First-order Singularity Research Articles

Behaviors of Eigenvalues and Eigenfunctions of The Singular Shrödinger Operator

Let us show the boundary value problem $L\left( q\right) $ with the $-y^{^{\prime\prime}}+q(x)y=\lambda y$ differential equation in the $\left[0,1\right] $ interval, and the $y(0)=0,y(1)=0$ boundary conditions in $\sigma\left( x\right) \equiv\int\limits_{0}^{x}q(t)dt.$ It is important to examine this operator as the solution to many problems of quantum physics is closely linked to the learning of the spectral properties of the operator $L\left( q\right) $. Singular Shr\"{o}dinger operators are characterized by the assumption that, in classical theory, the function $q(x)$ is not summable in the interval $\left[ a,b\right] $ for example it has singularity that cannot be integrated in at least one of the end points of the interval or at one of its internal points, or that the interval $\left( a,b\right) $ is infinite interval. In the present study, firstly, the operator of $L\left( q\right) $ will be proved to be well-defined in the class of distribution functions with first-order singularity, which is the larger class of functions. In the following step, the concepts of eigenvalue and eigenfunctions are defined for the well-defined $L\left( q\right) $ operator and the representations for their behaviour are obtained.

Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients

We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved.
 In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U .
 Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem
 of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].

Задачи типа Гильберта для уравнения Коши–Римана с сингулярной окружностью в младших коэффициентах

The Cauchy-Riemann system of equations occupies a special place in the class of first-order elliptic systems. Such a system with lowest terms and a right-hand side is called the generalized Cauchy-Riemann system (GCRS), which can be conveniently studied by making a transition from a real space into a complex space. There are several different mathematical theories of equations that generalize the methods of the theory of functions of a complex variable. In this regard, the work of L. Bers, in which the integration operations with respect to a complex variable for a generalized system of Cauchy-Riemann type are generalized, should primarily be mentioned. This approach has received a well-known finalization in the theory of pseudo-analytic functions. In the works of G.N. Polozhiy, the theory of p-analytic functions was developed, which is close in its ideas to the works of L. Bers. Another, more progressive research area, called «generalized analytical functions», was developed by the school of I.N. Vekua and his followers (B.V. Boyarsky and others). Here, the idea of correspondence between the functions of a complex variable and the solutions of the generalized Cauchy–Riemann equation using the functional analysis techniques is developed. The Vekua theory is constructed on the assumption that the coefficients of the function’s lowest terms belong to the space of summable functions with degree р > 2. The coefficients of such systems may admit «weak» singularities limited by the requirement of p-integrability. Thus, the Vekua theory does not cover even equations with coefficients having first-order singularities. However, other problems, the lowest coefficients of which admit first- order singularities or «strong singularities», boil down to GCRS. In this study, Hilbert type problems are solved for the GCRS the lowest coefficients of which admit strong singularity in a circle. &nbsp

Full-wave method for the analysis of the radiation characteristics of a VLF source in the atmosphere

The basis of the full-wave method is the propagation matrix method and a plane wave expansion, which is generally used for the analysis of radiation problems in layered media. The radiation field in the spatial domain is represented as an integral in the wavenumber domain. By integrating the vertical wavenumber, the radiation can be represented as a superposition of plane waves with different directions. If the atmosphere and ionosphere are regarded as layered media, the vertical wavenumber integral contains first-order singularities when the radiation source is located in the ionosphere, which are relatively easy to handle. When the radiation source is located in the air, the vertical wavenumber integral contains second-order singularities, which are more complicated. Therefore, there are few full-wave studies on radiation characteristics in the air and ionosphere when a very-low-frequency (VLF) source is located in the air. In this paper, air is considered as an isotropic medium with a micro-conductivity, the vertical wave number integral that has second-order singularities is solved, and the radiation fields in the air and the ionosphere are calculated. The numerical results show the correctness and effectiveness of the proposed algorithm.

On the sharpness of Jackson’s inequality in the spaces L p on the half-line with power weight

We obtain asymptotic representations as $\lambda \to \infty$ in the upper and lower half-planes for the solutions of the Sturm--Liouville equation $$ -y"+p(x)y'+q(x)y= \lambda ^2 \rho(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ under the condition that $q$ is a distribution of the first-order singularity, $\rho$ is a positive absolutely continuous function, and $p$ belongs to the space $L_2[a,b]$. In supplementary part, the results are generilized on equation of the following type $$-(r^2y')'+py'+qy=\lambda ^2 \rho^2 y, \qquad x\in [a,b] \subset \mathbb{R},$$ where $\lambda^2$ is the large parameter, $r$ and $\rho$ are positive functions, while $p$ and $q$ are complex valued ones. It is assumed that $p\in L_1[a,b],\quad q\in W_2^{-1}[a,b], \quad \rho ,r \in AC[a,b] =W_1^1[a,b],$ moreover, \begin{equation*} \rho'u, r'u, pu \in L_1[a,b], \quad \text{where }\, u=\int q \, dx, \end{equation*} and the antiderivative is understood in the sense of distributions.

Asymptotics of the solutions of the Sturm–Liouville equation with singular coefficients

We obtain asymptotic representations as λ→∞in the upper and lower half-planes for the solutions of the Sturm–Liouville equation $$- y'' + p\left( x \right)y' + q\left( x \right)y = {\lambda ^2}p\left( x \right)y$$ , x ∈ [a,b] ⊂ R under the condition that q is a distribution of first-order singularity, ρ is a positive absolutely continuous function, and p belongs to the space L2[a, b].

The Widom line as the crossover between liquid-like and gas-like behaviour in supercritical fluids

By conventional definition, a supercritical fluid is one that doesn’t exhibit distinct liquid- or gas-like states. This may need to be revised in light of measurements that show a sharp change in the speed of sound in supercritical argon when it crosses a well-defined line on its pressure versus temperature phase diagram. According to textbook definitions1, there exists no physical observable able to distinguish a liquid from a gas beyond the critical point, and hence only a single fluid phase is defined. There are, however, some thermophysical quantities, having maxima that define a line emanating from the critical point, named ‘the Widom line’2 in the case of the constant-pressure specific heat. We determined the velocity of nanometric acoustic waves in supercritical fluid argon at high pressures by inelastic X-ray scattering and molecular dynamics simulations. Our study reveals a sharp transition on crossing the Widom line demonstrating how the supercritical region is actually divided into two regions that, although not connected by a first-order singularity, can be identified by different dynamical regimes: gas-like and liquid-like, reminiscent of the subcritical domains. These findings will pave the way to a deeper understanding of hot dense fluids, which are of paramount importance in fundamental and applied sciences.

Path-independent integrals to identify localized plastic events in two dimensions

We use a power expansion representation of plane-elasticity complex potentials due to Kolossov and Muskhelishvili to compute the elastic fields induced by a localized plastic deformation event. Far from its center, the dominant contributions correspond to first-order singularities of quadrupolar and dipolar symmetry which can be associated, respectively, with pure deviatoric and pure volumetric plastic strain of an equivalent circular inclusion. By construction of holomorphic functions from the displacement field and its derivatives, it is possible to define path-independent Cauchy integrals which capture the amplitudes of these singularities. Analytical expressions and numerical tests on simple finite-element data are presented. The development of such numerical tools is of direct interest for the identification of local structural reorganizations, which are believed to be the key mechanisms for plasticity of amorphous materials.

Optimal fed-batch operation of recombinant cells subject to plasmid instability and death

Optimal feed rate strategy is studied for fed-batch culture of recombinant cells with plasmid instability and with different death rates for the plasmid-free cells (PFC) and plasmid-bearing cells (PBC). Most of the fed-batch fermentation is known to have first-order singularity and therefore a single singular arc. However, this study shows that a singular arc with second-order singularity and therefore two distinct singular arcs are possible for a recombinant cell process if PFC and PBC are subjected to death, and their specific growth rates are proportional to each other. Two types of singular arcs are elucidated and analyzed. The optimal policies over the singular arcs are theoretically explored as these findings reveal qualitative information on the singular arc, which is critically important in providing the optimal initial conditions in numerical computation of optimal feed rate profile.

Random-lattice models and simulation algorithms for the phase equilibria in two-dimensional condensed systems of particles with coupled internal and translational degrees of freedom.

In this work we concentrate on phase equilibria in two-dimensional condensed systems of particles where both translational and internal degrees of freedom are present and coupled through microscopic interactions, with a focus on the manner of the macroscopic coupling between the two types of degrees of freedom. First, an unconventional description of the translational degrees of freedom is developed, in which the randomly varying spatial connectivity of the particles is represented by a random lattice whose dynamic structure is given by triangulating the spatial configurations. Based on this random-lattice description, a series of three statistical-mechanical models are then constructed. All of the three models are in essence spin-\textonehalf{} Ising models where the spins, representing internal degrees of freedom, are associated with hard-disk particles and nearest-neighbor particles interact through spin-spin interactions that may have spatial dependence. The fluctuating number of nearest neighbors and the possible spatial dependence of the spin-spin interactions couple microscopically the spin degrees of freedom to the translational degrees of freedom. The first model (I) is a random-lattice Ising model with conventional nearest-neighbor spin-spin interactions. The second model (II) is an extension of this model to include a spatial dependence of the nearest-neighbor spin-spin interactions. The third model (III) is a modification of the second model that accounts for spin states with different internal degeneracy. Monte Carlo simulation techniques, including both a special algorithm for the random-lattice description and histogram and finite-size scaling analysis, are used to investigate the phase behavior of all three models. It is shown that the order-disorder spin transition in model I is decoupled from a first-order singularity---lattice melting---associated with the translational degrees of freedom and remains critical and falls in the universality class of the standard two-dimensional Ising model on regular lattices. Model II is shown to exhibit a phase diagram that has a region where the spin degrees of freedom are slaved by the translational degrees of freedom and develop a first-order singularity in the order-disorder transition that accompanies the lattice-melting transition. The internal degeneracy of the spin states in model III implies that the spin order-disorder singularity can be of first order throughout the phase diagram. It is found that this first-order singularity can be either coupled to or decoupled from the lattice-melting singularity, depending on the strength of the microscopic coupling. The calculated phase diagram and the associated thermodynamic transitional properties for model III are discussed in relation to experiments on planar bilayers of lipid-chain molecules whose properties are determined by a subtle coupling between the translational variables and the intrachain conformational states.

The Interface Anticrack and Green’s Functions for Interacting Anticracks and Cracks/Anticracks

Solution is obtained for an anticrack-a bonded rigid lamella inclusion-at the interface between two isotropic elastic solids. The problem is formulated in terms of distributed line-loads at the anticrack which constitute the Green’s functions and the system of the governing coupled integral equations is solved analytically in closed form for the cases of uniform biaxial tension and of anticracks loaded by concentrated forces or moment. Solutions are also obtained by the interaction of an interface anticrack with a first-order singularity (concentrated force and dislocation) and second-order singularity (doublet of forces) at the interface. In the latter case the limit as the second-order singularity approaches the tip of the anticrack does not exist, but neither can a finite limit be obtained by reseating as in the homogeneous material. The solution of the interface anticrack exhibits the oscillatory singularities that appear at interface cracks which indicates that the overlapping of the displacement on the crack faces is not’the reason for this anomalous behavior. Moreover, it should be pointed out that the material condition that the stress does not exhibit oscillatory behavior is not the same as for interface cracks: for anticracks it is κ1 (1−β) = κ2 (1 + β) while for cracks it is β = 0.

Partial differential equations with solutions having movable first-order singularities

This paper deals with the examination of nonlinear partial differential equations with solutions which have movable singularities of first order. Multi-phase and rational solutions of the family of nonintegrable equations are demonstrated. The equations can be integrable in other cases depending on the correlations of the coefficients of these equations.