Raymond Toy wrote:
> If you have some simple examples, can you send them to me?
There is a very simple example of general interest. Take the
Schroedinger equation (also called Sturm Liouville problem)
f'' +(E-V(x))f = 0
with e.g. boundary conditions f(0)=f(%pi)=0
Since this is a second order equation, it depends on 2 constants,
that one can fix by asking f(0)=0 f'(0)=1 for example.
In general the solution f does not satisfy f(%pi)=0 except for
specific values E_n of the energy, the eigenvalues.
This problem is amenable to colnew by adding the differential
equation E'= 0
The system becomes non linear, since we have the product E.f
but now it depends on 3 constants, and we have 3 equations
f(0)=1, f'(0)=1, f(%pi)=0.
The number of equations, equal to the number of unknowns is called
mstar in colnew.
When the potential V(x) is 0 the solution is trivial:
f(x)=1/n sin(nx) for n=1,2,... and so E_n=n^2.
Hence we can solve the general case by introducing a continuation parameter
t and replacing V(x) by tV(x) for 0<t<1. One has only to continue from t=0
to t=1. One can get rid of t by rescaling x, but then t appears in the
boundary conditions, it is a genuine deformation parameter.
There is a "theorem" in quantum mechanics called the adiabatic theorem
saying that if one varies slowly an external parameter t, the system remains
on the same level E_n. For small hbar this reduces to the adiabatic theorem
in classical mechanics which says that int p dq on a cycle remains constant,
coupled to the Bohr-Sommerfeld approximation of energy levels given
by int pdq = n hbar
Hence one necessarily stays on the same level n. Experimentally one sees
when performing continuations with colnew that one has to make small enough
steps so that the system does not hop from level to level.
Anyways, slowly varying t from 0 to 1, and using old solution as a guess for
new solution one sees that at t=1 one gets the correct energy levels and
wave function f for the given potential.
This works for arbitrary regular potential (for example a + b cos(x) will
give interesting Mathieu functions). If the potential becomes singular in
the interval, or the boundary conditions move to infinity, then the problem
becomes a singular boundary value problem, much more complicated,
with all sorts of continuous spectrum, etc. see Coddington and Levinson.
But for regular boundary value problem, i think this technique is obviously
working.
--
Michel Talon