Symmetric Potential Research Articles

Supersymmetry of 𝒫𝒯-symmetric tridiagonal Hamiltonians

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian [Formula: see text] and its supersymmetric partner [Formula: see text] in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to [Formula: see text] can be recovered from those polynomials arising from the same problem for [Formula: see text] with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of [Formula: see text]-symmetric complex potentials. Finally, we solve the shifted [Formula: see text]-symmetric Morse oscillator exactly in the tridiagonal representation.

New methods of isochrone mechanics

Isochrone potentials are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Whereas all previous results on isochrone mechanics have been established using classical analysis and geometry, in this article, we revisit the isochrone problem of motion using tools from Hamiltonian dynamical systems. In particular, we (1) solve the problem of motion using a well-adapted set of angle-action coordinates and generalize the notion of eccentric anomaly to all isochrone orbits, and (2) we construct the Birkhoff normal form for a particle orbiting a generic radial potential and examine its Birkhoff invariants to prove that the class of isochrone potentials is in correspondence with parabolas in the plane. Along the way, several fundamental results of celestial mechanics, such as the Bertrand theorem or the Kepler equation and laws, are obtained as special cases of more general properties characterizing isochrone mechanics.

Weak Feature Extraction of Local Gear Damage Based on Underdamped Asymmetric Periodic Potential Stochastic Resonance

The enhancement of the detection of weak signals against a strong noise background is a key problem in local gear fault diagnosis. Because the periodic impact signal generated by local gear damage is often modulated by high-frequency components, fault information is submerged in its envelope signal when demodulating the fault signal. However, the traditional bistable stochastic resonance (BSR) system cannot accurately match the asymmetric characteristics of the envelope signal because of its symmetrical potential well, which weakens the detection performance for weak faults. In order to overcome this problem, a novel method based on underdamped asymmetric periodic potential stochastic resonance (UAPPSR) is proposed to enhance the weak feature extraction of the local gear damage. The main advantage of this method is that it can better match the characteristics of the envelope signal by using the asymmetry of its potential well in the UAPPSR system and it can effectively enhance the extraction effect of periodic impact signals. Furthermore, the proposed method enjoys a good anti-noise capability and robustness and can strengthen weak fault characteristics under different noise levels. Thirdly, by reasonably adjusting the system parameters of the UAPPSR, the effective detection of input signals with different frequencies can be realized. Numerical simulations and experimental tests are performed on a gear with a local root crack, and the vibration signals are analyzed to validate the effectiveness of the proposed method. The comparison results show that the proposed method possesses a better resonance output effect and is more suitable for weak fault feature extraction under a strong noise background.

Graphene Oxide over-Layer on Liquid Metal Coated Cu Current Collector for Improved Cycle Performance of Lithium Metal Anode

Lithium metal is a promising anode material for Li-ion batteries because of its high specific capacity and negative electrochemical potential. However, there are two major challenges that need to be overcome before it can be used in practical applications: (1) Formation of dendrites that can cause short circuit of the cell leading to safety issues and (2) decomposition of electrolyte when contacting the reactive lithium metal that shortens the cycle lifetime. Keys to realize the lithium metal anode for practical application is to enable uniform deposition of lithium metal preventing dendrites and develop stable solid electrolyte interface (SEI) layer between lithium metal and the electrolyte.In this work, we propose a liquid metal (LM) coated on a copper current collector with graphene oxide (GO) overlayer to grow a stable SEI to uniformly deposit Li and protect it from direct contact with the electrolyte by SEI grown on GO. First, preconditioning was performed to develop SEI on the top layer of the anode. Ga-Sn-In alloy, which is in a liquid state at room temperature, inproves wetting of lithium promoting the uniform deposition and prevents the nucleation of the dendrites. Due to the high wettability of the LM against lithium and the SEI layer, the lithium deposition takes place between the LM and the to SEI layer preventing direct contact of lithium with the electrolyte and undesired chemical reactions.The effect of GO/LM coating was verified using a Li-Li symmetric cell. The result showed significant reduction in the reaction overpotentials on compared with LM coated and bare Cu current collectors. The low and symmetric reaction potentials were observed even after increasing the current density and areal capacity to 4.0 mA/cm2 and 8.0 mAh/cm2, respectively. Scanning electron microscopy (SEM) analysis of GO/LM coating before and after the conditioning and Li pre-deposition revealed even after the large amount of Li plating (> 6 mAh/cm2) the surface remains flat without any dendrites. This indicates a stable artificial SEI layer was formed and Li was deposited between the LM and the SEI layer. A full-cell test using LiNi1/3Co1/3Mn1/3O2 cathode with GO/LM anode showed much improvement in the cycle lifetime compared with the simple LM-coated and bare Cu. Figure caption : SEM images of SEI development and lithium plating on GO/LM coated Cu current collector. Figure 1

Solving forward and inverse problems of the nonlinear Schrödinger equation with the generalized -symmetric Scarf-II potential via PINN deep learning

In this paper, based on physics-informed neural networks (PINNs), a good deep learning neural network framework that can be used to effectively solve the nonlinear evolution partial differential equations (PDEs) and other types of nonlinear physical models, we study the nonlinear Schrödinger equation (NLSE) with the generalized -symmetric Scarf-II potential, which is an important physical model in many fields of nonlinear physics. Firstly, we choose three different initial values and the same Dirichlet boundary conditions to solve the NLSE with the generalized -symmetric Scarf-II potential via the PINN deep learning method, and the obtained results are compared with those derived by the traditional numerical methods. Then, we investigate the effects of two factors (optimization steps and activation functions) on the performance of the PINN deep learning method in the NLSE with the generalized -symmetric Scarf-II potential. Ultimately, the data-driven coefficient discovery of the generalized -symmetric Scarf-II potential or the dispersion and nonlinear items of the NLSE with the generalized -symmetric Scarf-II potential can be approximately ascertained by using the PINN deep learning method. Our results may be meaningful for further investigation of the nonlinear Schrödinger equation with the generalized -symmetric Scarf-II potential in the deep learning.

Light inflaton model in a metastable universe

We minimally extend the Standard Model (SM) with a ${\mathrm{Z}}_{2}$ symmetric potential containing a single scalar field, serving as our inflaton with a quartic self-coupling. In the model we have symmetry breaking in both sectors, and with the addition of an inflaton-Higgs portal, the universe is able to efficiently reheat via 2-2 inflaton-Higgs scattering. Assuming that the universe with a positive cosmological constant should be metastable, only one particular symmetry breaking pattern in the vacuum is possible, without the need to finely tune the Higgs quartic self-coupling. Inflatons with masses in the range $O({10}^{\ensuremath{-}3})\ensuremath{\le}{m}_{\ensuremath{\chi}}\ensuremath{\le}{m}_{h}$ and mixing angles that span ${\ensuremath{\theta}}_{m}^{2}=O({10}^{\ensuremath{-}11}--{10}^{\ensuremath{-}2})$ evade all current cosmological, experimental, and stability constraints required for a metastable electroweak vacuum. Upgraded particle physics experiments may be able to probe the parameter space with ${\ensuremath{\theta}}_{m}^{2}\ensuremath{\ge}O({10}^{\ensuremath{-}4})$, where we would observe trilinear Higgs couplings suppressed by up to 2% compared to the SM value. However, to access the parameter space of very weakly coupled inflatons, we rely on the proposals to build experiments that target the hidden sector.

Distorted light bullet in a tapered graded-index waveguide with PT symmetric potentials

The light bullet is obtained in tapered graded-index waveguide with parity-time-symmetric potentials. Different kind of parity-time (PT) symmetric potentials admits different shape of light bullet. The result shows the light bullet may be distorted with Rosen Morse PT and sech-type PT-symmetrical potential. Moreover, the size and position of the light bullet can be controlled by changing some key parameters. These results are useful for the potential applications in the tapered graded-index waveguide amplifier with PT symmetric potentials and three-dimensional optical information storage.

“Striped” rectangular rigid box with Hermitian and non-Hermitian PT symmetric potentials

Eigenspectra of a spinless particle inside a rigid rectangular box subject to diverse inner potential distributions are investigated under both Hermitian and non-Hermitian antiunitary PT (composite parity and time-reversal) symmetric regimes. Four “stripes” conjoined widthwise, spanning the box bearing piecewise constant potentials, are studied whose exact energy eigenspectra are obtained employing matrix mechanics. Diverse real-Hermitian and complex non-Hermitian PT symmetric potential compositions are considered separately and in conjunction, unraveling peculiar retention and breakdown scenarios engendered by PT symmetry. Some states exhibit remarkable crossovers of symmetry “making” and “breaking”: a broken PT gets reinstated, while higher levels “collude” to continue with symmetry breaking. Furthermore, a charged particle in a PT symmetric electric field imposed on the striped potential backdrop reveals peculiar symmetry retention and breakdown scenarios. Depictions of prominent probability-density redistributions under the norm conserving unitary regime, as well as non-conserving post-PT-breakdown, are presented.

Vacuum stability conditions and potential minima for a matrix representation in lightcone orbit space

The orbit space for a scalar field in a complex square matrix representation obtains a Minkowski space structure from the Cauchy–Schwarz inequality. It can be used to find vacuum stability conditions and minima of the scalar potential. The method is suitable for fields such as a bidoublet, an SU(2) triplet or SU(3) octet. We use the formalism to find the vacuum stability conditions for the left-right symmetric potential of a bidoublet and left and right Higgs doublets.

Height function delocalisation on cubic planar graphs

The interest is in models of integer-valued height functions on shift-invariant planar graphs whose maximum degree is three. We prove delocalisation for models induced by convex nearest-neighbour potentials, under the condition that each potential function is an excited potential, that is, a convex symmetric potential function $V$ with the property that $V(\pm1)\leq V(0)+\log2$. Examples of such models include the discrete Gaussian and solid-on-solid models at inverse temperature $\beta\leq\log2$, as well as the uniformly random $K$-Lipschitz function for fixed $K\in\mathbb N$. In fact, $\beta V$ is an excited potential for any convex symmetric potential function $V$ whenever $\beta$ is sufficiently small. To arrive at the result, we develop a new technique for symmetry breaking, and then study the geometric percolation properties of sets of the form $\{\varphi\geq a\}$ and $\{\varphi\leq a\}$, where $\varphi$ is the random height function and $a$ a constant. Along the same lines, we derive delocalisation for models induced by convex symmetric nearest-neighbour potentials which force the parity of the height of neighbouring vertices to be distinct. This includes models of uniformly random graph homomorphisms on the honeycomb lattice and the truncated square tiling, as well as on the same graphs with each edge replaced by $N$ edges linked in series. The latter resembles cable-graph constructions which appear in the analysis of the Gaussian free field.

Plasma screening of nuclear fusion reactions in liquid layers of compact degenerate stars: a first-principle study

ABSTRACT A reliable description of nuclear fusion reactions in inner layers of white dwarfs and envelopes of neutron stars is important for realistic modelling of a wide range of observable astrophysical phenomena from accreting neutron stars to Type Ia supernovae. We study the problem of screening of the Coulomb barrier impeding the reactions by a plasma surrounding the fusing nuclei. Numerical calculations of the screening factor are performed from the first principles with the aid of quantum-mechanical path integrals in the model of a one-component plasma of atomic nuclei for temperatures and densities typical for dense liquid layers of compact degenerate stars. We do not rely on various quasi-classic approximations widely used in the literature, such as factoring out the tunnelling process, tunnelling in an average spherically symmetric mean-force potential, usage of classic free energies and pair correlation functions, linear mixing rule, and so on. In general, a good agreement with earlier results from the thermonuclear limit to Γ ∼ 100 is found. For a very strongly coupled liquid 100 ≲ Γ ≤ 175, a deviation from currently used parametrizations of the reaction rates is discovered and approximated by a simple analytic expression. The developed method of nuclear reaction rate calculations with account of plasma screening can be extended to ion mixtures and crystallized phases of stellar matter.

Calculation of reaction rate constants from a classical Wigner model based on a Feynman path integral open polymer

The Open Polymer Classical Wigner (OPCW) method [J. Chem. Phys. 152, 094111 (2020)] is applied to a flux-Heaviside trace for calculating reaction rate constants. The obtained expression for the reaction rate constant is tested on a symmetric Eckart potential. The OPCW method is shown to converge toward the exact classical Wigner result. The OPCW method shows some promise for rate constant calculations and suggestions for future tests are made.

Vibrational Excitation Cross-Section by Positron Impact: A Wave-Packet Dynamics Study

The vibrational excitation cross-section of a diatomic molecule by positron impact is obtained using wave-packet propagation techniques. The dynamics study was carried on a two-dimensional potential energy surface, which couples a hydrogenlike harmonic oscillator to a positron via a spherically symmetric correlation polarization potential. The cross-section for the excitation of the first vibrational mode is in good agreement with previous reports. Our model suggests that a positron couples to the target vibration by responding instantly to an interaction potential, which depends on the target vibrational coordinate.

Giant and tunable Goos-Hänchen shift with a high reflectance induced by PT-symmetry in atomic vapor.

The Goos-Hänchen (GH) shifts of light beams reflected from conventional passive optical systems could be enhanced using the Brewster angle effect or resonance effect, but the maximum GH shift is located at the reflectance minima, which is difficult for experimental detection. In this paper, we present an efficient and flexible scheme to realize complex parity-time (PT)-symmetric periodic optical potentials (complex crystals) in helium atomic vapor. The GH shifts of probe light reflected from the complex crystal are theoretically investigated and large GH shifts could be obtained inside the high-reflection band. When the complex crystal is operated near the coherent perfect absorption-laser point, the maximum GH shift of probe light is exactly located at the reflectance peak. Moreover, the GH shifts could be easily controlled by adjusting the intensity of control light.

Towards a symmetric reversible single-molecule switch: Amino-imino-cyclo-n-enes

We propose cyclic 5- and 7-ring structures with alternating single and double bonds and adjacent imino and amino groups as candidates for switches in molecular electronics, with amino-imino tautomerisation as the switching mechanism. Due to the C2V-symmetric transition state, the molecules exhibit a symmetric double-well potential with identical energies for the two states, which is a desirable property for a functioning molecular switch. Calculations at the double hybrid mPW2PLYP-D2/def2-TZVP level show barriers of 1.07 and 0.52 eV for the 5-ring and 7-ring, respectively (zero-point corrected: 0.97 and 0.41 eV, respectively). The corresponding 9-ring structure is not suitable as a molecular switch, due to ring puckering and the existence of multiple minima. Attachment of ethyne groups to the nitrogens and the opposite carbon, as models for molecular wires, only slightly changes the barrier heights. The 5- and 7-ring structures are promising switch candidates for further investigation.

Stability of one and two-dimensional spatial solitons in a cubic–quintic–septimal nonlinear Schrödinger equation with fourth-order diffraction and [formula omitted]-symmetric potentials

In this paper, the existence and stability of solitons in parity-time (PT)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with fourth-order diffraction (FOD) coefficient and higher-order nonlinearities have been investigated. For the linear case, we have demonstrated numerically that, the FOD parameter can alter the PT-breaking points. Exact analytical expressions of the localized modes are obtained respectively, in one and two dimensional nonlinear Schrödinger (NLS) equation with both self-focusing and self-defocusing Kerr, and higher-order nonlinearities for nonlinear case. The effects of both FOD and higher-order nonlinearities on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Some stable and unstable solutions have been given and, it has been seen how higher-order self-focusing and self-defocusing nonlinearities can influence the stability/instability of the system.

On nonexistence of continuous families of stationary nonlinear modes for a class of complex potentials

AbstractThere are two cases when the nonlinear Schrödinger equation with an external complex potential is well known to support continuous families of localized stationary modes: the ‐symmetric potentials and the Wadati potentials. Recently, Kominis et al. have suggested that the continuous families can be also found in complex potentials of the form , where is an arbitrary real and is a real‐valued and bounded differentiable function. Here we study in detail nonlinear stationary modes that emerge in complex potentials of this type (for brevity, we call them W‐dW potentials). First, we assume that the potential is small and employ asymptotic methods to construct a family of nonlinear modes. Our asymptotic procedure stops at the terms of the order, where small characterizes amplitude of the potential. We therefore conjecture that no continuous families of authentic nonlinear modes exist in this case, but “pseudo‐modes” that satisfy the equation up to ‐error can indeed be found in W‐dW potentials. Second, we consider the particular case of a W‐dW potential well of finite depth and support our hypothesis with qualitative and numerical arguments. Third, we simulate the nonlinear dynamics of found pseudo‐modes and observe that, if the amplitude of W‐dW potential is small, then the pseudo‐modes are robust and display persistent oscillations around a certain position predicted by the asymptotic expansion. Finally, we study the authentic stationary modes that do not form a continuous family, but exist as isolated points. Numerical simulations reveal dynamical instability of these solutions.

Ground-state-energy universality of noninteracting fermionic systems

When noninteracting fermions are confined in a $D$-dimensional region of volume $\mathrm{O}(L^D)$ and subjected to a continuous (or piecewise continuous) potential $V$ which decays sufficiently fast with distance, in the thermodynamic limit, the ground state energy of the system does not depend on $V$. Here, we discuss this theorem from several perspectives and derive a proof for radially symmetric potentials valid in $D$ dimensions. We find that this universality property holds under a quite mild condition on $V$, with or without bounded states, and extends to thermal states. Moreover, it leads to an interesting analogy between Anderson's orthogonality catastrophe and first-order quantum phase transitions.

Compact formulation of the statistical moment method for the solution of the Fokker–Planck equation for two coupled macrospins

The statistical moment method hitherto formulated for single-macrospin relaxation problems is extended to the Fokker–Planck equation for two coupled macrospins describing the time evolution of the joint distribution function of magnetization orientations of an ensemble of such nanoparticles. The method is based on the reduction of that equation via expansion of the joint distribution function in an appropriate orthogonal basis to the solution of a hierarchy of multi-term differential-recurrence equations for the statistical moments (averaged products of spherical harmonics alias the Fourier coefficients), which may then be solved via standard matrix methods. Two particular examples of circularly symmetric potentials are mentioned, which simplify the hierarchy of statistical moment equations. The simplified equations, almost wholly expressed in terms of Clebsch-Gordan coefficients, which may be automatically calculated, agree with those previously derived via direct averaging of the governing Langevin equations with change of the random variables to products of spherical harmonics however leading to cumbersome expressions as originally formulated.

Gray solitons in parity-time-symmetric localized potentials with fractional-order diffraction

The properties of existence and stability of gray solitons in parity-time (PT)-symmetric localized potentials with fractional-order diffraction are studied. This system can support stable gray solitons. The influences of the Lévy index and the real and imaginary parts of the complex potentials on existence, stability, and grayness of these solitons are carefully investigated. Especially, if some coefficients of PT-symmetric potentials in nonlinear fractional Schrödinger equation are changed, the transition between gray soliton and anti-dark soliton will occur. In addition, we also discuss the transverse energy flow in the gray and anti-dark solitons with fractional-order diffraction.