluster algorithm in a much reduced dimension. [ABSTRACT FROM AUTHOR]

With the inhomogeneity of optical fiber media taken into account, under investigation in this paper is the variable coefficient Kundu-nonlinear Schrödinger equation, which describes the pulses propagation in optical fibers. Based on Lax pair, the Nth-order Darboux transformation is constructed. Depending on plane wave solution, the first- and second-order breather solutions are derived and the interactions between breathers are graphically analyzed. The Kuznetsov–Ma breather, Akhmediev breather, and spatial-temporal breather have been obtained. Moreover, the first-, second-, and third-order rogue wave solutions have been constructed. The usual rogue waves and first- and second-order line rogue waves are observed. The weak and strong interactions between the first-, second-order rogue waves, and spatial-temporal period breather are studied. Furthermore, variable coefficient δ (t) causes rogue waves to produce some interesting evolutionary phenomena, which have been systematically analyzed. In addition, the influences of parameters for the properties of solutions are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of quasilinear Schrödinger equations arising from a model of a self-trapped electrons in quadratic or hexagonal lattices. By variational methods, we prove that this problem admits a positive solution for any positive parameter. [ABSTRACT FROM AUTHOR]

A Kundu–nonlinear Schrödinger equation that can be utilized to simulate the pulse propagation in optical fibers is researched in this paper. First, the Lax integrability of the above equation is proved and its modulational instability (i.e., the main mechanism for producing the rogue wave solutions and the breather solutions) is calculated. Subsequently, using the generalized perturbation (n , N − n) -fold Darboux transformation, the rogue waves, breathers, and mixed interaction solutions are acquired, as well as the impact of various parameters on the solutions is examined. In particular, when we assume that the coefficient of the equation is θ = m x + d t , some new wave structures are found based on parameter variations, such as the rotational separation of first-order rogue waves, scale-like structures generated by second-order breathers, etc., which offer novel ideas for producing different signals via optical fibers. Ultimately, the classification numbers of mixed solutions of rogue waves and breathers are provided, which can better observe how the two types of the local waves are combined, in accordance with the distributions of the increasing numbers of algebraic equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the breathers in the framework of the sixth-order nonlinear Schrödinger equation by using the Darboux transformation. The primary objective of this research is twofold. First, we consider the nonlinear superposition of breathers on the plane wave background. Based on the concept that rogue waves are formed from colliding Akhmediev breathers, we obtain rogue wave sequences and a first-order Akhmediev breather with a central second-order rogue wave peak. Second, we consider the formation of breathers on the periodic wave background. The difficulty of solving the Lax pair is overcome, and we successfully construct the breathers on the cn- and dn-periodic wave background. [ABSTRACT FROM AUTHOR]

This paper studies the multi-component derivative nonlinear Schrödinger (n -DNLS) equations featuring nonzero boundary conditions. Employing the Darboux transformation method, we derive higher-order vector rogue wave solutions for the n -DNLS equations. Specifically, we focus on the distinctive scenario where the (n + 1) -order characteristic polynomial possesses an explicit (n + 1) -multiple root. Additionally, we provide an in-depth analysis of the asymptotic dynamic behaviors and pattern classification inherent to the higher-order vector rogue wave solution of the n -DNLS equations, mainly when one of the internal arbitrary parameters is extremely large. These patterns are related to the root structures in the generalized Wronskian–Hermite polynomial hierarchies. [ABSTRACT FROM AUTHOR]

The paper develops a finite-time quantum state reduction framework via the use of Lévy random bridges (LRBs) that can be understood as Doob h-transformations on Lévy processes. Building upon the non-anticipative semimartingale representation of LRBs, we propose a family of energy-driven stochastic Schrödinger equations that go beyond the purely-continuous Brownian motion setup, and enter the scope of quantum systems involving discontinuities and heavy-tails. In doing so, we allow collapse dynamics to be governed intrinsically by the Markovian statistics of LRBs. The framework can host progressively convoluted stochastic state reduction dynamics in a tractable way and encapsulates jump-diffusion Schrödinger evolutions by use of the Lévy-Itô decomposition. Our construct motivates the usage of more general Markov processes and Doob h-transformations in producing consistent wave function collapse dynamics in finite-time. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Kirchhoff type logarithmic Schrödinger equations involving the critical or supercritical Sobolev exponent. Such problems cannot be studied by applying variational methods in a standard way, because the nonlinearities do not satisfy the Ambrosetti-Rabinowitz condition and change sign. Moreover, the appearance of the logarithmic term makes the associated energy functional lose differentiable in the sense of Gateaux. By analyzing the structure of the Nehari manifold and developing some analysis techniques, the above obstacles are overcome in subtle ways and several existence result are obtained. Furthermore, we investigate the regularity, the monotonicity, and the symmetric properties of the solutions via the iterative technique and the moving plane method. [ABSTRACT FROM AUTHOR]

This is the second progress report on the study of the structure of the exact wave function. First, Theorem II of Paper I (H. Nakatsuji, J. Chem. Phys. 113, 2949 (2000)) is generalized: when we divide the Hamiltonian of our system into N[sub D] (number of division) parts, we correspondingly have a set of N[sub D] equations that is equivalent to the Schro¨dinger equation in the necessary and sufficient sense. Based on this theorem, the iterative configuration interaction (ICI) method is generalized so that it gives the exact wave function with the N[sub D] number of variables in each iteration step. We call this the ICIND method. The ICIGSD (general singles and doubles) method is an important special case in which the GSD number of variables is involved. The ICI methods involving only one variable [ICION(one) or S(simplest)ICI] and only general singles (GS) number of variables (ICIGS) are also interesting. ICIGS may be related to the basis of the density functional theory. The convergence rate of the ICI calculations would be faster when N[sub D] is larger and when the quality of the initial guess function is better. We then study the structure of the ICI method by expanding its variable space. We also consider how to calculate the excited state by the ICIGSD method. One method is an ICI method aiming at only one exact excited state. The other is to use the higher solutions of the ICIGSD eigenvalues and vectors to compute approximate excited states. The latter method can be improved by extending the variable space outside of GSD. The underlying concept is similar to that of the symmetry-adapted-cluster configuration-interaction (SAC-CI) theory. A similar method of calculating the excited state is also described based on the ICIND method. © 2001 American Institute of Physics. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving nonlinearity with subcritical and critical growth. To apply the variational method and circumvent the "lack of compactness" of the problem, we combine the dual approach developed by Colin–Jeanjean [Nonlinear Anal. 56, 213–226 (2004)], Fang–Szulkin [J. Differ. Equations, 254, 2015–2032 (2013)], and Liu–Wang–Wang [J. Differ. Equations 187, 473–493 (2003)] with Del Pino–Felmer's penalization technique [Calc. Var. Partial Differ. Equations 4, 121–137 (1996)], Moser's iteration method, and an adaptation of Alves' arguments [J. Elliptic Parabol. Equations 1, 231–241 (2015)] of the semilinear case. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear Schrödinger equation with an almost-periodic potential, subject to periodic boundary conditions. Under appropriate assumptions on the potential, we obtain many small-amplitude, time-almost-periodic solutions for the equation. We improve the Kolmogorov–Arnold–Moser iteration to reduce the almost-periodic potential, especially depending on the space variables. The analysis of the nonlinear perturbation is also the key argument. [ABSTRACT FROM AUTHOR]

In this paper, we study the discrete nonlinear random Schrödinger equation i ∂ ∂ t u = (ε Δ + V) u + δ | u | 2 p u (p ∈ N + ) on Z d × [ 0 , ∞) , where 0 < ɛ, δ ≪ 1, Δ is the discrete Laplacian, and V is the random potential. We fix the random potential V in a good set. Then, we use small amplitudes as parameters to construct quasiperiodic solutions of the nonlinear random Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear Schrödinger equations. Let A (x) ≔ [ V (x) ] p p − 2 − N 2 [ K (x) ] − 2 p − 2 . Under some conditions on A , we show the local uniqueness of positive multi-peak solutions concentrating near k(k ≥ 2) distinct non-degenerate critical points of A by using the local Pohozaev identity. We generalize Cao–Li–Luo's results to the competing potential cases and show how these two potentials impact the uniqueness of concentrated solutions. [ABSTRACT FROM AUTHOR]

Bohmian mechanics is a non-relativistic quantum theory based on a particle approach. In this paper, we study the Schrödinger equation with a rapidly oscillating potential and the associated Bohmian trajectory. We prove that the corresponding Bohmian trajectory converges locally in a measure, and the limit coincides with the Bohmian trajectory for the effective Schrödinger equation on a finite time interval. This is beneficial for efficient simulation of the Bohmian trajectories in oscillating potential fields. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problem for the mixed Schrödinger equation. For the Schwartz initial data q 0 (x) ∈ S (R) , by defining a general analytical domain and two reflection coefficients, we ever found an unified long-time asymptotic formula via the Deift–Zhou nonlinear steepest descent method. In this paper, under essentially minimal regularity assumptions on initial data in a much weak weighted Sobolev space q 0 (x) ∈ H 2 , 2 (R) , we apply the ∂ ̄ steepest descent method to obtain long-time asymptotics for the mixed Schrödinger equation. In the asymptotic expression, the leading order term O ( t − 1 / 2 ) comes from the dispersive part qt + iqxx and the error order O ( t − 3 / 4 ) comes from a ∂ ̄ equation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a pruned, nondirect product multi-configuration time dependent Hartree (MCTDH) method for solving the Schrödinger equation. MCTDH uses optimized 1D basis functions, called single particle functions, but the size of the standard direct product MCTDH basis scales exponentially with D, the number of coordinates. We compare the pruned approach to standard MCTDH calculations for basis sizes small enough that the latter are possible and demonstrate that pruning the basis reduces the CPU cost of computing vibrational energy levels of acetonitrile (D = 12) by more than two orders of magnitude. Using the pruned method, it is possible to do calculations with larger bases, for which the cost of standard MCTDH calculations is prohibitive. Pruning the basis complicates the evaluation of matrix-vector products. In this paper, they are done term by term for a sum-of-products Hamiltonian. When no attempt is made to exploit the fact that matrices representing some of the factors of a term are identity matrices, one needs only to carefully constrain indices. In this paper, we develop new ideas that make it possible to further reduce the CPU time by exploiting identity matrices. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of Schrödinger equation coupled with a neutral scalar field. By constructing solutions with multiple peaks, we prove that the number of non-radial solutions of this problem goes to infinity as the Maxwell coupling constant tends to infinity. The Chern–Simons limit of those solutions is also discussed. [ABSTRACT FROM AUTHOR]

This paper is concerned with a one-dimensional quasi-periodically forced nonlinear Schrödinger equation under Dirichlet boundary conditions. The existence of the quasi-periodic solutions for the equation is verified. By infinitely many symplectic transformations of coordinates, the Hamiltonian of the linear part of the equation can be reduced to an autonomous system. By utilizing the measure estimation of small divisors, there exists a symplectic change of coordinate transformation of the Hamiltonian of the equation into a nice Birkhoff normal form. By an abstract KAM (Kolmogorov-Arnold-Moser) theorem, the existence of a class of small-amplitude quasi-periodic solutions for the above equation is verified. [ABSTRACT FROM AUTHOR]

In this paper, we show a connection between fractional Schrödinger equations with power-law nonlinearity and fractional Schrödinger equations with logarithm-law nonlinearity. We prove that ground state solutions of power-law fractional equations, as p → 2+, converge to a ground state solution of logarithm-law fractional equations. In particular, we provide a new proof to the existence of a ground state of logarithm-law fractional Schrödinger equations. [ABSTRACT FROM AUTHOR]

In this paper, using the algorithm due to Ablowitz et al. [Phys. Rev. Lett. 128, 184101 (2022); J. Phys. A: Math. Gen. 55, 384010 (2022)], we explore the anomalous dispersive relations, inverse scattering transform, and fractional N -soliton solutions of the integrable fractional higher-order nonlinear Schrödinger (fHONLS) equations, containing the fractional third-order NLS (fTONLS), fractional complex mKdV (fcmKdV), and fractional fourth-order nonlinear Schrödinger (fFONLS) equations, etc. The inverse scattering problem can be solved exactly by means of the matrix Riemann–Hilbert problem with simple poles. As a consequence, an explicit formula is found for the fractional N -soliton solutions of the fHONLS equations in the reflectionless case. In particular, we analyze the fractional one-, two-, and three-soliton solutions with anomalous dispersions of fTONLS and fcmKdV equations. The wave, group, and phase velocities of these envelope fractional one-soliton solutions are related to the power laws of their amplitudes. Moreover, we also deduce the formula for the fractional N -soliton solutions of all fHONLS equations and analyze some velocities of the one-soliton solution. These obtained fractional N -soliton solutions may be useful to explain the related super-dispersion transports of nonlinear waves in fractional nonlinear media. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to quantum evolution in the presence of a quantum bath. We develop an equation of motion for an observed system evolving under the influence of an unobserved quantum bath. The methodology we follow uses operator expansions of the Feynman propagator. Corrections to the zeroth order approximation are corrections to an adiabatic approximation. In this paper we explicitly develop an approximation which is infinite order in bath and system coupling, but first order in system degree of freedom. This infinite order approximation is found through a resummation of an infinite class of terms in the operator expansion. We first present a simplified single time (Markovian) version of the theory. We then present a derivation for including the effects of memory. The approach developed in this paper also has the potential for systematic improvement. In other words, while the bath and system coupling in this calculation is treated to infinite order, the system itself is only treated to first order. We will briefly discuss how these higher order corrections can be found. Finally, we present a test calculation of the our approach with comparison to exact results. For a two-dimensional test problem with potential much like that for collinear H+H2 the effective one-dimensional approximation we apply produces essentially exact results. [ABSTRACT FROM AUTHOR]

Geometric integrators of the Schrödinger equation conserve exactly many invariants of the exact solution. Among these integrators, the split-operator algorithm is explicit and easy to implement but, unfortunately, is restricted to systems whose Hamiltonian is separable into kinetic and potential terms. Here, we describe several implicit geometric integrators applicable to both separable and nonseparable Hamiltonians and, in particular, to the nonadiabatic molecular Hamiltonian in the adiabatic representation. These integrators combine the dynamic Fourier method with the recursive symmetric composition of the trapezoidal rule or implicit midpoint method, which results in an arbitrary order of accuracy in the time step. Moreover, these integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable and, in contrast to the split-operator algorithm, conserve energy exactly, regardless of the accuracy of the solution. The order of convergence and conservation of geometric properties are proven analytically and demonstrated numerically on a two-surface NaI model in the adiabatic representation. Although each step of the higher order integrators is more costly, these algorithms become the most efficient ones if higher accuracy is desired; a thousand-fold speedup compared to the second-order trapezoidal rule (the Crank-Nicolson method) was observed for a wavefunction convergence error of 10−10. In a companion paper [J. Roulet, S. Choi, and J. Vaníček, J. Chem. Phys. 150, 204113 (2019)], we discuss analogous, arbitrary-order compositions of the split-operator algorithm and apply both types of geometric integrators to a higher-dimensional system in the diabatic representation. [ABSTRACT FROM AUTHOR]

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schrödinger equation. In this paper, we present three strategies for extending any sixth-order scheme for the Schrödinger equation with time-independent potential to a sixth-order method for the Schrödinger equation with laser potential. As demonstrated via numerical examples, these schemes prove effective in the atomic regime as well as the semiclassical regime and are a particularly appealing alternative to time-ordered exponential splittings when the laser potential is highly oscillatory or known only at specific points in time (on an equispaced grid, for instance). These schemes are derived by exploiting the linear in space form of the time dependent potential under the dipole approximation (whereby commutators in the Magnus expansion reduce to a simpler form), separating the time step of numerical propagation from the issue of adequate time-resolution of the laser field by keeping integrals intact in the Magnus expansion and eliminating terms with unfavorable structure via carefully designed splittings. [ABSTRACT FROM AUTHOR]

Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH. [ABSTRACT FROM AUTHOR]

This paper presents in detail our fast semistochastic heat-bath configuration interaction (SHCI) method for solving the many-body Schrödinger equation. We identify and eliminate computational bottlenecks in both the variational and perturbative steps of the SHCI algorithm. We also describe the parallelization and the key data structures in our implementation, such as the distributed hash table. The improved SHCI algorithm enables us to include in our variational wavefunction two orders of magnitude more determinants than has been reported previously with other selected configuration interaction methods. We use our algorithm to calculate an accurate benchmark energy for the chromium dimer with the X2C relativistic Hamiltonian in the cc-pVDZ-DK basis, correlating 28 electrons in 76 spatial orbitals. Our largest calculation uses two billion Slater determinants in the variational space and semistochastically includes perturbative contributions from at least trillions of additional determinants with better than 10−5 Ha statistical uncertainty. [ABSTRACT FROM AUTHOR]