Short-range interaction in three-dimensional quantum mechanics

We show that it is possible to define shape-independent three-dimensional short-range quantum interactions in two-parameter form for non-spherical angular momentum channels through double rescaling of potential strength. Unlike the special case of l=0, where the zero-range limit of the system is renormalizable, the effective ranges diverge for l≠0 channels, and the system becomes trivial at zero-size limit. It is also shown that the two-parameter representation with finite interaction range is useful in analyzing phase shifts and describing resonances with accuracy in non-spherical scatterings.     

广义Gelfand模型的正解

在不要求极限limτ→0f(τ）/τ,limτ→∞f（τ）τ存在的情况下,讨论了二阶边值问题u″(t) λh(t)f（u（t））=0,0≤t≤1,u（0）=u（1）=0的正解存在性和多重性。     

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric tor-sional contact problem of a half-space coated with func-tionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thick-ness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral trans-form technique and reduce the problem to a singular integral equation, and then numerically determine the unknown con-tact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.     

Cavitation,Invertibility, and Convergence of Regularized Minimizers in Nonlinear Elasticity

We prove that energy minimizers for nonlinear elasticity in which cavitation is allowed only at a finite number of prescribed flaw points can be obtained, in the limit as ε→0, by introducing micro-voids of radius ε in the domain at the prescribed locations and minimizing the energy without allowing for cavitation. This extends the result by Sivaloganathan, Spector, and Tilakraj (SIAM J. Appl. Math. 66:736–757, 2006) to the case of multiple cavities, and constitutes a first step towards the numerical simulation of cavitation (in the nonradially-symmetric case).      

Spectral behavior of pulsed Bessel beams focused by a dispersive lens

Starting from the Huygens–Fresnel diffraction integral, the analytical expression for the power spectrum of pulsed Bessel beams focused by a dispersive aperture lens is derived and used to study the spectral anomalies of pulsed Bessel beams in the focused field. Numerical calculation results are given to illustrate the dependence of spectral anomalous behavior on the pulse parameters, truncation parameter and material dispersion of the lens. It is shown that near the phase singularities the spectral anomalies may take place. The potential applications of spectral anomalies of ultrashort pulsed beams in information encoding and information transmission are considered.     

Infinitely many singular interactions on noncompact manifolds

We show that the ground state energy is bounded from below when there are infinitely many attractive delta function potentials placed in arbitrary locations, while all being separated at least by a minimum distance, on two dimensional non-compact manifold. To facilitate the reading of the paper, we first present the arguments in the setting of Cartan–Hadamard manifolds and then subsequently discuss the general case. For this purpose, we employ the heat kernel techniques as well as some comparison theorems of Riemannian geometry, thus generalizing the arguments in the flat case following the approach presented in Albeverio et al. (2004).     

On the correlation dimension of the spectral measure for the thue-morse sequence

Using the relationship between the decay rate of autocorrelation and the characteristics of singular Fourier spectra, we show that the correlation dimension of the spectral measure for the infinite Thue-Morse symbolic sequence equals 3 − log(1 + √17)/log 2=0.64298….     

A second-sound based, hyperbolic SIR model for high-diffusivity spread

We present an analytical study of one-dimensional (1D) kinematic wave phenomena under a hyperbolic SIR model based not on Fick's diffusion law, but rather on the inertial-type II flux law of second-sound theory. Unlike in the Ficken context, we are able to derive exact traveling wave solutions (TWS)s, as well as explicit asymptotic/approximate expressions, for both the susceptibles and infectives. We also determine, using singular surface theory, how shock-fronts resulting from initial jump discontinuities propagate and evolve under this model. In particular, critical values and special cases are examined and possible mitigation methods, which take the form of parameter-value manipulation(s), are noted.     

Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities

This is a subsequent work of our previous one in [25]. Let the nonlinear terms of non-autonomous parabolic problems with singular initial data satisfy subcritical and critical growth conditions. We first establish the existence of uniform attractors in W^1,r(Ω)$W^{1,r}(\Omega )$, 1<r<N$1<r<N$, for the family of processes corresponding to the equations with external forces being translation bounded but not translation compact. Then, we prove the existence of pullback attractors in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively, for the process corresponding to the equation with the weaker assumption on the external force than previous one. Finally, we investigate the robust of attractors and establish the existence of pullback exponential attractors for the process acting in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively.