Representation of the solution of a Cauchy problem for the one-dimensional Schrödinger equation with bounded smooth potential in terms of quasi-Feynman formulae

2021 ◽  
Vol 85 (1) ◽  
Author(s):  
Denis Valerievich Grishin ◽  
Yan Yurievich Pavlovskiy
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
One Dimensional ◽  
Smooth Potential ◽  
Bounded Smooth ◽  
The One

scholarly journals The Cauchy problem of the one dimensional Schrödinger equation with non-local potentials

1993 ◽  
Vol 16 (4) ◽  
pp. 791-794 ◽  
Author(s):  
I. E. Kougias
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Unit Disc ◽  
Schrodinger Equation ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Non Local ◽  
Local Potentials ◽  
The One ◽  
Spaces Of Analytic Functions

For a large class of operatorsA, not necessarily local, it is proved that the Cauchy problem of the Schrödinger equation:−d2f(z)dz2+Af(z)=s2f(z), f(0)=0, f′(0)=1possesses a unique solution in the Hilbert(H2(Δ))and Banach(H1(Δ))spaces of analytic functions in the unit discΔ={z:|z|<1}.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

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