Terms Of Eigenfunctions Research Articles

Bound and Scattering States Solution of the Relativistic Spinless Particles in View of the Multiparameter Potential

In this paper, we explore the analytical solutions for both bound and scattering states of the Klein–Gordon equation with the multiparameter potential which describes atomic, diatomic and polyatomic molecular structures via the standard method by applying a Pekeris-type approximation to the centrifugal potential. For the bound states, we obtain the energy eigenvalues and the corresponding normalized eigenfunctions in terms of hypergeometric functions. In the scattering states, the phase shift relation is derived. Besides, we investigate the special potentials, which are defined in the literature and derived from to the multiparameter potential. Finally, by using the obtained relations, we give the results both for bound and scattering states numerically and graphically.

The continuous classical Heisenberg ferromagnet equation with in-plane asymptotic conditions. I. Direct and inverse scattering theory

We develop the direct and inverse scattering theory of the linear eigenvalue problem associated with the classical Heisenberg continuous equation with in-plane asymptotic conditions. In particular, analyticity of the scattering eigenfunctions and scattering data, and their asymptotic behaviours are derived. The inverse problem is formulated in terms of Marchenko equations, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided.

Interaction of waves with a body floating on polynya between two semi-infinite ice sheets

The interaction of waves with a two-dimensional body floating on polynya between two semi-infinite ice sheets is investigated, based on a hybrid method utilizing a simple source function and eigenfunction matching. The ice sheet is modelled as a continuous thin elastic plate with uniform properties, while the fluid flow is described by the velocity potential. In the polynya, an integral equation is established by using the simple source function. In the two exterior ice covered regions, the potential is expanded in terms of eigenfunctions which satisfy the governing equation and all boundary conditions apart from that on the interface with the inner region. The unknown coefficients in the expansion and the boundary integral equation in the inner region are solved together by enforcing the continuity conditions of the pressure and normal velocity on the interface. The effectiveness and accuracy of the hybrid method is demonstrated through comparison with published results for a submerged cylinder and a floating rectangular body. Simulations are then carried out for a floating elliptical cylinder. Extensive results for the hydrodynamic force and motion response are provided, and the effects of ice draught as well as the body shape are investigated.

The total angular momentum algebra related to the [formula omitted] Dunkl Dirac equation

We consider the symmetry algebra generated by the total angular momentum operators, appearing as constants of motion of the S3 Dunkl Dirac equation. The latter is a deformation of the Dirac equation by means of Dunkl operators, in our case associated to the root system A2, with corresponding Weyl group S3, the symmetric group on three elements. The explicit form of the symmetry algebra in this case is a one-parameter deformation of the classical total angular momentum algebra so(3), incorporating elements of S3. This was obtained using recent results on the symmetry algebra for a class of Dirac operators, containing in particular the Dirac–Dunkl operator for arbitrary root system. For this symmetry algebra, we classify all finite-dimensional, irreducible representations and determine the conditions for the representations to be unitarizable. The class of unitary irreducible representations admits a natural realization acting on a representation space of eigenfunctions of the Dirac Hamiltonian. Using a Cauchy–Kowalevski extension theorem we obtain explicit expressions for these eigenfunctions in terms of Jacobi polynomials.

Inverse scattering transform and soliton solutions for square matrix nonlinear Schrödinger equations with non-zero boundary conditions

The inverse scattering transform (IST) with non-zero boundary conditions at infinity is developed for an m×m matrix nonlinear Schrödinger-type equation which, in the case m=2, has been proposed as a model to describe hyperfine spin F=1 spinor Bose–Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). The IST for this system was first presented by Ieda et al. (2007) , using a different approach. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann–Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the soliton solutions is analyzed in detail in the 2 × 2 self-focusing case, including some special solutions not previously discussed in the literature.

On the Asymptotic Behavior of Eigenfunctions of the Continuous Spectrum at Infinity in Configuration Space for the System of Three Three-Dimensional Like-Charged Quantum Particles

To our knowledge there are no complete results, even not rigorously mathematically justified, related to a system of three and more quantum particles, interacting by Coulomb pair potentials, and expressed in terms of eigenfunctions. For the system of three such identical particles, asymptotic formulas describing the behavior of eigenfunctions at infinity in configuration space are suggested.

Mathematical modelling of wave impact on floating breakwater

The impact of breaking wave on shoreline can be lessen or prevented by placing some kind of protection before the wave to reduce the speed of the wave before attacking the shoreline. Such protection can be in the form of a breakwater which is a structure designed to help reducing the wave intensity in whether in inshore waters or relatively shallow water. Thus, a mathematical model of Pressure Impulse, P is used to model the effect of waves exerted on a wall of a breakwater. A two-dimensional field of equations is derived for P which are applicable in three regions of breakwater problems by expressing this in terms of eigenfunctions that satisfy the boundary conditions apart from that the impact region and the matching of the three regions (before the breakwater, under the breakwater and after the breakwater). As in Cooker, we found that the equations of P in region 1 and region 3 are same as Cooker only that equation in region 3 has to include a secular term.

Numerical resolution of an electromagnetic inverse medium problem at fixed frequency

The aim of this paper is to solve numerically the inverse problem of determining the complex refractive index of an electromagnetic medium from partial boundary field measurements at a fixed frequency. The governing equations are the time-harmonic Maxwell equations formulated in electric field in a two-dimensional bounded domain. We express the inverse problem as the minimization of a cost function representing the difference between the measured and predicted fields. Our numerical reconstruction algorithm combines the BFGS method and an iterative process, called the Adaptive Eigenspace Inversion. The unknown complex coefficient is expanded in terms of eigenfunctions of an elliptic operator. Both the eigenspace and the mesh are iteratively adapted during the minimization procedure. Numerical experiments illustrate the performance of the reconstruction for various configurations.

New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

We consider hypo-elliptic diffusion and convection–diffusion on R3⋊S2, the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels, that is directly related to the exact solutions.

An upper bound theory to approximate the natural frequencies and parameters identification of composite beams

For the free vibration of composite beams and non-uniform beams we propose a new upper bound theory to approximate the first few natural frequencies. The Rayleigh quotient is expressed in terms of boundary functions, instead of that in terms of eigenfunctions. The boundary function satisfies all boundary conditions of the given beam, and is at least the fourth-order polynomial with leading coefficient to be one. We prove that the maximality of the Rayleigh quotient in the space of the kth order boundary functions is equivalent to the orthogonality of the kth order boundary function to lower order optimal boundary functions. Hence, we can easily find the kth order natural frequency through an orthogonalization technique provided. When the first three natural frequencies are compared with the exact or numerically found ones, good results are obtained, which confirm the applicability of the present upper bound theory. We address the inverse problems of composite beam equations, where we use the orthogonal system of boundary functions as bases to expand the unknown functions and derive linear algebraic equations to determine the expansion coefficients. As a consequence, we can fast and accurately estimate the unknown rigidity function and planar inertial function with the help of the first three natural frequencies, and the supplemented measured data of recovered function on two boundaries. The robustness of the present inversion methods is demonstrated by numerical examples.

Phase-space representations of symmetric informationally complete positive-operator-valued-measure fiducial states

The problem of finding symmetric informationally complete POVMs (SIC-POVMs) has been solved numerically for all dimensions $d$ up to 67 (A.J. Scott and M. Grassl, {\it J. Math. Phys.} 51:042203, 2010), but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM which is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct an Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.

Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium

For the $S$ states of two-electron atoms, we introduce an exact and unique factorization of the internal eigenfunction in terms of a marginal amplitude, which depends functionally on the electron-nucleus distances $r_1$ and $r_2$, and a conditional amplitude, which depends functionally on the interelectronic distance $r_{12}$ and parametrically on $r_1$ and $r_2$. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schr\"odinger equation in a form akin to the Born-Oppenheimer separation of nuclear and electronic degrees of freedom in molecules. The marginal equation involves an effective radial Hamiltonian, which contains a nonadiabatic potential energy surface that takes into account all interparticle correlations in an averaged way, and whose unique eigenvalue is the internal energy. At each point $(r_1,r_2)$, such surface is, in turn, the unique eigenvalue in the conditional equation. Employing the ground state of He as prototype, we show that the nonadiabatic potential energy surface affords a molecularlike interpretation of the structure of the atom, and aids in the analysis of energetic and spatial aspects of the Coulomb correlation, in particular correlation-induced symmetry breaking and quantum phase transition.

Study of five-dimensional potential-energy surfaces for actinide isotopes by the macroscopic-microscopic method

In this work, the nuclear potential-energy of the deformed nuclei as a function of shape coordinates is calculated in a five-dimensional (5D) parameter space of the axially symmetric generalized Lawrence shapes, on the basis of the macroscopic-microscopic method. The liquid-drop part of the nuclear energy is calculated according to the Myers-Swiatecki model and the Lublin-Strasbourg-drop (LSD) formula. The Woods-Saxon and the folded-Yukawa potentials for deformed nuclei are used for the shell and pairing corrections of the Strutinsky-type. The pairing corrections are calculated at zero temperature, T, related to the excitation energy. The eigenvalues of Hamiltonians for protons and neutrons are found by expanding the eigen-functions in terms of harmonic-oscillator wave functions of a spheroid. Then the BCS pair is applied on the smeared-out single-particle spectrum. By comparing the results obtained by different models, the most favorable combination of the macroscopic-microscopic model is known as the LSD formula with the folded-Yukawa potential. Potential-energy landscapes for actinide isotopes are investigated based on a grid of more than 4,000,000 deformation points and the heights of static fission barriers are obtained in terms of a double-humped structure on the full 5D parameter space. In order to locate the ground state shapes, saddle points, scission points and optimal fission path on the calculated 5D potential-energy surface, the falling rain algorithm and immersion method are designed and implemented. The comparison of our results with available experimental data and others' theoretical results confirms the reliability of our calculations.

Spectral Theory for Random Poincaré Maps

We consider stochastic differential equations, obtained by adding weak Gaussian white noise to ordinary differential equations admitting $N$ asymptotically stable periodic orbits. We construct a discrete-time, continuous-space Markov chain, called a random Poincaré map, which encodes the metastable behavior of the system. We show that this process admits exactly $N$ eigenvalues which are exponentially close to $1$, and we provide expressions for these eigenvalues and their left and right eigenfunctions in terms of committor functions of neighborhoods of periodic orbits. The eigenvalues and eigenfunctions are well-approximated by principal eigenvalues and quasi-stationary distributions of processes killed upon hitting some of these neighborhoods. The proofs rely on Feynman--Kac-type representation formulas for eigenfunctions, Doob's $h$-transform, spectral theory of compact operators, and a recently discovered detailed-balance property satisfied by committor functions.

In-medium covariant propagator of baryons under a strong magnetic field: Effect of the intrinsic magnetic moments

We obtain the covariant propagator at finite temperature for interacting baryons immersed in a strong magnetic field. The effect of the intrinsic magnetic moments on the Green function are fully taken into account. We make an expansion in terms of eigenfunctions of a Dirac field, which leads us to a compact form of its propagator. We present some simple applications of these propagators, where the statistical averages of nuclear currents and energy density are evaluated.

Analysis of the modified point-matching method in the electrostatic problem for axisymmetric particles

An integral modification of the generalized point-matching method (GPMMi) in the electrostatic problem for axisymmetric particles is developed. Scalar potentials that determine electric fields are represented as expansions in terms of eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAEs), which are obtained from the requirement of a minimum of the integrated residual in the boundary conditions on the particle surface. Matrix elements of ISLAEs and expansion coefficients of the “scattered” field at large index values are analyzed analytically and numerically. It is shown analytically that the applicability condition of the GPMMi coincides with that for the extended boundary conditions method (ЕВСМ). As model particles, oblate pseudospheroids \(r\left( \theta \right) = a\sqrt {1 - {^2}{{\cos }^2}\theta } ,\;{^2} = 1 - {\raise0.7ex\hbox{${{b^2}}$} \!\mathord{\left/ {\vphantom {{{b^2}} {{a_2}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{a_2}}$}} \geqslant 0\) with semiaxes a = 1 and b ≤ 1 are considered, which are obtained as a result of the inversion of prolate spheroids with the same semiaxes with respect to the coordinate origin. For pseudospheroids, the range of applicability of the considered methods is determined by the condition \({\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$b$}} < \sqrt 2 + 1\). Numerical calculations show that, as a rule, the ЕВСМ yields considerably more accurate results in this range, with the time consumption being substantially shorter. Beyond the ЕВСМ range of applicability, the GPMMi approach can yield reasonable results for the calculation of the polarizability, which should be considered as approximate and which should be verified with other approaches. For oblate nonconvex pseudospheroids (i.e., at \({\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$b$}} \geqslant \sqrt 2 \)), it is shown that the spheroidal model works well if pseudospheroids are replaced with ordinary “effective” oblate spheroids. Semiaxes aef and bef of the effective spheroids are determined from the requirement of the particle volumes, as well as from the equality of the maximal longitudinal and transverse dimensions of particles or their lengths. As a result, the polarizability of pseudospheroids can be calculated by simple explicit formulas with an error of about 0.5–2%.

Scattering amplitudes for the rationally extended [formula omitted] symmetric complex potentials

In this paper, we consider the rational extensions of two different classes of PT symmetric complex potentials namely the asymptotically vanishing Scarf II and asymptotically non-vanishing Rosen–Morse II [ RM-II] and obtain the accompanying bound state eigenfunctions in terms of the exceptional Xm Jacobi polynomials and a certain class of orthogonal polynomials. By considering the asymptotic behavior of the exceptional polynomials, we also derive the reflection and transmission amplitudes for them and discuss the various novel properties of the corresponding amplitudes.

Algebraic solutions of shape-invariant position-dependent effective mass systems

Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and Lévy-Leblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class of non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.

Feynman Propagator for a Spinless Relativistic Particle in Feshbach–Villars Representation

In this paper we used the Feynman path integral technique to calculate propagators for the Feshbach–Villars equation and the generalized Feshbach–Villars equation. We calculate these propagators for three cases: for a free particle, for a particle in a constant magnetic field and in an inhomogeneous magnetic field. Some special relations between Pauli matrices, which appear in the Feshbach–Villars equation, allow us to express the Feynman propagator for the free particle in a compact form, and in the one-dimensional case we can calculate this propagator in explicit form in terms of Bessel functions. For the second case, i.e. for a particle in a constant magnetic field, using a relation between the Feshbach–Villars Hamiltonian and a certain associated Schrodinger Hamiltonian, which appears in the Feshbach–Villars representation, we are able to express the path integral propagator for Feshbach–Villars equation in explicit form by means of Landau levels for the associated Schrodinger Hamiltonian. In the third case, i.e. for inhomogeneous magnetic field, in the same way as in the second case, we can express the path integral propagator for Feshbach–Villars equation in the explicit form in terms of eigenfunctions of problem.

On the uniform convergence of the Fourier series for one spectral problem with a spectral parameter in a boundary condition

In this paper, we investigate the uniform convergence of the Fourier series expansions in terms of eigenfunctions for the spectral problem urn:x-wiley:mma:media:mma3640:mma3640-math-0001 where λ is a spectral parameter, q(x) is a real‐valued continuous function on the interval [0,1], and a1,b0,b1,c1,d0, and d1 are real constants that satisfy the conditions urn:x-wiley:mma:media:mma3640:mma3640-math-0002 Copyright © 2015 John Wiley &amp; Sons, Ltd.