scholarly journals Uniform boundary controllability of a discrete 1-D Schrodinger equation

2015 ◽  
Vol 7 (2) ◽  
pp. 259-270
Author(s):  
Z. Hajjej ◽  
M. Balegh
Keyword(s):  
Schrödinger Equation ◽  
High Frequency ◽  
Discrete System ◽  
Schrodinger Equation ◽  
Dimensional System ◽  
Boundary Controllability ◽  
Finite Dimensional ◽  
Boundary Observability ◽  
Spurious Modes ◽  
Uniform Controllability

In this paper we study the controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D Schrodinger equation with a boundary control. As for other problems, we can expect that the uniform controllability does not hold in general due to high frequency spurious modes. Based on a uniform boundary observability estimate for filtered solutions of the corresponding conservative discrete system, we show the uniform controllability of the projection of the solutions over the space generated by the remaining eigenmodes.

QUANTUM DYNAMICS OF WAVE PACKET IN HARMONIC POTENTIAL

2005 ◽  
Vol 19 (24) ◽  
pp. 3745-3754
Author(s):  
ZHAN-NING HU ◽  
CHANG SUB KIM
Keyword(s):  
Schrödinger Equation ◽  
Wave Packet ◽  
Quantum Dynamics ◽  
Schrodinger Equation ◽  
Analytic Solution ◽  
Nonlinear Differential Equations ◽  
Time Dependent ◽  
Dimensional System ◽  
Two Dimensional ◽  
Vibrational States

In this paper, the analytic solution of the time-dependent Schrödinger equation is obtained for the wave packet in two-dimensional oscillator potential. The quantum dynamics of the wave packet is investigated based on this analytic solution. To our knowledge, this is the first time we solve, analytically and exactly this kind of time-dependent Schrödinger equation in a two-dimensional system, in which the Gaussian parameters satisfy the coupled nonlinear differential equations. The coherent states and their rotations of the system are discussed in detail. We find also that this analytic solution includes four kinds of modes of the evolutions for the wave packets: rigid, rotational, vibrational states and a combination of the rotation and vibration without spreading.

SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

2001 ◽  
Vol 16 (31) ◽  
pp. 5061-5084 ◽  
Author(s):  
GUY JUMARIE
Keyword(s):  
Brownian Motion ◽  
Schrödinger Equation ◽  
Fractional Order ◽  
Schrodinger Equation ◽  
Relativistic Quantum ◽  
Relativistic Quantum Mechanics ◽  
Dimensional System ◽  
One Dimensional ◽  
Klein Gordon ◽  
Complex Valued

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

Chatter Stability Analysis of the Variable Speed Face-Milling Process

2000 ◽  
Vol 123 (4) ◽  
pp. 753-756 ◽  
Author(s):  
Sridhar Sastry, ◽  
Shiv G. Kapoor, ◽  
Richard E. DeVor, and ◽  
Geir E. Dullerud
Keyword(s):  
Discrete System ◽  
Milling Process ◽  
Face Milling ◽  
Solution Technique ◽  
Dimensional System ◽  
Multiple Time ◽  
Time Varying ◽  
Variable Speed ◽  
Finite Dimensional ◽  
Industrial Grade

In this study, a solution technique based on a discrete time approach is presented to the stability problem for the variable spindle speed face-milling process. The process dynamics are described by a set of differential-difference equations with time varying periodic coefficients and time delay. A finite difference scheme is used to discretize the system and model it as a linear time varying (LTV) system with multiple time delays. By considering all the states over one period of speed variation, the infinite dimensional periodic time-varying discrete system is converted to a finite dimensional time-varying discrete system. The eigenvalues of the state transition matrix of this finite dimensional system are then used to propose criteria for exponential stability. Predicted stability boundaries are compared with lobes generated by numerical time-domain simulations and experiments performed on an industrial grade variable speed face-milling testbed.

Calculation of high-frequency spectral wings by solution of the Schrödinger equation in the complex plane

1990 ◽  
Vol 175 (1-2) ◽  
pp. 105-110 ◽  
Author(s):  
J. Yang ◽  
C.G. Gray ◽  
B.G. Nickel ◽  
J.D. Poll ◽  
A.G. Basile
Keyword(s):  
Schrödinger Equation ◽  
Complex Plane ◽  
High Frequency ◽  
Schrodinger Equation
Sign in / Sign up

Export Citation Format

Share Document