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Sunday, April 19, 2020 | History

3 edition of Asymptotic solutions of the one-dimensional Schrödinger equation found in the catalog.

Asymptotic solutions of the one-dimensional Schrödinger equation

S. Yu Slavianov

# Asymptotic solutions of the one-dimensional Schrödinger equation

Written in English

Subjects:
• Schrödinger equation.,
• Differential equations -- Asymptotic theory.

• Edition Notes

The Physical Object ID Numbers Statement S. Yu. Slavyanov ; [translated by Vadim Khidekel]. Series Translations of mathematical monographs -- vol.151 Contributions American Mathematical Society. Pagination xvi,190p. ; Number of Pages 190 Open Library OL22455201M ISBN 10 0821805363

A. Contreras and D. Pelinovsky, Stability of multi-solitons in the cubic NLS equation, J. Hyperbolic Differ. Equations 11(02) () – Link, Google Scholar; 6. S. Cuccagna and R. Jenkins, On asymptotic stability of N-solitons of the Gross-Pitaevskii equation, preprint (), arXivv1. Google Scholar; 7. by: 1. The Schrodinger equation is given above. 1. The wavefunction Ψcan be complex, so we should remember to take the Real part of Ψ. 2. Time-harmonic solutions to Schrodinger equation are of the form: 3. Ψ(x,t) is a measurable quantity and represents the probability distribution of File Size: 1MB. The one-dimensional time-independent Schrödinger equation for the potential energy discussed in Problem is d^2 Psi/dx^2 + 2m/h^2 (E - a|x|) Psi = 0 Define E = epsilon (h^2 a^2/m)^1/3 and x = z(h^2/ma)^1/3. (a) Show that e and z are dimensionless. This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of Brand: Springer International Publishing.

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### Asymptotic solutions of the one-dimensional Schrödinger equation by S. Yu Slavianov Download PDF EPUB FB2

Buy Asymptotic Solutions of the One-dimensional Schrodinger Equation (Translations of Mathematical Monographs) on FREE SHIPPING on qualified orders Asymptotic Solutions of the One-dimensional Schrodinger Equation (Translations of Mathematical Monographs): S.

Slavyanov: : Books5/5(1). Abstract: This book is devoted to asymptotic analysis of solutions of second order ordinary differential equations with a small parameter. The main emphasis is on various constructive schemes of obtaining asymptotic solutions, their advantages and.

Asymptotic solutions of the one-dimensional Schrödinger equation. Request This. Author Slavi͡anov, S. I͡U. (Sergeĭ I͡Urʹevich) Title Asymptotic solutions of the one-dimensional Schrödinger equation / S. Slavyanov ; [translator Vadim Khidekel]. Format Book Published Providence, RI: American Mathematical Society,   We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation − y ″ + q (x) y = 0 with oscillating potential q (x)= x β P (x 1+α)+ cx −2 as x → +∞.

The real parameters α and β satisfy the inequalities β − α ≥ −1, 2 α − β > 0 and c is an arbitrary real by: 5. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (), doi: / Google Scholar [20].

In this paper, we study the existence and asymptotic behavior of radial solutions for a class of nonlinear Schrödinger elliptic equations on infinite domains describing the gyre of geophysical fluid flows. The existence theorem and asymptotic properties of radial positive solutions are established by using a new renormalization by: THE ASYMPTOTIC SOLUTIONS OF THE GENERAL DIFFERENTIAL EQUATION 2.

The given equation. A change of variables may be made to reduce the differential equation as given above to the normal form (1) u"(z)+ {p24>2(z) -x(z)}u(z) =0, and simultaneously to transfer to the origin the point at which the coefficient 4>2 vanishes.

In fact, the general form of the Schrodinger Equation is known as the Time-Dependent Schrodinger Equation (TDSE): Asymptotic solutions of the one-dimensional Schrödinger equation book 2 2m ∂2Ψ(x,t) ∂x2 +U(x,t)Ψ(x,t)=i ∂Ψ(x,t) ∂t.

() When the potential energy is independent of time (true Asymptotic solutions of the one-dimensional Schrödinger equation book many interesting systems), wave functions satisfying the TDSE can always be written as (in 1 D)File Size: 2MB.

PART I: A SIMPLE SOLUTION OF THE TIME-INDEPENDENT SCHRÖDINGER EQUATION IN ONE DIMENSION H. Erbil a Ege University, Science Faculty, Physics Department Bornova - IZMIRTURKEY We found a simple procedure for the solution of the time-independent Schrödinger equation in one dimension without making any approximation.

Get this from a library. Asymptotic solutions of the one-dimensional Schrödinger equation. [S I︠U︡ Slavi︠a︡nov] -- This book is devoted to asymptotic analysis of solutions of second order ordinary differential equations with a small parameter.

The main emphasis is on various constructive schemes of obtaining. the asymptotic stability of solitons was studied in the works of Buslaev and author [4, 5]. We considered the Cauchy problem (), () and proved that in the case wherethespectrumof thelinearizationof equation() at theinitial solitonhas the simplest possible structure in some natural sense, the solution ψhas an asymptotic behavior of the.

Asymptotic solutions of the one-dimensional Schrödinger equation. [S I︠U︡ Slavi︠a︡nov] General theory. Asymptotic solutions on the complex plane. Method of comparison equations for equations with one transition point. Method of comparison equations for equations with two transition points.

# Differential. Abstract. We consider the blow-up problem for the nonlinear Schrödinger equation with critical power nonlinearity on ℝ show that every blow-up solution concentrates its L 2 mass at most finite points in some sense. We also estimate the amount of concentrated L 2 mass, and consider the asymptotic profiles of blow-up by: 2.

We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation − y″ +q(x)y = 0 with oscillating potential q(x)=x β P(x 1+α)+cx −2 as x→ +∞. The real parameters α and β Author: Pavel Nesterov. Asymptotics of the Solutions of the Random Schrödinger Equation Article in Archive for Rational Mechanics and Analysis (2) May with 27 Reads How we measure 'reads'.

• Newton’s equations of motion evolve x,v as functions of time • The Schrödinger equationevolves in time • There are energy eigenstates of the Schrodinger equation - for these, only a phase changes with time Y(x,t) In quantum mechanics, x and v cannot be precisely known simultaneously (the uncertainty principle).

A particle isFile Size: KB. We prove the WKB asymptotic behavior of solutions of the differential equation −d 2 u/dx 2 +V(x) u=Eu for a.e. E>A where V=V 1 +V 2, V 1 ∈L p (R), and V 2 is bounded from above with A=lim sup x→∞ V(x), while V′ 2 (x)∈L p (R), 1⩽pasymptotic Cited by:   Derivation of the Bohr-Sommerfeld Quantum, Conditions from an Asymptotic Solution, of the Schroedinger Equation was written by Joseph B.

Keller. This is a 26 page book, containing words. Search Inside is enabled for this title. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic : Joseph B.

Keller. [3] Solutio of Schrodinger equatio 55n 9 n where C is an arbitrary constant of integration. If A is not a positive integer, then s0 = a/2 fro; m ( u(r)) is not a square-integrable function, sinc its e asymptotic form involvears.

ei On the other hand if A is a positive integer, s0 = —a/2 an,d () shows that u(r) i square-integrables. Thus, for th eigenvalue e problem the, explici t Cited by: 6. The Schroedinger Equation by F. Berezin,available at Book Depository with free delivery worldwide.3/5(2). The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system.: 1–2 It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the equation is named after Erwin Schrödinger, who postulated the equation inand published it in.

Rao and Kagali obtain analytic solutions of a relativistic spinless particle in a one-dimensional screened Coulomb potential; they also illustrate the existence of several genuine bound states and make comparisons to the energy levels of bound states between the Schrödinger equation, Klein–Gordon equation, and Dirac by: 4.

Multiple solutions for a nonlinear Schrödinger systems. Communications on Pure & Applied Analysis,19 (2): doi: /cpaa [8] Chuangye Liu, Zhi-Qiang Wang. Synchronization of positive solutions for coupled Schrödinger by: 5.

It is shown that, contrary to common experience and opinion, the exact solutions to Schrödinger's equation in one dimension (or any similar ordinary linear second-order differential equation) can be numerically computed at a speed characterized by the variations of the potential function, i.e.

at effectively the speed of solving Hamilton's by: Solving The Stationary One Dimensional Schr odinger Equation With The Shooting Method by Marie Christine Ertl The Schr odinger equation is the fundamental quantum mechanical equation. However, only for a handful of cases it can be solved analytically, requiring a decent numerical method for systems where no analytical solution exists.

Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces Nakao Hayashi 1 and Pavel I.

Naumkin 2 1 Department of Mathematics, Graduate School of Science, Osaka University, Osaka Cited by: 1. Central to the method is the construction of an appropriate bounded reproducing kernel (RK) [Formula Presented] from the linear operator [Formula Presented] where [Formula Presented] is the N-dimensional Laplacian, [Formula Presented] is a parameter related to the binding energy of the system under study, and the real number [Formula Presented Cited by: 8.

From the above solution of the Schrödinger equation, we know the outgoing wave is a spherical one, so in a particular direction the amplitude decreases.

But that doesn’t happen with a plane wave. The clearest way to handle this is to put the system in a big box, a cube of side $$L$$, with periodic boundary conditions. is a solution of the Schrodinger equation, the function a (where a is a constant) is also a solution. The multiplicative factor a therefore has to be chosen such that the function a satisfies (3).

This process is called normalizing the wavefunction. In general, there will be solutions to the Schrodinger equation (1) whose integral is infinite. Hence, a single Taylor series representation of a function can span the entire range needed for a corresponding independent variable.

We decided to find eigenvalues to 14 decimal digits by solving one-dimensional Schrödinger equations by the "shooting method" by employing a single Talylor series in each case. A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the one-dimensional Schrödinger equation.

Our approximation is based on the Padé expansion of the square root function in the complex by: 2. By inserting these variables into the Schr odinger equation, we nd d2 d˘2 = 2 " ˘2 2.

(˘)() which is written in adimensional units. Exact solution One can easily verify that for large ˘(such that "can be neglected) the solutions of Eq.() must have an asymptotic behavior like (˘) ˘˘ne ˘2=2 () where nis any nite Size: KB.

This volume contains 18 papers related to the asymptotic analysis and qualitative research paper concerning the problems of nonlinear wave equations and nonlinear dispersive equations, such as nonlinear Schrödinger equations, the Hartree equation, the Camassa-Holm equation, and the Ginzburg-Landau equations.

The Time Independent Schrödinger Equation Second order differential equations, like the Schrödinger Equation, can be solved by separation of variables.

These separated solutions can then be used to solve the problem in general. Assume that we can factorize the solution between time and space. In quantum physics, you can break the three-dimensional Schrödinger equation into three one-dimensional Schrödinger equations to make it easier to solve 3D problems.

In one dimension, the time-dependent Schrödinger equation (which lets you find a wave function) looks like this: And you can generalize that into three dimensions like this: Using the Laplacian operator, you [ ].

The Schrodinger equation. The one-dimensional, steady-state, dimensionless, Schrodinger equation is wit (x) rightarrow 0 as x rightarrow plusminus infinity. For example, if V(x) = plusminus x^2, then the solution is given by where H_n = (-1)^ne^x^2 d^n/dx^n e^-x^2 is. Asymptotic behavior of the nonlinear Schr odinger equation with harmonic trapping Zaher Hani, Laurent Thomann To cite this version: Zaher Hani, Laurent Thomann.

Asymptotic behavior of the nonlinear Schr odinger equation with harmonic trapping. Communications on Pure and Applied Mathematics, Wiley,69 (9), pp{ Slavyanov S.

Asymptotic solutions of the one-dimensional Schrödinger equation / [transl. by V. Khidekel]. - Providence: American Mathematical Society, - xvi. The purpose of this work is to investigate the asymptotic behaviours of solutions for the discrete Klein–Gordon–Schrödinger type equations in one-dimensional lattice.

We first establish the global existence and uniqueness of solutions for the corresponding Cauchy problem. According to the solution's estimate, it is shown that the semi-group generated by the solution Cited by: 2. We obtain local well-posedness for the one-dimensional Schrödinger-Debye interactions in nonlinear optics in the spaces L^2× L^p, 1≤p solutions extend globally.

In the focusing regime, we consider a family of solutions {(u_{τ }, v_{τ })}_{τ >0} in H^1× H^1 associated to an initial data family {(u_{τ _0},v_{τ _0})}_{τ >0} Author: Adán J. Corcho, Juan C. Cordero. Physically acceptable solutions require that n must be greater than or equal to $$l +1$$.

The smallest value for $$l$$ is zero, so the smallest value for n is 1. The angular momentum quantum number affects the solution to the radial equation because it appears in the radial differential equation, Equation \ref{}.In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean.

It is worth mentioning that the numerical solution of the Schrödinger equation in undergraduate courses was discussed by several authors in the past [2–9] and recently we have resorted to one of the available shooting methods for the discussion of the Wronskians that enable us to take into account the asymptotic behaviour of the Author: Francisco M Fernández.