MATLAB codes

We provide two easy to use MATLAB codes for computing resoances in one dimension. A more complete package which allows refinement of parameters (necessary for some computations) is available at

http://www.cims.nyu.edu/$ \sim$dbindel/matscat

The complete and easily downloadable codes are


squarepot.m


splinepot.m


An example of the use of the first code is given in Fig.1. The data is a vector of values,

$\displaystyle [V_1, V_2, \cdots , V_N ]  , $

and a vector of points

$\displaystyle [ x_0 , x_1 , \cdots , x_N ]  .$

and

squarepot( $ [V_1 , \cdots, V_N], [x_0, \cdots , x_N]$)

computes the first $ 40 $ (or so, depending on $ V$'s and $ x$'s) resonances for $ H_V $ with

$\displaystyle V ( x ) = V_j  ,   x_{j-1} \leq x_j  ,   j = 1, \cdots , N  .$

An example of using the second code is given in Fig.8. The data now are two vectors of the same length and

splinepot( $ [V_1 , \cdots, V_N], [x_1, \cdots , x_N]$)

computes the first $ 40 $ (or so, depending on $ V$'s and $ x$'s) resonances for $ V$ given by a cubic spline interpolating between the values $ V_j $ at $ x_j $, clamped at $ x_1 $ and $ x_N $ (that is $ V'( x_1 + ) = V' ( x_N - ) =0$), and $ 0 $, for $ x \notin [x_1, x_N]$. If $ V_1 = V_N = 0 $

$\displaystyle V\in C^1 ( \mathbb{R})  .$

The approximation of a smooth potential can still provide a good accuracy of resonances relatively close to the real axis - see Fig.8 and [4].

David Bindel 2006-10-04