Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I

We consider the Zakharov equation in space dimension two

Regularity of weak solutions to a two-dimensional modified Dirac-Klein-Gordon system of equations

We show that solutions to the modified Dirac-Klein-Gordon system in standard notation

An expansion of Hartree-Fock and correlation energies,relativistic corrections and the dipole matrix element in the powers of of 1/Z

Feynman diagrammatic technique was used for the calculation of Hartree-Fock and correlation energies, relativistic corrections, dipole matrix element. The whole energy of atomic system was defined as a polen-electron Green function. Breit operator was used for the calculation of relativistic corrections. The Feynman diagrammatic technique was developed for 〈H_B>. Analytical expressions for the contributions from diagrams were received. The calculations were carried out for the terms of such configurations as 1s² 2sⁿ¹ 2pⁿ² (2 ≧n₁≧ 0, 6≧ n₂ ≧ 0). Numerical results are presented for the energies of the terms in the form $$E = E_0 Z^2 + \Delta {\rm E}_2 + \frac{1}{Z}\Delta {\rm E}_3 + \frac{{\alpha ^2 }}{4}(E_0^r + \Delta {\rm E}_1^r Z^3 )$$ and for fine structure of the terms in the form $$\begin{gathered} \left\langle {1s^2 2s^{n_1 } 2p^{n_2 } LSJ|H_B |1s^2 2s^{n_1 \prime } 2p^{n_2 \prime } L\prime S\prime J} \right\rangle = \hfill \\ = ( - 1)^{\alpha + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 1} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 [E^{(0)} (Z - B) + \varepsilon _{co} ] + \hfill \\ + ( - 1)^{L + S\prime + J} \left\{ {\begin{array}{*{20}c} {L S J} \\ {S\prime L\prime 2} \\ \end{array} } \right\}\frac{{\alpha ^2 }}{4}(Z - A)^3 \varepsilon _{cc} . \hfill \\ \end{gathered} $$ Dipole matrix elements are necessary for calculations of oscillator strengths and transition probabilities. For dipole matrix elements two members of expansion by 1/Z have been obtained. Numerical results were presented in the form P(a,a′) = a/Z(1+τ/Z).     

Long-Time Asymptotics of Solutions to the Cauchy Problem for the Defocusing Nonlinear Schrödinger Equation with Finite-Density Initial Data. II. Dark Solitons on Continua

For Lax-pair isospectral deformations whose associated spectrum, for given initial data, consists of the disjoint union of a finitely denumerable discrete spectrum (solitons) and a continuous spectrum (continuum), the matrix Riemann–Hilbert problem approach is used to derive the leading-order asymptotics as of solutions to the Cauchy problem for the defocusing nonlinear Schrödinger equation ( NLSE), , with finite-density initial data

A nonlinear instability for 3×3 systems of conservation laws

The phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2-periodic solutionsu ^N C ([0,t _N ] × witht _N bounded, satisfying

The equation of motion and gravitational radiation of a small extended spherically symmetric mass

On the basis of an approximation method developed in a previous paper the motion of an extended small mass on a gravitational background is investigated. The mass is described by a spherically symmetric rest mass distribution with some form of rigidity; the smallness of the mass is defined by the assumption that the radius of the mass is small compared with the change of the background . The equation of motion is yielded by integrating Einstein's conservation law of energy and momentum over the world tube of the mass. In the lowest mixed order (mixed of the background and the retarded potentials of the mass in lowest order) this equation is identical with the geodesic line linearized in . In the case when the motion on a static background generated by a localized matter distribution is finite, the gravitational radiation of the mass in lowest order is given.     

Unitary dressing transformations and exponential decay below threshold for quantum spin systems. Parts III and IV

This is the continuation of a series of articles concerning a class of quantum spin systems with Hamiltonian operators of the form

Nonpersistence of breather families for the perturbed sine Gordon equation

We show that, up to one exception and as a consequence of first order perturbation theory only, it is impossible that a large portion of the well-known family of breather solutions to the sine Gordon equation could persist under any nontrivial perturbation of the form

A Note on the Integrability of Hamiltonian Systems

We consider a family of Hamiltonian systems

On the Davey–Stewartson and Ishimori Systems

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations i u_t + u = N(v), (t, x, y) R³, u(0, x, y) = u₀(x, y), (x, y) R² (A), where the Laplacian = ² _x + ² _y, the solution u is a complex valued function, the nonlinear term N = N₁ + N₂ consists of the local nonlinear part N₁(v) which is cubic with respect to the vector v=(u,u_x,u_y,\overline{u},\overline{u}_{x},\overline{u}_{y}) in the neighborhood of the origin, and the nonlocal nonlinear part N₂(v) =(v, ^{– 1} _x K_x(v)) + (v, ^{– 1} _y K_y(v)), where (, ) denotes the inner product, and the vectors K_x (C⁴(C⁶; C))⁶ and K_y (C⁴(C⁶; C))⁶ are quadratic with respect to the vector v in the neighborhood of the origin. We assume that the components K⁽²⁾ _x = K⁽⁴⁾ _x 0, K⁽³⁾ _y = K⁽⁶⁾ _y 0. In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic–hyperbolic Davey–Stewartson system can be reduced to Equation (A) with , and all the rest components of the vectors K_x and K_y are equal to zero. The elliptic–hyperbolic Ishimori system is involved in Equation (A), when , and . Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely if N(eⁱ v) = eⁱ N(v) for all R.     

On the problem of quantum tunneling of a singular potential

The problem of quantum tunneling through the singular potential barrier \(V(x) = \left\{ {\begin{array}{*{20}c} {V_0 (b/x - x/a)^2 ,} & {0 < \left| x \right| \leqslant \sqrt {ab} } \\ {0,} & {\left| x \right| > \sqrt {ab} } \\ \end{array} } \right.\) is discussed on the subject of the possibility to replace the singular behavior of the problem at the point x = 0 by a limiting process at the top of the truncated potential. The validity of such a replacement and, on this basis, the zero transparency of the quantum potential barrier are shown.     

A remark on tensor representations

The identity $$\sum\limits_{v = 0} {\left( {\begin{array}{*{20}c} {n + 1} \\ v \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} {n - v} \\ v \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} {n - v} \\ {v - 1} \\ \end{array} } \right)} \right] = ( - 1)^n } $$ is proved and, by means of it, the coefficients of the decomposition ofD ₁ ⁿ into irreducible representations are found. It holds: ifD ₁ ⁿ \(\mathop {\sum ^n }\limits_{m = 0} A_{nm} D_m \) , then $$A_{nm} = \mathop \sum \limits_{\lambda = 0} \left( {\begin{array}{*{20}c} n \\ \lambda \\ \end{array} } \right)\left[ {\left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda } \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} \lambda \\ {n - m - \lambda - 1} \\ \end{array} } \right)} \right].$$      

Levinson's Theorem for the Nonlocal Interaction in One Dimension

The Levinson theorem for the one-dimensional Schrödinger equation with both local and the nonlocal symmetric potentials is established by the Sturm–Liouville theorem. The critical case where the Schro;audinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n ₊ (n _–) of bound states with even (odd) parity is related to the phase shift ₊(0)[_–(0)] of the scattering states with the same parity at zero momentum as

Coherent pion production in neutrino and antineutrino interactions on nuclei of heavy freon molecules

Studying the coherent diffractive production of pions in neutrino and antineutrino scattering off the nuclei of freon molecules we have observed for the first time in one experiment all three states of the isospin triplet of the axial part of the weak charged and neutral currents. For the corresponding cross sections we derive $$\begin{array}{*{20}c} {\sigma _{coh}^v (\pi ^ + ) = (106 \pm 16) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ {\sigma _{coh}^{\bar v} (\pi ^ - ) = (113 \pm 35) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}and} \\ {\sigma _{coh}^v (\pi ^0 ) = (52 \pm 19) \cdot 10^{ - 40} {{cm^2 } \mathord{\left/ {\vphantom {{cm^2 } {\left\langle {nucl.} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {nucl.} \right\rangle }}} \\ \end{array} $$ . Comparing our data with theoretical predictions based on the standard model of weak interactions we find reasonable agreement. Independently from any model of coherent pion production we determine the isovector axial vector coupling constant to be |β|=0.99±0.20.     

Covariant differential calculus on the quantum exterior vector space

We formulate a differential calculus on the quantum exterior vector space spanned by the generators of a non-anticommutative algebra satisfying

Simultaneous geometrization of classical and quantum mechanics of spinless particles

It is shown that classical and quantum equations of motion of a relativistic spinless particle (the Lorentz and Klein-Gordon equations) allow for a geometrization on the same manifold ₄. A classical particle on ₄ is described as a free particle ( p=0), while the quantum particle, as a free wave ( s=0).Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 70–74, September, 1990.     

On the Modified Korteweg–De Vries Equation

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg–de Vries equation , with initial data . We assume that the coefficient is real, bounded and slowly varying function, such that , where . We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space . In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395–418), here we exclude the condition that the integral of the initial data u ₀ is zero. We prove the time decay estimates and for all , where . We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.     

Gleichzeitige Messung von Hyperfeinstruktur,Starkeffekt und Zeemaneffekt des133Cs19F mit einer Molekülstrahl-Resonanzapparatur

An electric molecular beam resonance spectrometer has been used to measure simultaneously the Zeeman- and Stark-effect splitting of the hyperfine structure of¹³³Cs¹⁹F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. Electrically induced (Δ m _J =±1)-transitions have been measured in theJ=1 rotational state, υ=0, 1 vibrational state. The obtained quantities are: The electric dipolmomentμ _el of the molecule for υ=0, 1; the rotational magnetic dipolmomentμ _J for υ=0, 1; the anisotropy of the magnetic shielding (σ _⊥-σ‖) by the electrons of both nuclei as well as the anisotropy of the molecular susceptibility (ξ _⊥-ξ‖), the spin rotational interaction constantsc _Cs andc _F, the scalar and the tensor part of the nuclear dipol-dipol interaction, the quadrupol interactioneqQ for υ=0, 1. The numerical values are:

$$\begin{gathered} \mu _{el} \left( {\upsilon = 0} \right) = 73878\left( 3 \right)deb \hfill \\ \mu _{el} \left( {\upsilon = 1} \right) - \mu _{el} \left( {\upsilon = 0} \right) = 0.07229\left( {12} \right)deb \hfill \\ \mu _J /J\left( {\upsilon = 0} \right) = - 34.966\left( {13} \right) \cdot 10^{ - 6} \mu _B \hfill \\ \mu _J /J\left( {\upsilon = 1} \right) = - 34.823\left( {26} \right) \cdot 10^{ - 6} \mu _B \hfill \\ \left( {\sigma _ \bot - \sigma _\parallel } \right)_{Cs} = - 1.71\left( {21} \right) \cdot 10^{ - 4} \hfill \\ \left( {\sigma _ \bot - \sigma _\parallel } \right)_F = - 5.016\left( {15} \right) \cdot 10^{ - 4} \hfill \\ \left( {\xi _ \bot - \xi _\parallel } \right) = 14.7\left( {60} \right) \cdot 10^{ - 30} erg/Gau\beta ^2 \hfill \\ c_{cs} /h = 0.638\left( {20} \right)kHz \hfill \\ c_F /h = 14.94\left( 6 \right)kHz \hfill \\ d_T /h = 0.94\left( 4 \right)kHz \hfill \\ \left| {d_s /h} \right|< 5kHz \hfill \\ eqQ/h\left( {\upsilon = 0} \right) = 1238.3\left( 6 \right) kHz \hfill \\ eqQ/h\left( {\upsilon = 1} \right) = 1224\left( 5 \right) kHz \hfill \\ \end{gathered} $$     

Absence of symmetry breaking for systems of rotors with random interactions

We prove that Gibbs states for the Hamiltonian , with thes _x varying on theN-dimensional unit sphere, obtained with nonrandom boundary conditions (in a suitable sense), are almost surely rotationally invariant if withJ _xy i.i.d. bounded random variables with zero average, 1 in one dimension, and 2 in two dimensions.