THE GRADIENT ESTIMATE OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL

In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian foliations.Under some suitable conditions,we give the gradient estimates of these maps and establish a Liouville type result.

Photodetachment dynamics of H^- ion in a harmonic potential plus a time-dependent oscillating electric field

The photodetachment dynamics of H^- ion in a harmonic potential plus an oscillating electric field is studied using the time-dependent closed orbit theory. An analytical formula for calculating the photodetachment cross section of this system is put forward. It is found that the photodetachment cross section of this system is nearly unaffected for the weak oscillating electric field strength, but oscillates complicatedly when the oscillating electric field strength turns strong. In addition, the frequency of the harmonic potential and the oscillating electric field （the frequency of the harmonic potential and the frequency of the oscillating electric field are the same in the paper, unless otherwise stated.） can also affect the photodetachment dynamics of this system. With the increase of the frequency in the harmonic potential and the oscillating electric field, the number of the closed orbits for the detached electrons increased, which makes the oscillatory structure in the photodetachment cross section much more complex. Our study presents an intuitive understanding of the photodetachment dynamics driven by a harmonic potential plus an oscillating electric field from a space and time dependent viewpoint. This study is very useful in guiding the future experimental research for the photodetachment dynamics in the electric field both changing with space and time.

Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear SchrSdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.

Oscillation properties of matter–wave bright solitons in harmonic potentials

We investigate the oscillation periods of bright soliton pair or vector bright soliton pair in harmonic potentials. We demonstrate that periods of low-speed solitons are greatly affected by the position shift during their collisions. The modified oscillation periods are described by defining a characterized speed, with the aid of asymptotic analysis on related exact analytic soliton solutions in integrable cases. The oscillation period can be used to distinguish the inter-and intra-species interactions between solitons. However, a bright soliton cannot oscillate in a harmonic trap, when it is coupled with a dark soliton(without any trapping potentials). Interestingly, it can oscillate in an anti-harmonic potential, and the oscillation behavior is explained by a quasi-particle theory. The modified period of two dark-bright solitons can be also described well by the characterized speed. These results address well the effects of position shift during soliton collision, which provides an important supplement for previous studies without considering phase shift effects.

Quantum photodetachment of hydrogen negative ion in a harmonic potential subjected to static electric field

Photodetachment of negative ions has attracted immense interest owing to its fundamental nature and practical implications with regard to technology. In this study, we explore the quantum dynamics of the photodetachment cross section of negative ion of hydrogen H-in the perturbed one dimensional linear harmonic potential via static electric field. To this end,the quantum formula for total photodetachment cross section of the H-ion is derived by calculating the dipole matrix element in spherical coordinates. In order to obtain the detached electron wave function, we have solved the time-independent Schr¨odinger wave equation for the perturbed Hamiltonian of the harmonic oscillator in momentum representation. To acquire the corresponding normalized final state detached electron wave function in momentum space, we have employed an approach analogous to the WKB(Wenzel–Kramers–Brillouin) approximation. The resulting analytical formula of total photodetachment cross section depicts interesting oscillator structure that varies considerably with incident-photon energy,oscillator potential frequency, and electric field strength as elucidated by the numerical results. The current problem having close analogy with the Stark effect in charged harmonic oscillator may have potential implications in atomic and molecular physics and quantum optics.

Quantum and semiclassical studies on photodetachment cross sections of H^- in a harmonic potential

The photodetachment cross section of H- in a linear harmonic oscillator potential is investigated. This system pro- vides a rare example that can be studied analytically by both quantum and semiclassical methods with some approxi- mations. The formulas of the cross section for different laser polarization directions are explicitly derived by both the traditional quantum approach and closed-orbit theory. In the traditional quantum approach, we calculate the cross sections in coordinate representation and momentum representation, and get the same formulas. We compare the quantum formulas with closed-orbit theory formulas, and find that when the detachment electron energy is larger than hco, where co is the frequency of the oscillator potential, the quantum results are shown to be in good agreement with the semiclassical results.

Exact Radial Solution of the Non-relativistic Schrdinger Equation for the Helium Atom with the Potential Harmonic Method

We proposed a simple potential harmonic(PH) scheme for calculating the non\|relativistic radial correlation energies of atomic systems. The scheme was applied to the low\|lying \%n\%\+1\%S\%(\%n\%=1,2) and \%n\%\+3\%S\%(\%n\%=2,3) states of the helium atom. The results exhibit a very stable convergence characterization in both the angular and radial directions with PH and generalized Laguerre functions(GLF) respectively, even though the method is non\|variational one. The ninth significant figure of the non\|relativistic radial energy(NRE) calculated for the ground state exactly agrees with that of the most accurate literature data from the modified configuration interaction method. The convergent NRE′s for the excited states 2\+1\%S\%, 2\+3\%S\% and 3\+3\%S\% with the similar accuracy were also obtained.

The Harmonic Potential Theorem for a Quantum System with Time-Dependent Effective Mass

We investigate the many-body wave function of a quantum system with time-dependent effective mass, confined by a harmonic potential with time-dependent frequency, and perturbed by a time-dependent spatially homogeneous electric field. It is found that the wave function is comprised of a phase factor times the solution to the unperturbed time-dependent Schr6dinger equation with the latter being translated by a time-dependent value that satisfies the classical driven equation of motion. The wave function reduces to that of the harmonic potential theorem wave function when both the effective mass and frequency are static. An example of application is also given.

The Coherence of a Dipolar Condensate in a Harmonic Potential Superimposed to a Deep Lattice

Within the mean-field model, the coherent matter waves of a dipolar condensate in a harmonic potentiM super- imposed to a deep lattice are investigated by the variational princip]e. It is shown that, in a harmonic potential superimposed to a deep lattice, it is possible to control the decoherence of Bloch oscillations due to the fact that the on-site and the inter-site dipolar interactions can not only damp out Bloch oscillations but also maintain long-lived Bloch oscillations under the certain condition. In particular, long-lived Bloch oscillations of dipolar condensate can be realized when the dipolar interaction, the contact interaction, the frequency of the harmonic potentiM and initial width of the wave packet satisfy an analytical condition. Thus the decoherence of Bloch os- cillation can be controlled by adjusting the dipolar interaction, the contact interaction, the frequency of harmonic potentiM and the initial width of the wave packet.

Global Existence and Blow-up for a Dissipative Nonlinear Schrdinger Equation with Harmonic Potential

This paper is concerned with a dissipative nonlinear Schroedinger equation with a harmonic potential. By using some intricate inequalities and the argument of priori estimates, some behaviors of the solution are investigated.

The stability of Bose-Einstein condensates in a two-dimensional shallow trap

The stability of Bose Einstein condensates （BECs） loaded into a two-dimensional shallow harmonic potential well is studied. By using the variational method, the ground state properties for interacting BECs in the shallow trap are discussed. It is shown that the possible stable bound state can exist. The depth of the shallow well plays an important role in stabilizing the BECs, The stability of BECs in the shallow trap with the periodic modulating of atom interaction by using the Feshbach resonance is also discussed. The results show that the collapse and diffusion of BECs in a shallow trap can be controlled by the temporal modulation of the scattering length.

Sharp Criterion of Global Existence for Nonlinear Schrodinger Equation with a Harmonic Potential

This paper discusses nonlinear SchrSdinger equation with a harmonic potential. By constructing a different cross-constrained variational problem and the so-called invariant sets, we derive a new threshold for blow-up and global existence of solutions.

THE ELASTIC FIELD CASED BY TWO INCLUSIONS WITH UNI- FORM DILATATIONAL EIGENSTRAINS IN DISSIMILAR MEDIA

The displacement and the stress states cased by single inclusion are achieved from the fundamental solutions such as nuclei of strain in bimaterals. The elastic field induced by multiple inclusions in dissimilar media could be found from the superstition of that of individual precipitate. In this paper, the effect of the planner interface with parameters of depth from the interface, both pairs of elastic moduli and also shapes of the inclusion are also given, which are of great significance in physical applications.

CONCENTRATION PHENOMENA TO THE NONLINEAR SCHRDINGER EQUATION WITH HARMONIC POTENTIAL IN GENERAL DATA

We analyze the blowup problems to the nonlinear Schrodinger equation with har-monic potential. This equation always models the Bose-Einstein condensation in lower dimensions. It is known that the mass of the blowup solutions from radially symmet-ric initial data can concentrate on the point of blowup. In this paper based on the refined compactness lemma, we extend the result to general data.

On the accuracy of the potential harmonic functions

The matrix elements of the correlation function between symmetric potential harmonics am simplified into the analytical summations of the grand angular momenta by dy using the recurrence and coupling relations of the potential harmonics. 'The correlation-function potential-harmonic and generalized-Laguerre-function method (CFPHGLF), recently developed by us, is applied to the S states of the helium-like systems for Z = 2 to 9. The results exhibit good convergence with the bases in tern of both the angular and radial directions. The final eigen-energies agree excellently with the best s-limits of the variational configuration interaction (CI) method for the involved low-lying S states. The accuracy of the potential harmonic (PH) expansion scheme is discussed relative to the exact Hylleraas CI results (HCI), and Hartree-Fock results. Moreover, suggestion is given for the future improvement of the PH scheme.

An improved simple model for α decay half-lives

In this paper,using theαparticle preformation probabilities Pαfrom Xu et al.[Xu and Ren,Nucl.Phys.A 760,303(2005)],which were extracted by fitting experimental half-lives ofαdecay,based on a phenomenological harmonic oscillator potential model(HOPM)[Bayrak,J Phys G 47,025102(2020)],refitting 178αdecay half-lives of even-even nuclei obtained from the latest nuclear property table NUBASE2020,we obtain the only one adjustable parameter V0-162.6 MeV in the HOPM,i.e.,the depth of nuclear potential.The corresponding root-mean-square(rms)deviation isσ-0.322.Furthermore,to consider the contribution of centrifugal potential to unfavoredαdecay half-lives,adding a new term■(d and l are the adjustable parameter and orbital angular momentum carried away by emittedαparticle)to the logarithmic form of favoredαdecay half-lives under the HOPM framework,we propose an improved simple model(ISM)for calculating favored and unfavoredαdecay half-lives.Fitting the experimental half-lives of 205 unfavoredαdecay,we obtain d-0.381.The ISM is used to calculate the unfavoredαdecay half-lives of 128 odd-A and 77 odd-odd nuclei.The results improve by 54.2%and 53.6%,respectively,compared with HOPM.In addition,we extend the ISM to predict theαdecay half-lives of 144 nuclei with Z=117,118,119,and 120.For comparison,the improved model with eight parameters(DUR)proposed by Deng et al.[Deng,Phys.Rev.C 101,034307(2020)]and the modified universal decay law(MUDL)proposed by Soylu et al.[Soylu,Nucl.Phys.A 1013,122221(2021)]are also used.The predictions of these models and/or formulas are generally consistent with each other.

Multiquark cluster form factors in the relativistic harmonic oscillator model

A QCD multiquark cluster system is studied in the relativistic harmonic oscillator potential model （RHOPM）, and the electromagnetic form factors of the pion, proton and deuteron in the RHOPM are predicted. The calculated theoretical results are then compared with existing experimental data, finding very good agreement between the theoretical predictions and experimental data for these three target particles. We claim that this model can be applied to study QCD hadronic properties, particularly neutron properties, and to find six-quark cluster and/or nine-quark cluster probabilities in light nuclei such as helium 3He and tritium 3H. This is a problem of particular importance and interest in quark nuclear physics.

Limiting profile of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Gross-Pitaevskii equation. Using the concentration compact principle and the variational characterization of the corresponding ground state, we obtain the limiting profile of blow-up solutions with critical mass in the corresponding weighted energy space. Moreover, we extend this result to small super-critical mass case by the variational methods and scaling technique.

Dirac oscillator in noncommutative space

We study the Dirac oscillator problem in the presence of the Aharonov-Bohm effect with the harmonic potential in commutative and noncommutative spaces in S-= V and S =-V symmetry limits. We calculate exact energy levels and the corresponding eigenfunctions by the Nikiforov-Uvarov （NU） method and report the impact of the spin and the magnetic flux on the problem. Helpful numerical data is included.

Exact Solutions of the Two-Dimensional Cubic-Quintic Nonlinear Schrdinger Equation with Spatially Modulated Nonlinearities

Applying the similarity transformation, we construct the exact vortex solutions for topological charge S≥1 and the approximate fundamental soliton solutions for S = 0 of the two-dimensional cubic-quintic nonlinear Schrodinger equation with spatially modulated nonlinearities and harmonic potential. The linear stability analysis and numerical simulation are used to exam the stability of these solutions. In different profiles of cubic-quintic nonlinearities, some stable solutions for S 〉 0 and the lowest radial quantum number n = 1 are found. However, the solutions for n ≥ 2 are all unstable.