ON FINITE NORMAL SERIES SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH IRREGULAR SINGULARITY

We compute all potentials with the following property: The one-dimensional nonrelativistic Schrödinger equation for these potentials has irregular singular points at infinity and/or zero and is solved by a finite normal series. We restrict to expansion order zero, discuss some properties of the potentials obtained and, as an application, calculate for some given potentials exact solutions and energies. The aim of this paper is to provide a tool for finding exact solutions of the Schrödinger equation for a large class of singular potentials.     

On integrable rational potentials of the Dirac equation

The one-dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.     

Compacton-Like Solutions in a Camassa-Holm Type Equation

This paper shows that a Camassa-Holm type equation can be reduced to Hamiltonian system by transformation of variables. The hamiltonian system is studied by making use of the dynamical systems theory and the qualitative behavior of degenerate singular points is presented. In particular, new type of compacton-like solutions is obtained by setting the partial differential equation under boundary condition →±∞ψ(ξ)=A.     

Note on wavefront dislocation in surface water waves

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the ‘nonlinear dispersion relation’ for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the ‘soliton on finite background’, an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.     

Exact results for the Kardar-Parisi-Zhang equation with spatially correlated noise

We investigate the Kardar-Parisi-Zhang (KPZ) equation in d spatial dimensions with Gaussian spatially long-range correlated noise -- characterized by its second moment -- by means of dynamic field theory and the renormalization group. Using a stochastic Cole-Hopf transformation we derive exact exponents and scaling functions for the roughening transition and the smooth phase above the lower critical dimension . Below the lower critical dimension, there is a line marking the stability boundary between the short-range and long-range noise fixed points. For , the general structure of the renormalization-group equations fixes the values of the dynamic and roughness exponents exactly, whereas above , one has to rely on some perturbational techniques. We discuss the location of this stability boundary in light of the exact results derived in this paper, and from results known in the literature. In particular, we conjecture that there might be two qualitatively different strong-coupling phases above and below the lower critical dimension, respectively. Received 5 August 1998     

Some Schrödinger Operators with Power-Decaying Potentials and Pure Point Spectrum

We construct (deterministic) potentials such that the Schr?dinger equation on has dense pure point spectrum in for almost all boundary conditions at . As a by-product, we also obtain power-decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions. Received: 8 November 1996 / Accepted: 8 January 1997     

NLIE of Dirichlet sine-Gordon model for boundary bound states

We investigate boundary bound states of sine-Gordon model on the finite-size strip with Dirichlet boundary conditions. For the purpose we derive the nonlinear integral equation (NLIE) for the boundary excited states from the Bethe ansatz equation of the inhomogeneous XXZ spin 1/2 chain with boundary imaginary roots discovered by Saleur and Skorik. Taking a large volume (IR) limit we calculate boundary energies, boundary reflection factors and boundary Lüscher corrections and compare with the excited boundary states of the Dirichlet sine-Gordon model first considered by Dorey and Mattsson. We also consider the short distance limit and relate the IR scattering data with that of the UV conformal field theory.     

Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy

In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.     

A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method

Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.     

Burton-Miller改进型边界方程的多频计算方法

提出了一种采用Burton-Miller改进型边界积分方程进行多频计算的方法。将Burton-Miller方程中的高奇异积分转化为弱奇异积分形式,获得Burton-Miller改进型边界积分方程;将方程中格林函数进行Taylor级数展开,并把波数从方程中分离出来,从而使随波数变化的计算矩阵表示为波数的矩阵级数形式。数值分析表明,本方法不仅保证了解在全波数范围内的唯一性,并且计算频率点数较多时可以节约大量时间,提高计算效率。      

Dynamics of the universe with disformal coupling between the dark sectors

We use a dynamical analysis to study the evolution of the universe at late time for the model in which the interaction between dark energy and dark matter is inspired by a disformal transformation. We extend the analysis in the existing literature by assuming that the disformal coefficient depends both on the scalar field and its kinetic terms. We find that the dependence of the disformal coefficient on the kinetic term of scalar field leads to two classes of the scaling fixed points that can describe the acceleration of the universe at late time. The first class exists only for the case where the disformal coefficient depends on the kinetic terms. The fixed points in this class are saddle points unless the slope of the conformal coefficient is sufficiently large. The second class can be viewed as the generalization of the fixed points studied in the literature. According to the stability analysis of these fixed points, we find that the stable fixed point can take two different physically relevant values for the same value of the parameters of the model. These different values of the fixed points can be reached for different initial conditions for the equation of state parameter of dark energy. We also discuss the situations in which this feature disappears.     

A boundary condition capturing immersed interface method for 3D rigid objects in a flow

To simulate the flow around an object, we can replace the object with the fluid enclosed by a singular force. We can then simulate the flow on a fixed domain with a fluid–fluid interface supporting the singular force. In this paper, we present a boundary condition capturing approach to determine the singular force for a 3D rigid object. We apply a discontinuous body force to enforce the rigid motion of the fluid replacing the object and compute the singular force based on the kinematics of the object. Due to the singular force and the body force, the flow is not smooth across the interface. We solve the flow using the immersed interface method. Our boundary condition capturing immersed interface method is very efficient and stable, and its accuracy based on the infinity norm is near second order for the velocity and above first order for the pressure.     

Phase transitions and ordering of confined dipolar fluids

We apply a modified mean-field density functional theory to determine the phase behavior of Stockmayer fluids in slit-like pores formed by two walls with identical substrate potentials. Based on the Carnahan-Starling equation of state, a fundamental-measure theory is employed to incorporate the effects of short-ranged hard-sphere-like correlations while the long-ranged contributions to the fluid interaction potential are treated perturbatively. The liquid-vapor, ferromagnetic-liquid-vapor, and ferromagnetic-liquid-isotropic-liquid first-order phase separations are investigated. The local orientational structure of the anisotropic and inhomogeneous ferromagnetic liquid phase is also studied. We discuss how the phase diagrams are shifted and distorted upon varying the pore width.     

Path integral for Koenigs spaces

I discuss a path-integral approach for the quantum motion on two-dimensional spaces according to Koenigs, for short “Koenigs spaces”. Their construction is simple: one takes a Hamiltonian from a two-dimensional flat space and divides it by a two-dimensional superintegrable potential. These superintegrable potentials are the isotropic singular oscillator, the Holt potential, and the Coulomb potential. In all cases, a nontrivial space of nonconstant curvature is generated. We can study free motion and the motion with an additional superintegrable potential. For possible bound-state solutions, we find in all three cases an equation of the eighth order in the energy E. The special cases of the Darboux spaces are easily recovered by choosing the parameters accordingly. The text was submitted by the authors in English.     

Boundary integral solutions of coupled Stokes and Darcy flows

An accurate computational method based on the boundary integral formulation is presented for solving boundary value problems for Stokes and Darcy flows. The method also applies to problems where the equations are coupled across an interface through appropriate boundary conditions. The adopted technique consists of first reformulating the singular integrals for the fluid quantities as single and double layer potentials. Then the layer potentials are regularized and discretized using standard quadratures. As a final step, the leading term in the regularization error is eliminated in order to gain one more order of accuracy. The numerical examples demonstrate the increase of the convergence rate from first to second order and show a decrease in magnitude of the error. The coupled problems require the computation of the gradient of the Stokes velocity at the common interface. This boundary condition is also written as a combination of single and double layer potentials so that the same approach can be used to compute it accurately. Extensive numerical examples show the increased accuracy gained by the correction terms.     

Scattering in graphene with impurities: A low energy effective theory

We analyze the scattering sector of the Hamiltonians for both gapless and gapped graphene in the presence of a charge impurity using the 2D Dirac equation, which is applicable in the long wavelength limit. We show that for certain range of the system parameters, the combined effect of the short range interactions due to the charge impurity can be modelled using a single real parameter appearing in the boundary conditions. The phase shifts and the scattering matrix depend explicitly on this parameter. We argue that this parameter for graphene can be fixed empirically, through measurements of observables that depend on the scattering data.     

On Uniqueness in the General Inverse Transmisson Problem

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.     

Spectral problem for the Schrodinger equation with singular potential polynomial of even degree

For the stationary Schrodinger equation with singular potential U(x)=x^2r, where r∈Z₊, that describes situations associated with phase transition in quantum systems, discovery of symmetries can be used to reduce problems to algebraic problems for finding spectra. We construct analytic procedures for the convergent theory of perturbations to find eigenvalues for potentials of the indicated type. We discuss the fine structure of Bohr—Sommerfeld quantum conditions. Moscow State Institute of Radio Engineering, Electronics, and Automation. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 55–70, May, 1996.