Exact Solutions of Klein-Gordon Equation with Exponential Scalar and Vector Potentials

We obtain the exact analytical solution of the Klein-Gordon equation for the exponential vector and scalar potentials by using the asymptotic iteration method. For the scalar potential greater than the vector potential case, the exact bound state energy eigenvalues and corresponding eigenfunctions are presented. The bound state eigenfunction solutions are obtained in terms of the confluent hypergeometric functions.     

Exact nuclear level density formula for periodic spectra

The total nuclear level density is obtained for periodic single particle spectra. First, we express the canonical partition function in terms of Jacobi theta functions for degenerated fermion systems. Secondly, we use a relation for the number of representations of an integer number as sums of smaller numbers to find an exact formula for the nuclear level density. The shell structure is incorporated in the result in a simple way. An asymptotic relation can be easily obtained from the exact result and shown to be the well known back-shifted Bethe-formula.     

Study on high-temperature spectra of asymptotic asymmetric-top radical SiO2

Total internal partition sums are calculated in the product approximation at temperatures up to 6000 K for the asymptotic asymmetric-top SiO2 molecule.The rotational partition function and the vibrational partition function are calculated with the rigid-top model and in the harmonic oscillator approximation,respectively.Our values of the total internal partition sums are consistent with the calculated value in the Gaussian program within 0.137% at 296 K.Using the calculated partition functions and the rotationless transition dipole moment squared as a constant,we calculate the line intensities of 001-000 band of SiO2 at normal,medium and high temperatures.Simulated spectra of the 001-000 band of the asymptotic asymmetric-top SiO2 molecule at 2000,5000 and 6000 K are also obtained.     

A spatial finite size scaling method for calculation of critical parameters for weakly bound states

We present a finite size scaling approach to calculate critical parameters and exponents of quantum few-body system for which a bound state energy becomes absorbed or degenerates with the continuum. The scaling is done by introducing a cutoff radius for the potential. The asymptotic behavior for large values of the cutoff parameter is determined by the exact critical parameters.     

Production of electron-positron pairs on nuclei in the field of a bichromatic plane electromagnetic wave

The exact solution of the Dirac equation for an electron in the field of a bichromatic plane electromagnetic wave — with components that are monochromatic waves propagating in one direction — is used to calculate the probability of e⁺e^– pair production on a nucleus in the field of such a wave. This influence of the nucleus is treated perturbatively as an external Coulomb field. The cases of small and large values of the intensity parameters of the wave components are considered and the passage to the limit of a crossed field is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 33–40, September, 1978.We are very grateful to V. R. Khalilov for a fruitful discussion of the results.     

What Does It Take to Solve the 3D Ising Model? Minimal Necessary Conditions for a Valid Solution

An exact solution of the Ising model on the simple cubic lattice is one of the long-standing open problems in rigorous statistical mechanics. Indeed, it is generally believed that settling it would constitute a methodological breakthrough, fomenting great prospects for further application, similarly to what happened when Lars Onsager solved the two-dimensional model eighty years ago. Hence, there have been many attempts to find analytic expressions for the exact partition function Z, but all such attempts have failed due to unavoidable conceptual or mathematical obstructions. Given the importance of this simple yet paradigmatic model, here we set out clear-cut criteria for any claimed exact expression for Z to be minimally plausible. Specifically, we present six necessary—but not sufficient—conditions that Z must satisfy. These criteria will allow very quick plausibility checks of future claims. As illustrative examples, we discuss previous mistaken “solutions”, unveiling their shortcomings.     

On the Newtonian limit of general relativity

We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einstein's equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einstein's equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones).Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.     

The effect of high momentum transfer on scattering from oscillators and crystals

We examine two aspects of scattering at high energy and momentum transfer; asymptotic forms of scattering laws and comparison of approximate and exact results for a particular system, a harmonic lattice.By treating model systems of harmonically bound particles, an asymptotic analysis shows that, under conditions of high momentum transfer the particle looks as though it is unbound, in the sense that the weighting (envelope) of the various inelastic channels is that of a free particle, although the discreteness of the allowed energy transfers reflects the fact that the particles are, in reality, bound. The discreteness of the energy transfers requires an extension of the usual impulse approximation, which had been realised by Wick for a total cross section calculation, and this is treated in an appendix.An exact numerical technique for handling multiphonon effects directly, that is, without expansions, for the case of scattering from a lattice is used as a check of the Sjölander approximation. The latter treats elastic and one-phonon processes exactly, the remainder in a Gaussian-like approximation. As expected this scheme works well at low and high momentum transfer. At intermediate values the scattering law turns out to have a lot of structure which isnot reproduced. Simple interpretations would then give split peaks and such effects which in reality are simply manifestations of multiphonon processes.     

Line intensities of the asymptotic asymmetric-top radical HO2 at high temperatures

The total internal partition sums were calculated in the product approximation at temperatures up to 5000 K for the asymptotic asymmetric-top HO2 molecule. The calculations of the rotational partition function and the vibrational partition function were carried out with the rigid-top model and in the harmonic oscillator approximation, respectively. Our values of the total internal partition sums are consistent with the data of HITRAN database with -0.14% at 296 K. Using the calculated partition functions, we have calculated the line intensities of υ2 band of HO2 at several high temperatures. The results showed that the calculated line intensities are in very good agreement with those of HITRAN database at temperatures up to 3000 K, which provides a strong support for the calculations of partition functions and line intensities at high temperatures. Then we have extended the calculation to higher temperatures. The simulated spectra of υ2 band of the asymptotic asymmetric-top HO2 molecule at 4000 and 5000 K are also obtained.     

Coulomb collisions and the bound-electron recombination distribution function

A detailed analysis is made of how the distribution function of bound electrons is formed in the presence of three-bodyi—e—e recombination. The kinetic K matrix is obtained, characterizing the number of transitions from one energy range to another due to Coulomb collisions. It is found to have a pole of the third order in the magnitude of the transferred energy. This makes it possible to transform from the balance equation described by the K matrix to the Fokker — Planck equation describing diffusion and drift along the energy axis. Numerical solution of the time-dependent Fokker — Planck equation demonstrates that the characteristic relaxation time of the bound-electron distribution function is the time interval between Coulomb collisions. A comparison with previous computer modeling of the distribution function from first principles demonstrates that these results cannot be in agreement without rejecting the principle of detailed balance in its present formulation for a Coulomb plasma. This agrees with the conclusions previously reached in a computer modeling of the properties of a classical Coulomb plasma.General Physics Institute, Russian Academy of Sciences, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 3–19, September, 1994.     

Radial wave functions of a discrete spectrum

The system of equations for radial wave functions is written in a form allowing both relativistic and non relativistic wave functions to be obtained. In the case of a discrete spectrum, an asymptotic solution of this system is obtained for a potential which includes not only a Coulomb term but also terms corresponding to dipole and quadrupole polarization. The normalizing factor of the asymptote is determined in the approximation of the relativistic quantum-defect method, which offers the possibility, in principle, of using the functions in semiempirical calculations analogous to nonrelativistic calculations by the Bates — Damgaard — Seaton method. The calculation scheme is illustrated for the example of the calculation of Si IV wave functions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 39–44, December, 1978.     

Study of systematic errors of bound-state parameters in SVZ sum rules

We study systematic errors of the ground-state parameters obtained from Shifman—Vainshtein—Zakharov sum rules, making use of the harmonic-oscillator potential model as an example. In this case, one knows the exact solution for the polarization operator, which allows one to obtain both the OPE to any order and the parameters (masses and decay constants) of the bound states. We determine the parameters of the ground state making use of the standard procedures of the method of sum rules and compare the obtained results with the known exact values. We show that, in the situation when the continuum contribution to the polarization operator is not known and is modeled by an effective continuum, the method of sum rules does not allow one to control the systematic uncertainties of the extracted ground-state parameters. The text was submitted by the authors in English.     

Analytic continuation of scattering data to the region of negative energies for systems that have one and two bound states

An exactly solvable potential model is used to study the possibility of deducing information about the features of bound states for the system under consideration (binding energies and asymptotic normalization coefficients) on the basis of data on continuum states. The present analysis is based on an analytic approximation and on the subsequent continuation of a partial-wave scattering function from the region of positive energies to the region of negative energies. Cases where the system has one or two bound states are studied. The α+d and α+¹²C systems are taken as physical examples. In the case of one bound state, the scattering function is a smooth function of energy, and the procedure of its analytic continuation for different polynomial approximations leads to close results, which are nearly coincident with exact values. In the case of two bound states, the scattering function has two poles—one in the region of positive energies and the other in the region of negative energies between the energies corresponding to the two bound states in question. Padéapproximants are used to reproduce these poles. The inclusion of these poles proves to be necessary for correctly describing the properties of the bound states.     

Adiabatic similarity and dependence of the energy of molecular systems on the masses of particles

The dependence of the energy of three-particle molecules on their masses is examined. It is shown that such molecules with the same values of the ratio of the reduced masses for motion in a “fast” and “slow” Jacobi coordinates have the property of adiabatic similarity: In the adiabatic approximation, their energies are proportional to the reduced masses. This allows information on the energy of molecules symmetric in the masses of particles to be extended to asymmetric molecules adiabatically similar to the symmetric molecules. For molecules with arbitrary masses of the particles, an analytic expression for the adiabatic energy and a formula approximating the exact energy are constructed using the principle of adiabatic similarity. Along with the adiabatic energy, which is the lower bound of the exact energy, a simple procedure is considered for determining the upper bound of the energy of asymmetric molecules from the energy of their symmetric counterparts. Based on these results, values of the lower and upper energy bounds are calculated and an approximation of the exact energy is obtained for 43 three-particle molecular systems.     

Asymptotic Heat Kernel Expansion in the Semi-Classical Limit

Let H_(^h/2p) = (^h/_2p)²L +V{H_\hbar = \hbar^{2}L +V}, where L is a self-adjoint Laplace type operator acting on sections of a vector bundle over a compact Riemannian manifold and V is a symmetric endomorphism field. We derive an asymptotic expansion for the heat kernel of H_(^h/2p){H_\hbar} as (^h/_2p) \searrow 0{\hbar \searrow 0}. As a consequence we get an asymptotic expansion for the quantum partition function and we see that it is asymptotic to the classical partition function. Moreover, we show how to bound the quantum partition function for positive (^h/_2p){\hbar} by the classical partition function.     

Line intensities of the asymptotic asymmetric-top radical HO2 at high temperatures

The total internal partition sums were calculated in the product approximation at temperatures up to 5000\,K for the asymptotic asymmetric-top HO$_{2}$ molecule. The calculations of the rotational partition function and the vibrational partition function were carried out with the rigid-top model and in the harmonic oscillator approximation, respectively. Our values of the total internal partition sums are consistent with the data of HITRAN database with $-$0.14{\%} at 296\,K. Using the calculated partition functions, we have calculated the line intensities of $\nu _{2}$ band of HO$_{2}$ at several high temperatures. The results showed that the calculated line intensities are in very good agreement with those of HITRAN database at temperatures up to 3000\,K, which provides a strong support for the calculations of partition functions and line intensities at high temperatures. Then we have extended the calculation to higher temperatures. The simulated spectra of $\nu_{2}$ band of the asymptotic asymmetric-top HO$_{2}$ molecule at 4000 and 5000\,K are also obtained.     

Thermodynamic Instabilities in One Dimension: Correlations, Scaling and Solitons

Many thermodynamic instabilities in one dimension (e.g., DNA thermal denaturation, wetting of interfaces) can be described in terms of simple models involving harmonic coupling between nearest neighbors and an asymmetric on-site potential with a repulsive core, a stable minimum and a flat top. The paper deals with the case of the Morse on-site potential, which can be treated exactly in the continuum limit. Analytical expressions for correlation functions are derived; they are shown to obey scaling; numerical transfer-integral values obtained for a discrete version of the model exhibit the same critical behavior. Furthermore, it is shown in detail that the onset of the transition can be characterized by an entropic stabilization of an—otherwise unstable—nonlinear field configuration, a soliton-like domain wall (DW) with macroscopic energy content. The statistical mechanics of the DW provides an exact estimate of the critical temperature for a wide range of the discretization parameter; this suggests that the transition can be accurately viewed as being driven by a nonlinear entity.     

Residual Representations of Spacetime

Spacetime is modelled by binary relations—by the classes of the automorphisms of a complex two-dimensional vector space with respect to the definite unitary subgroup U(2). In extension of Feynman propagators for particle quantum fields representing only the tangent spacetime structure, global spacetime representations are given, formulated as residues using energy–momentum distributions with the invariants as singularities. The associated quantum fields are characterized by two invariant masses—for time and position—supplementing the one mass for the definite unitary particle sector with another mass for the indefinite unitary interaction sector without asymptotic particle interpretation.     

Simple nonrelativistic Gaunt factor for radiative recombination on fully ionized atoms

An algebraic expression for the Gaunt factor, obtained by analytical continuation of the asymptotic expansion of the hard photon Bremsstrahlung reduced matrix element, is in good agreement with the exact values, even for the lowest n up to energies of 15 Ry Z². Analytic expressions for the recombination rate and the power radiated by a maxwellian distribution of electrons are obtained.     

The dyadic Green's function for the cylindrical chirostrip antenna

Using the eigenfunction expansion of dyadic Green's function in unbounded chiral medium, and the method of scattering superposition, the expressions of dyadic Green's function for the one and two—layer cylindrical chirostrip antennas are derived, when the field sources are placed in any layer of cylindrical chirostrip antenna. Also, using the method of saddle point integration, the asymptotic expression of dyadic Green's function for the wraparound chirostrip antenna is obtained. The results can be directly used to analyse the radiation characteristics of cylindrical chirostrip dipolc antenna, et al.