scholarly journals On the asymptotic distribution of eigenvalues

1950 ◽  
Vol 200 (1063) ◽  
pp. 572-580 ◽  
Keyword(s):  
Approximate Solution ◽  
Schrödinger Equation ◽  
Asymptotic Distribution ◽  
Schrodinger Equation ◽  
Eigenvalue Distribution ◽  
Three Dimensions ◽  
One Dimension ◽  
One Dimensional ◽  
Distribution Of Eigenvalues ◽  
The One

It is well known that the asymptotic distribution of the eigenvalues of the one-dimensional Schrödinger equation is provided by the so-called W. K. B. formula. Most proofs of this depend on the approximate solution of the equation in two regions and the joining up of these solutions at the boundaries of the regions in a certain way. These methods are not easily generalized to the Schrödinger equation for dimensions greater than one. In the present paper the methods of Courant & Hilbert are applied to this problem and they lead very simply to a proof of the known result in one dimension and to analogous formulae for the eigenvalue distribution of the Schrödinger equation in two and three dimensions.

Stationary solutions to a nonlinear Schrödinger equation with potential in one dimension

2003 ◽  
Vol 133 (2) ◽  
pp. 409-437 ◽  
Author(s):  
Mihai Mariş
Keyword(s):  
Schrödinger Equation ◽  
Sound Velocity ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stationary Solutions ◽  
Nonlinear Schrodinger Equation ◽  
One Dimension ◽  
One Dimensional ◽  
Nonlinear Schrödinger ◽  
The One

We study the one-dimensional Gross-Pitaevskii-Schrödinger equation with a potential U moving at velocity v. For a fixed v less than the sound velocity, it is proved that there exist two time-independent solutions if the potential is not too big.

scholarly journals On the one dimensional cubic NLS in a critical space

2022 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Marco Bravin ◽  
Luis Vega
Keyword(s):  
Initial Data ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Three Dimensions ◽  
Critical Space ◽  
Initial Value ◽  
One Dimensional ◽  
Geometrical Problem ◽  
The One

<p style='text-indent:20px;'>In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.</p>

scholarly journals One-dimensional hydrogen atom

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150534 ◽  
Author(s):  
Rodney Loudon
Keyword(s):  
Hydrogen Atom ◽  
Schrödinger Equation ◽  
Ground State ◽  
Short Distance ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
One Dimension ◽  
Current Status ◽  
One Dimensional ◽  
The One

The theory of the one-dimensional (1D) hydrogen atom was initiated by a 1952 paper but, after more than 60 years, it remains a topic of debate and controversy. The aim here is a critique of the current status of the theory and its relation to relevant experiments. A 1959 solution of the Schrödinger equation by the use of a cut-off at x = a to remove the singularity at the origin in the 1/| x | form of the potential is clarified and a mistaken approximation is identified. The singular atom is not found in the real world but the theory with cut-off has been applied successfully to a range of four practical three-dimensional systems confined towards one dimension, particularly their observed large increases in ground state binding energy. The true 1D atom is in principle restored when the short distance a tends to zero but it is sometimes claimed that the solutions obtained by the limiting procedure differ from those obtained by solution of the basic Schrödinger equation without any cut-off in the potential. The treatment of the singularity by a limiting procedure for applications to practical systems is endorsed.

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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