Solutions of the SchrödingerEquation forPT-Symmetric Coupled Quintic Potentialsin Two Dimensions

This paper deals with the solutions of time independent Schrödinger wave equation for a two-dimensional PT-symmetric coupled quintic potential in its most general form. Employing wavefunction ansatz method, general analytic expressions for eigenvalues and eigenfunctions for first four states are obtained. Solutions of a particular case are also presented.     

Non-hypergeometric Type of Polynomialsand Solutions of Schrödinger Equation with Position-Dependent Mass

Using the coordinate transformation method, we study the polynomial solutions of the Schrödinger equation with position-dependentmass (PDM). The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The issues related to normalization of the wavefunctions and Hermiticity of the Hamiltonian are also analyzed.     

An Extended Subequation Rational Expansion Method and Solutions of (2+1)-Dimensional Cubic Nonlinear Schrödinger Equation

An extended subequation rational expansion method is presented and used to construct some exact analytical solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation. From our results, many known solutions of the (2+1)-dimensional cubic nonlinear Schrödinger equation can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. With computer simulation, the properties of new non-travelling wave and coefficient function's soliton-like solutions, and elliptic solutions are demonstrated by some plots.     

Exact Solutions of the Morse-like Potential,Step-Up and Step-Down Operators via Laplace Transform Approach

We intend to realize the step-up and step-down operators of the potential V (x) = V₁ e ^2βx + V₂ e ^βx. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential (β → -β).     

Deep learning neural networks for the third-order nonlinear Schrödinger equation: bright solitons,breathers, and rogue waves

The dimensionless third-order nonlinear Schrödinger equation (alias the Hirota equation) is investigated via deep leaning neural networks. In this paper, we use the physics-informed neural networks (PINNs) deep learning method to explore the data-driven solutions (e.g. bright soliton, breather, and rogue waves) of the Hirota equation when the two types of the unperturbated and perturbated (a 2% noise) training data are considered. Moreover, we use the PINNs deep learning to study the data-driven discovery of parameters appearing in the Hirota equation with the aid of bright solitons.     

Qualitative Analysis and Explicit Exact Solitary,Kink and Anti-Kink Wave Solutions of the Generalized Nonlinear Schrdinger Equation with Parabolic Law Nonlinearity

The generalized nonlinear Schrdinger equation with parabolic law nonlinearity is studied by using the factorization technique and the method of dynamical systems.From a dynamic point of view,the existence of smooth solitary wave,kink and anti-kink wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given.Also,all possible explicit exact parametric representations of the waves are presented.     

Structure of quantum disordered wave functions: weak localization, far tails, and mesoscopic transport

We report on the comprehensive numerical study of the fluctuation and correlation properties of wave functions in three-dimensional mesoscopic diffusive conductors. Several large sets of nanoscale samples with finite metallic conductance, modeled by an Anderson model with different strengths of diagonal box disorder, have been generated in order to investigate both small and large deviations (as well as the connection between them) of the distribution function of eigenstate amplitudes from the universal prediction of random matrix theory. We find that small, weak localization-type, deviations contain both diffusive contributions (determined by the bulk and boundary conditions dependent terms) and ballistic ones which are generated by electron dynamics below the length scale set by the mean free path ℓ. By relating the extracted parameters of the functional form of nonperturbative deviations (“far tails”) to the exactly calculated transport properties of mesoscopic conductors, we compare our findings based on the full solution of the Schr?dinger equation to different approximative analytical treatments. We find that statistics in the far tail can be explained by the exp-log-cube asymptotics (convincingly refuting the log-normal alternative), but with parameters whose dependence on ℓ is linear and, therefore, expected to be dominated by ballistic effects. It is demonstrated that both small deviations and far tails depend explicitly on the sample size--the remaining puzzle then is the evolution of the far tail parameters with the size of the conductor since short-scale physics is supposedly insensitive to the sample boundaries. Received 19 August 2002 Published online 19 November 2002     

Resolved fluorescence spectra of NiH. Electronic structure,electronic energy transfer,and the Zeeman effect in low-lying states

Fourier transform spectra of collisionally induced fluorescence following isotopically selective laser excitation of NiH at ～550?nm have located an excited Ω?=?1/2 state of NiH lying 17900?cm^?1 above the electronic ground state. This is identified as v?=?0 of a ²Π_1/2 state originating from an Ni⁺ 3d⁸4s^{1 2}F configuration. Emission from this Ω′?=?1/2 state occurs predominantly to v″?=?0 and 1 of the ²Σ⁺ and W₂ ²Π_1/2 ligand field states, locating elusive f parity levels of W₂ ²Π_1/2 up to 5600?cm^?1 above the first rotational level of the electronic ground state, X ₁ ²Δ_5/2. Collisionally induced fluorescence following laser excitation at lower energies has also been recorded in the presence of a magnetic field (0.7–1?T), at Doppler limited resolution. Effective Landé factors g _J for rotational levels of the v?=?0 and 1 levels of the low-lying Ω″?=?5/2 and 3/2 components of the ²Δ and ²Π states of NiH have been derived from partially resolved Zeeman patterns. About 1600 transitions recorded in field-free conditions have been reduced to term energies relative to the lowest level of the ground state. They confirm strong spin-orbit mixing between the low-lying ligand-field states.     

Proton and Carbon‐13 NMR Studies of Some Tryptamines,Precursors, and Derivatives: Ab Initio Calculations for Optimized Structures

Abstract

Proton and carbon‐13 NMR data are presented for 5‐methoxytryptamine, 1; 6‐methoxytryptamine, 2; N,N‐diisopropyl‐5‐methoxytryptamine, 3, (5‐MeO‐DIPT); and N,N‐diisopropyl‐5‐methoxyindole‐3‐glyoxylamide, 4, at 300 MHz (¹H) and 75 MHz (¹³C) in CDCl₃ at ambient temperature. Compound 3, considered a potential hallucinogen, had been placed into Schedule I of the Controlled Substances Act, effective April 4, 2003, by the U.S. Drug Enforcement Administration. Compound 4 can serve as a possible precursor to 3. We believe that these are the first proton NMR assignments obtained at medium field (7 tesla) using selective homodecoupling and two‐dimensional homonuclear chemical shift correlation spectra (using one or more of the COSY45, COSY90, and COSYLR experiments) for rigorous aryl proton assignments in this group of compounds. Significant observed differences in the proton and carbon‐13 NMR spectra should allow facile distinction of the 5‐methoxy series, 1 and 3, from the 6‐methoxy series, 2. Energy minimizations to obtain optimized structures for each compound were performed at the Hartree–Fock level with the 6‐31G* basis set, and the resulting geometries are discussed. The presented geometry calculations appear to be the most accurate reported to date for 1 based on the basis set employed, and the first HF/6‐31G* structures for compounds 2, 3, or 4. Appreciable geometry differences in 3 and 4 for the pendant sidechain containing the N[CH(CH₃)₂]₂ moiety are noteworthy. Proximity of the carbonyl oxygens in 4 to H2 and H4 is suggested as a possible contributing factor in the deshielding of these protons in the NMR spectrum.     

A comparative study on the hydrolysis of acetic anhydride and N,N‐dimethylformamide: Kinetic isotope effect,transition‐state structure,polarity, and solvent effect

Recent studies have shown that general‐base assisted catalysis is a viable mechanistic pathway for hydrolysis of smaller anhydrides. Therefore, it is the central purpose of the present work to compare and contrast the number of hydrogen atoms in‐flight and stationary in the transition state structure of the base‐catalyzed mechanisms of 2 hydrolytic reactions as well as determine if any solvent effects occur on the mechanisms. The present research focuses on the hydrolytic mechanisms of N,N‐dimethylformamide (DMF) and acetic anhydride in alkali media of varying deuterium oxide mole fractions. Acetic anhydride has been included in this study to enable comparisons with DMF hydrolysis. Comparative studies may give synergistic insight into the detailed structural features of the activated complexes for both systems. Hydrolysis reactions in varying deuterium oxide mole fractions were conducted in concentrations of 2.0M , 2.5M , and 3.0M for DMF and 0.10M for acetic anhydride at 25°C. Studies in varying deuterium mole fractions allow for proton inventory analysis, which sheds light on the number and types of hydrogen atoms involved in the activated complex. For these systems, this type of study can distinguish between direct nucleophilic attack of the hydroxide ion on the carbonyl center and general‐base catalysis by the hydroxide ion to facilitate a water molecule attacking the carbonyl center. The numerical data are used to discuss 3 possible mechanisms in the hydrolysis of DMF.     

Spin 3/2 fermions with attractive interactions in a one-dimensional optical lattice: phase diagrams,entanglement entropy,and the effect of the trap

We study spin 3/2 fermionic cold atoms with attractive interactions confined in a one-dimensional optical lattice. Using numerical techniques, we determine the phase diagram for a generic density. For the chosen parameters, one-particle excitations are gapped and the phase diagram is separated into two regions: one where the two-particle excitation gap is zero, and one where it is finite. In the first region, the two-body pairing fluctuations (BCS) compete with the density ones. In the other one, a molecular superfluid (MS) phase, in which bound-states of four particles form, competes with the density fluctuations. The properties of the transition line between these two regions is studied through the behavior of the entanglement entropy. The physical features of the various phases, comprising leading correlations, Friedel oscillations, and excitation spectra, are presented. To make the connection with experiments, the effect of a harmonic trap is taken into account. In particular, we emphasize the conditions under which the appealing MS phase can be realized, and how the phases could be probed by using the density profiles and the associated structure factor. Lastly, the consequences on the flux quantization of the different nature of the pairing in the BCS and MS phases are studied in a situation where the condensate is in a ring geometry.     

Theoretical studies of the optical and EPR spectra for Fe ion in the MF3:Fe (M=Al, Ga) systems

By analyzing the EPR spectra of Fe³⁺ ion in the fluorinde glasses, the local lattice structures around impurity Fe³⁺ ion in MF₃:Fe³⁺ (M=Al, Ga) systems have been studied by means of diagonalizing the complete energy matrices of the electron-electron repulsion, the ligand-field and the spin-orbit coupling for a d⁵ configuration ion in a trigonal ligand-field. Both the second-order and fourth-order EPR parameters D and (a−F) are taken simultaneously in the structural investigation. The results indicate that the local lattice structure around octahedral Fe³⁺ center has an expansion distortion for Fe³⁺ in MF₃:Fe³⁺ (M=Al, Ga). The expansion distortion may be ascribed to the fact that the radius of Fe³⁺ ion is larger than that of Al³⁺ ion and Ga³⁺ ion, and the Fe³⁺ ion will push the fluoride ligands upwards and downwards, respectively. The local lattice structure parameters R=1.927 A, θ=55.538° for Fe³⁺ in AlF₃:Fe³⁺ and R=1.931 A, θ=56.09° for Fe³⁺ in GaF₃:Fe³⁺ are determined, respectively, and the EPR spectra of the MF₃:Fe³⁺ (M=Al, Ga) systems are satisfactorily explained.