VIII.—The Schrödinger Equation with a Periodic Potential

1971 ◽  
Vol 69 (2) ◽  
pp. 125-131
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Equation ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Floquet Theory ◽  
Periodic Potential ◽  
Eigenvalue Problems ◽  
Alternative Proof ◽  
Conditional Stability ◽  
Image Position

SynopsisThe differential equationin N dimensions is considered, where q(x) is periodic. When N = 1, it is known that the conditional stability set coincides with the spectrum and that these also coincide with two other sets involving eigenvalues of associated eigenvalue problems. These results have been proved by means of the Floquet theory and the discriminant. Here an alternative proof is given which avoids the Floquet theory and which applies to the general case of N dimensions.

The Schrödinger equation as a singular perturbation problem

1978 ◽  
Vol 82 (1-2) ◽  
pp. 1-11 ◽  
Author(s):  
E. M. de Jager ◽  
T. Küpper
Keyword(s):  
Schrödinger Equation ◽  
Singular Perturbation ◽  
Schrodinger Equation ◽  
Eigenvalue Problems ◽  
Singular Perturbation Problem ◽  
Perturbation Problem ◽  
First Approximation ◽  
Image Position

SynopsisComparisons have been made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problemsandwith φ ∈ C(-∞, +∞) and 0≦φ(x)≦C|x|i+1(1+|x|1), −∞<x<+∞ where i and l are arbitrary positive numbers with i≧2k≧2, k integer. In first approximation the eigenvalues λ and λ− and the corresponding eigenfunctions ψ and ψ are the same for ε→0; the error decreases whenever the exponent i increases.

One-Dimensional Schrödinger Equation with an Almost Periodic Potential

1983 ◽  
Vol 50 (23) ◽  
pp. 1873-1876 ◽  
Author(s):  
Stellan Ostlund ◽  
Rahul Pandit ◽  
David Rand ◽  
Hans Joachim Schellnhuber ◽  
Eric D. Siggia
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Periodic Potential ◽  
Almost Periodic ◽  
One Dimensional

Painlevé analysis of the damped, driven nonlinear Schrödinger equation

1988 ◽  
Vol 109 (1-2) ◽  
pp. 109-126 ◽  
Author(s):  
Peter A. Clarkson
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Analytic Functions ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Completely Integrable ◽  
Real Analytic Functions ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equationwhere a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only ifwhere α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

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