scholarly journals ANALYTICAL SOLUTION OF THE SCHRÖDINGER INTEGRAL EQUATION

2020 ◽  
pp. 5-17
Author(s):  
Mikhail A. BUBENCHIKOV ◽  
◽  
Aleksey M. BUBENCHIKOV ◽  
Soninbayar JAMBAA ◽  
Alexander V. LUN-FU ◽  
...  
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Differential Operator ◽  
Exponential Function ◽  
Membrane Separation ◽  
Classical Equation ◽  
Spectral Functions ◽  
Double Barrier ◽  
Wave Dynamics ◽  
Representation Form

In this paper, the question about the use of wave dynamics for solving problems of membrane separation of helium isotopes in the gas state at cryogenic temperatures is considered. The dimensionless form of the stationary Schrödinger differential equation is obtained. Following that, the integral representation form of the wave function is written. This form, which is equivalent to the classical equation, is similar to the integral equation with a degenerate core; however, it contains a modulus of the argument with a shift along the real axis. Using the shift operator, the existing exponential function in the Schrödinger integral equation can be split into a differential operator and an exponential function of the argument module which does not contain a shift. The Fourier identity allows reducing the exponent of the modulus of the argument to a regular exponent. Next, based on the general property of a differential operator acting on an exponent, it is possible to calculate the spectral functions of the problem and write down the distribution for the wave function. This distribution is ultimately expressed through the spectra of the potential barrier. Thereafter, the structure and the spectrum of the composite barrier are considered. With the expression determining the reflection coefficient, it is found that the double-barrier system can have a resonant passage of one of the components in the sequence of distances between the layers of the membrane.

Nucleon-antinucleon interaction in the new Tamm-Dancoff approximation

1960 ◽  
Vol 257 (1288) ◽  
pp. 59-73
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Cross Sections ◽  
Nucleon Nucleon ◽  
Phase Shifts ◽  
Exchange Term ◽  
Total Cross Sections ◽  
Nucleon Wave Function ◽  
Usual Manner ◽  
Order Of Magnitude

The nucleon-antinucleon ( N-N ) problem is formulated in the new Tamm-Dancoff (NTD) approximation in the lowest order, and the integral equation for N-N̅ scattering derived, taking account of both the exchange and annihilation interactions. It is found convenient to represent the N-N̅ wave-function as a 4 x 4 matrix, rather than the usual 16 x 1 matrix for the nucleon-nucleon wave-function, and a complete correspondence is established between these two representations. The divergences associated with the annihilation interaction and their renormalization are discussed in detail in the following paper (Mitra & Saxena 1960; referred to as II). The integral equation with the exchange interaction alone, is then separated into eigenstates of T, J, L and S in the usual manner and the various phase shifts obtained. The results of II for the contribution of the annihilation term are then used to calculate the complete phase shifts from which the various cross-sections (scattering and charge exchange) are derived. The results indicate that while the exchange term alone gives too small values for the total cross-sections versus energy, inclusion of the annihilation interaction without renormalization effects makes the cross-sections nearly three times larger than those observed. On the other hand, inclusion of the finite effects of renormalization (which manifest themselves essentially as a suppression of the virtual meson propagator) brings down these cross-sections to the order of magnitude of the observed ones.

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

UNITARY TRANSFORMATION APPROACH FOR THE PHASE OF THE DAMPED DRIVEN HARMONIC OSCILLATOR

2003 ◽  
Vol 17 (26) ◽  
pp. 1365-1376 ◽  
Author(s):  
JEONG-RYEOL CHOI
Keyword(s):  
Wave Function ◽  
Harmonic Oscillator ◽  
Unitary Transformation ◽  
Operator Method ◽  
Classical Equation ◽  
Transformation Method ◽  
Invariant Operator ◽  
Harmonic Oscillators ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Function

Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.

Renormalization for nucleon-antinucleon interaction

1960 ◽  
Vol 257 (1288) ◽  
pp. 74-82 ◽  
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Angular Momentum ◽  
Vertex Operator ◽  
Renormalization Constant ◽  
Preceding Paper ◽  
Present Problem ◽  
Wave Interactions ◽  
Self Energy ◽  
Specific Effects

The divergences associated with the integral equation for nucleon-antinucleon ( N-N̅ ) interaction derived in the preceding paper (Mitra & Saxena 1960) are here analyzed in detail. The problem is found to be quite similar to that discussed by Dalitz & Dyson (1955) in connexion with the π-N problem. The part of the wave-function representing the specific effects of the annihilation interaction is found to be expressed in terms of the so-called vertex operator and a meson propagator modified by self-energy effects. The vertex operator is then renormalized and the finite (observable) effects of this renormalization on the N-N̅ wave-function evaluated. It is found that, unlike the case of π-N scattering where two different renormalization factors are needed for describing s - and p -wave interactions, the present problem requires a unique renormalization constant covering both odd and even angular momentum states.

Correlated Atomic Pair Functions by the e-ϱ-Method. I. Ground State 11S and Lowest Excited States n1S (n > 1) and n3S of Helium

1999 ◽  
Vol 54 (12) ◽  
pp. 711-717
Author(s):  
F. F. Seelig ◽  
G. A. Becker
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Helium Atom ◽  
Excited States ◽  
Ground State ◽  
Single Parent ◽  
Hartree Fock ◽  
Atomic Pair ◽  
Electronic Wave Function ◽  
S States

Abstract Some low n1S and n3S states of the helium atom are computed with the aid of the e-e method which formulates the electronic wave function of the 2 electrons ψ = e-e F, where ϱ=Z(r1+r2)–½r12 and here Z = 2. Both the differential and the integral equation for F contain a pseudopotential Ṽ instead of the true potential V that contrary to V is finite. For the ground state, F = 1 yields nearly the Hartree-Fock SCF accuracy, whereas a multinomial expansion in r1, r2 , r2 yields a relative error of about 10-7 . All integrals can be computed analytically and are derived from one single “parent” integral.

A canonical approach for computing the eigenvalues of the Schrödinger equation for double-well potentials

2000 ◽  
Vol 78 (11) ◽  
pp. 969-976 ◽  
Author(s):  
M Korek ◽  
K Fakhreddine
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Schrödinger Equation ◽  
Volterra Integral Equation ◽  
Schrodinger Equation ◽  
High Accuracy ◽  
Present Formulation ◽  
Numerical Application ◽  
Canonical Approach ◽  
Double Well Potential

The problem of obtaining the eigenvalues of the Schrödinger equation for a double-well potential function is considered. By replacing the differential Schrödinger equation by a Volterra integral equation the wave function will be given by [Formula: see text] where the coefficients ai are obtained from the boundary conditions and the fi are two well-defined canonical functions. Using these canonical functions, we define an eigenvalue function F(E) = 0; its roots E1, E2, ... are the eigenvalues of the corresponding double-well potential. The numerical application to analytical potentials (either symmetric or asymmetric) and to a numerical potential of the (2)1 [Formula: see text] state of the molecule Na2 shows the validity and the high accuracy of the present formulation. PACS No.: 03.65Ge

Integral equation approach to the problem of scattering by a number of fixed scatterers

1977 ◽  
Vol 55 (16) ◽  
pp. 1442-1452
Author(s):  
M. Hron ◽  
M. Razavy
Keyword(s):  
Integral Equation ◽  
Wave Function ◽  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Scattering Problem ◽  
Three Dimensional ◽  
Inhomogeneous Term ◽  
Integral Equation Approach ◽  
An Array Of Cylinders

In the derivation of the Lippmann–Schwinger integral equation for scattering of a wave ψ(r) by the potential ν(r), one constructs the Green's function for the operator [Formula: see text], and treats νψ as the inhomogeneous term. However, in certain cases, it is desirable to formulate the scattering problem in terms of an integral equation by obtaining the Green's function for the operator [Formula: see text], and by considering (−k2ψ) as the inhomogeneous term. An important aspect of this formulation is that the resulting integral equation can be used to generate a low energy expansion of the wave function for some separable and nonseparable systems. For two-dimensional scattering, if the geometry of the scatterers is simple enough, the Laplace equation with the prescribed boundary conditions on the surface of the scatterers is separable in a certain coordinate system, then one can write the solution of the wave equation as an inhomogeneous integral equation. In this way the problems of scattering by two cylinders, an array of cylinders, and a grating can be formulated in terms of integral equations. For three-dimensional scattering, one can consider either the spherically symmetric cases or nonseparable problems. In the former case, for certain types of force laws, a Volterra integral equation in one variable can be found for the wave function. In the latter case, integral equations in two or three variables can be obtained for scattering by two spheres or by a torus.

scholarly journals Series expansion of the Gamma function and its reciprocal

2021 ◽  
Vol 27 (4) ◽  
pp. 104-115
Author(s):  
Ioana Petkova ◽  
Keyword(s):  
Differential Operator ◽  
Explicit Form ◽  
Series Expansion ◽  
Exponential Function ◽  
Gamma Function ◽  
Linear Functional ◽  
Maclaurin Series

In this paper we give representations for the coefficients of the Maclaurin series for \Gamma(z+1) and its reciprocal (where \Gamma is Euler’s Gamma function) with the help of a differential operator \mathfrak{D}, the exponential function and a linear functional ^{*} (in Theorem 3.1). As a result we obtain the following representations for \Gamma (in Theorem 3.2): \begin{align*} \Gamma(z+1) & = \big(e^{-u(x)}e^{-z\mathfrak{D}}[e^{u(x)}]\big)^{*}, \\ \big(\Gamma(z+1)\big)^{-1} & = \big(e^{u(x)}e^{-z\mathfrak{D}}[e^{-u(x)}]\big)^{*}. \end{align*} Theorem 3.1 and Theorem 3.2 are our main results. With the help of the first theorem we give our approach for finding the coefficients of Maclaurin series for \Gamma(z+1) and its reciprocal in an explicit form.

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