Nonperturbative Time-Independent Green Function of Matrix Schrödinger Equation. General Formalism and Quasiclassical Representation

A Green function of time-independent multichannel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multichannel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multichannel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multipotential system within quasiclassical approximation. The limits of strong and weak interchannel interactions are studied.Alexander I. Pegarkov:On leave from Physics Faculty     

Accurate analytic presentation of solution of the Schrödinger equation with arbitrary physical potential: Excited states

The quasilinearization method (QLM) is used to approximate analytically, both the ground state and the excited state solutions of the Schrödinger equation for arbitrary potentials. The procedure of approximation was demonstrated on examples of a few often used physical potentials such as the quartic anharmonic oscillator, the Yukawa and the spiked harmonic oscillator potentials. The accurate analytic expressions for the ground and excited state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the Schrödinger equation into a nonlinear Riccati form and then solving that nonlinear equation analytically in the first QLM iteration. In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The method provides final and reasonable results for both small and large values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. The choice of zero iteration is based on general features of solutions near the boundaries. In order to estimate the accuracy of the QLM solutions, the exact numerical solutions were found as well. The first QLM iterate given by analytic expression allows to estimate analytically the role of different parameters and the influence of their variation on different characteristics of the relevant quantum systems.     

Numerical Analysis of Emission Line Broadening under Coulomb Interaction in an Electron–Hole System

A numerical analysis of the energy spectrum of an ensemble of interacting particles in a two-dimensional system has been performed by means of direct solution of the microparticle Schröedinger equation. Comparison of the numerical calculations of the form factor of the homogeneous broadening of spectral lines with the results obtained according to perturbation theory has been made. The range of applicability of the universal relation between the amplification and emission spectra is discussed.     

Quasilinearization method: Nonperturbative approach to physical problems

The general properties of the quasilinearization method (QLM), particularly its fast quadratic convergence, monotonicity, and numerical stability, are analyzed and illustrated on different physical problems. The method approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of linear ones and is not based on the existence of a small parameter. It is shown that QLM gives excellent results when applied to different nonlinear differential equations in physics, such as Blasius, Lane-Emden, and Thomas-Fermi equations, as well as in computation of ground and excited bound-state energies and wave functions in quantum mechanics (where it can be applied by casting the Schrödinger equation in the nonlinear Riccati form) for a variety of potentials most of which are not treatable with the help of perturbation theory. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and the first few iterations already yield extremely precise results. The QLM approximations, unlike the asymptotic series in perturbation theory and 1/N expansions, are not divergent at higher orders. The method sums many orders of perturbation theory as well as of the WKB expansion. It provides final and accurate answers for large and infinite values of the coupling constants and is able to handle even supersingular potentials for which each term of the perturbation series is infinite and the perturbation expansion does not exist.     

Perturbation of chiral solitons

This paper studies the perturbation of soliton due to the chiral nonlinear Schrödinger's equation by the aid of soliton perturbation theory. The perturbation term that is studied is the quantum potential perturbation of the chiral soliton that is known as Bohm potential. The stable fixed point of the chiral soliton parameters is obtained.     

Approximate Symmetry Reduction to the Perturbed One-Dimensional Nonlinear Schrödinger Equation

The one-dimensional nonlinear Schrödinger equation with a perturbation of polynomial type is considered. Using the approximate symmetry perturbation theory, the approximate symmetries and approximate symmetry reduction equations are obtained.     

Propagation constants and delay parameters of near parabolic optical fibres

The normalized propagation constants and delay parameters of near parabolic index fibres are obtained by first order perturbation theory. Departures from the parabolic variation in the form of gaussian dip or bump, periodic changes and polynomial functions are combined in a single term allowing the generalization of existing results. The relevant parameters for each of these special cases are also obtained separately in a closed form.     

Theory of vibronic optical spectra of impurity centers with violation of the Condon approximation

Expressions for non-Condon generating functions of absorption and luminescence of optical centers in crystals at zero temperature have been obtained in the adiabatic approximation. A solution to the Schrödinger equation for an electronic subsystem has been considered to the first order of the perturbation theory for a vibronic interaction linear in normal coordinates of the vibrational subsystem. The non-Condon form function of the optical transition has been obtained in the form of a convolution operator acting on the normalized Condon form function. It has been proved that, if the optical transition is forbidden in the Condon approximation due to symmetry, the non-Condon form function does not contain a zero-phonon line and the non-Condon form functions of absorption and luminescence are symmetric.     

Thermal spectral functions of strongly coupled N = 4 supersymmetric Yang-Mills theory

We use the gauge-gravity duality conjecture to compute spectral functions of the stress-energy tensor in finite-temperature N = 4 supersymmetric Yang-Mills theory in the limit of large N(c) and large 't Hooft coupling. The spectral functions exhibit peaks characteristic of hydrodynamic modes at small frequency, and oscillations at intermediate frequency. The nonperturbative spectral functions differ qualitatively from those obtained in perturbation theory. The results may prove useful for lattice studies of transport processes in thermal gauge theories.     

Invariant exchange perturbation theory for multicenter systems and its application to the calculation of magnetic chains in manganites

The formalism of exchange perturbation theory is presented with regard to the general principles of constructing an antisymmetric vector with the use of the Young diagrams and tableaux in which the coordinate and spin parts are not separated. The form of the energy and wave function corrections coincides with earlier obtained expressions, which are reduced in the present paper to a simpler form of a symmetry-adapted perturbation operator, which preserves all intercenter exchange contributions. The exchange perturbation theory (EPT) formalism itself is presented in the standard form of invariant perturbation theory that takes into account intercenter electron permutations between overlapping nonorthogonal states. As an example of application of the formalism of invariant perturbation theory, we consider the magnetic properties of perovskite manganites La_1/3Ca_2/3MnO₃ that are associated with the charge and spin ordering in magnetic chains of manganese. We try to interpret the experimental results obtained from the study of the effect of doping the above alloys by the model of superexchange interaction in manganite chains that is constructed on the basis of the exchange perturbation theory (EPT) formalism. The model proposed makes it possible to carry out a quantitative analysis of the effect of substitution of manganese atoms by doping elements with different electron configurations on the electronic structure and short-range order in a magnetic chain of manganites.     

Theoretical investigation into dipole-moment functions of HF, HCl, and HBr molecules at small internuclear separations

Ab initio calculations of the dipole moment functions are performed within the united atom approximation in the first-order perturbation theory for the ground electronic states of HF, HCl and HBr molecules in the range of small internuclear separations. The calculation results are used for correction of the semi-empirical dipole-moment functions of these molecules obtained earlier. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 71–75, November, 2006.     

Double wells: Perturbation series summable to the eigenvalues and directly computable approximations

We give a rigorous proof of the analyticity of the eigenvalues of the double-well Schrödinger operators and of the associated resonances. We specialize the Rayleigh-Schrödinger perturbation theory to such problems, obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method. By an approximation we obtain new series expansions directly computable, still summable, which, in the case of the Herbst-Simon model, can be given in an explicit form.Partially supported by Ministero della Pubblica Istruzione     

Vibrational intensities in methane

The absorption intensities of the two infra-red active vibrations in methane have been obtained from a perturbation calculation on the equilibrium wave functions derived in the preceding paper. The perturbation field is the change in the potential field due to the nuclei which results from moving the nuclei in the vibrational coordinate concerned, and a simplified form of second order perturbation theory, developed by Pople and Schofield, is used for the calculation. The main approximation involved is the neglect of f and higher harmonics in the spherical harmonic expansion of the nuclear field. The resulting dipole moment derivatives are approximately three times larger than the experimental values, but they show qualitative features and sign relationships which are significant.

The experimental intensity measurements are interpreted and discussed in relation to these results.     

Linear guidance properties of solitonic Y-junction waveguides

A linear Y-junction waveguide device is designed using a generalization of the theory of solitonic potentials for the linear Schrödinger equation. This Y-junction device, unlike other adiabatic Y-junctions, has the advantage that it may be directly written into a planar medium with homogeneous saturable nonlinearity by a strong light beam. The generalized theory provides the error terms that are introduced when the parameters of a solitonic potential are allowed to vary in the propagation direction, and shows that under certain adiabaticity conditions the error is small although the deformation of the potential is significant. At the operating wavelength for which the device is designed to function optimally, the Y-junction has two approximate bound modes that we find explicitly. Each mode has the property that when it is excited at the neck of the junction, it exits in only one of the two output ports. In this way, the device functions like a standard modal splitter in a multimode slab waveguide. When the wavelength is detuned, modal beating is introduced that degrades the optimal switching characteristics. We describe this effect in terms of four universal coupling functions using perturbation theory.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Relative to the interpretation of terms of the Schrödinger perturbation theory series

An approach different from those ordinarily expounded is proposed for the construction of the Schrödinger perturbation theory. This approach sets up a functional dependence between the energy corrections and the coefficients characterizing the corrections to the wave functions. It is shown that within the framework of the Schrödinger theory these coefficients are determined by extremum conditions of fourth-order corrections to the energy. In the Schrödinger approximation the fourth-order energy corrections reach their extremal values. The approach developed includes a proof of the Dalgarno and Stewart theorem that says that the energy can be estimated to the order (2n + 1) if the wave function is known to n-th-order accuracy.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 18–20, February, 1982.     

An exact solution to the multi-dimensional line transfer equation

The theory of generalized analytic functions is used to obtain an exact closed form analytical solution to a transfer problem for spectral line radiation in a multi-dimensional atmosphere. The multi-dimensional full-space and half-space Green's functions so obtained are quite general and may be used, along with the corresponding orthogonality relationships, to obtain solutions to any general multi-dimensional radiative transfer problem involving model two-level atoms. An application of the method using perturbation techniques is illustrated.     

Differential geometry of atomic structure and the binding energy of atoms

The geometric theory of partial differential equations due to E. Cartan is applied to atomic systems in order to solve the many-body problems and to obtain the binding energies of electrons in an atom. The procedure consists in defining a Schrödinger equation over an Euclidean patch which overlaps with other Euclidean patches in a specified way to form a manifold. If the energy of the system has to be a minimum, it is shown using the Dirichlet principle that the coordinate systems are related by the Cauchy-Riemann relations. The invariance of the Schrödinger equations in the overlapping region leads to a nonlinear second-order equation which is invariant to automorphic transformations and whose solutions are doubly periodic functions. There are only two possible single-valued solutions to this nonlinear partial differential equation and these correspond to lattices of points in the complex space, which are (a) corners of an array of equilateral triangles, and (b) corners of an array of isosceles right-angled triangles. The first solution was used in an earlier work to derive many static properties of nuclei. In this paper it is shown that the second solution gives binding energies of atoms in agreement of about 3% for the few experimental points that are available and also in good agreement with the binding energies of atoms obtained by the perturbation theory. It is also shown that this lattice under certain approximations is equivalent to a pure Coulomb law and the Bohr orbits of the hydrogen atom are correctly predicted. In obtaining the binding energies of atoms, no free parameters are required in the theory, except for the value of the binding energy of the He atom, as the theory is developed only for spinless systems. All other constants turn out to be fundamental constants.     

String Scattering from Decaying Branes

We develop the general formalism of string scattering from decaying D-branes in bosonic string theory. In worldsheet perturbation theory, amplitudes can be written as a sum of correlators in a grand canonical ensemble of unitary random matrix models, with time setting the fugacity. An approach employed in the past for computing amplitudes in this theory involves an unjustified analytic continuation from special integer momenta. We give an alternative formulation which is well-defined for general momenta. We study the emission of closed strings from a decaying D-brane with initial conditions perturbed by the addition of an open string vertex operator. Using an integral formula due to Selberg, the relevant amplitude is expressed in closed form in terms of zeta functions. Perturbing the initial state can suppress or enhance the emission of high energy closed strings for extended branes, but enhances it for D0-branes. The closed string two point function is expressed as a sum of Toeplitz determinants of certain hypergeometric functions. A large N limit theorem due to Szegö, and its extension due to Borodin and Okounkov, permits us to compute approximate results showing that previous naive analytic continuations amount to the large N approximation of the full result. We also give a free fermion formulation of scattering from decaying D-branes and describe the relation to a grand canonical ensemble for a 2d Coulomb gas.     

Perturbation expansion of quantum resonance energies

We present new results concerning the theoretical treatment of quantum resonances based on the use of optical potentials and perturbation theory. A new criterion is presented for selecting an optical potential. As a result, a wide class of optical potentials which lead to efficient asymptotic series is available. A numerical illustration for two shape resonances show that accurate resonance energies and lifetimes can be obtained from a very small number of real square-integrable functions.