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.

Asymptotic behaviour of solutions of hyperbolic conservation laws ut + (um)x = 0 as m → ∞ with inconsistent initial values

1989 ◽  
Vol 113 (1-2) ◽  
pp. 61-71 ◽  
Author(s):  
Xiangsheng Xu
Keyword(s):  
Singular Perturbation ◽  
Asymptotic Behaviour ◽  
Conservation Laws ◽  
Positive Measure ◽  
Hyperbolic Conservation Laws ◽  
Singular Perturbation Problem ◽  
Stationary Equation ◽  
Hyperbolic Conservation ◽  
Perturbation Problem ◽  
Image Position

SynopsisWe study the behaviour of solutions u = um of ut, + (um)x = 0 for t > 0, x ∊ R, u(x, 0) = u0(x), u0 ≧0, u0 ∊ L1(R) as m → ∞. This is a singular perturbation problem about m = ∞ if u0 > 1 on a set of positive measure. It is shown that the limit exists and satisfies the stationary equation

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.

scholarly journals A singular perturbation problem for the Lavrent'ev-Bitsadze equation

1983 ◽  
Vol 26 (1) ◽  
pp. 49-66
Author(s):  
R. J. Weinacht
Keyword(s):  
Singular Perturbation ◽  
Mixed Type ◽  
Singular Integral Equations ◽  
Singular Perturbation Problem ◽  
Equations Of Mixed Type ◽  
Perturbation Problem ◽  
Small Positive Parameter ◽  
Carry Over ◽  
Image Position ◽  
Asymptotic Validity

In this note we consider a singular perturbation problem for the equationwhere K(y) = sgn y and. Ε is a small (positive) parameter. This equation for ε≠O is elliptic for y<0 and hyperbolic for y>0. Many of the results carry over to more difficult and interesting problems for equations of mixed type. The particularly simple model treated here permits the elimination of some complications in the analysis involving singular integral equations while preserving the main qualitative features of more general cases. For a special Tricomi-like problem for (1.1) we construct asymptotic expansions in ε, including boundary layer corrections, of the solution. A proof of uniform asymptotic validity of the lowest order terms is given.

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