Is this approximation approach applicable?

[zhanhai]

A one dimensional potential field V(x), and its solution of SE, is divided into three regions. And the solution has two coefficients in each of these regions. There are six boundary conditions: two on each of the boundaries, plus two global boundary conditions.

Now, as I wish to refine the solution in the two side regions by introducing two new components (base vectors) in each of the two side regions, there would be totally ten coefficients, which is more than the 6 boundary conditions.

I would think of an approach like this:

Keeping the solution in the center region and one side region intact, and in the remaining side region the two new components along with their new coefficients are added. In this way, we have four coefficients to be determined in this region, with four available boundary conditions (two on its boundary, plus the two global). So the four coefficients can be determined.

Next, we would do the same for the other side region. (Or, we could also "loosen" the two original coefficients in the central region (but without adding new components/coefficients) and find their anewed values.)

Is this applicable? And, is this the so-called "recursion"?

 

[eljose79]

Yes, this approach is applicable and it can be considered a form of recursion. Recursion is a problem-solving technique where the solution to a larger problem can be obtained by solving smaller instances of the same problem. In this case, the larger problem is finding the solution to the entire one dimensional potential field, and the smaller instances are the individual regions with their own coefficients and boundary conditions. By solving for the coefficients in each region separately and then combining them, you are using recursion to find the solution to the larger problem. This approach is commonly used in physics and engineering to solve complex problems with multiple boundary conditions.