# Solving using schrodinger equation techniques

Hello,
I have problem I wish to solve, and I wonder if anyone already delt with it when solving the schrodinger 2D equation.

say E(x,y) is a scalar field function that complies with

( $$\frac{d}{dx}$$2+$$\frac{d}{dy}$$2 ) *E(x,y)+k(x,y)*E(x,y)=k1*E(x,y)

where k(x,y)={k2 for x2+y2<R2 and 0 otherwise}, i.e. a tube potential.
All is known but E(x,y).
I think it can be examined as a 2D schrodinger equation, even thow the eigenvalue k1 is known.

How can I get to start finding the solutions of this equation?
Can I expect to know how many are there? - one, two, many?

At least in 3D, a rotationally invariant system is best dealt with in spherical coordinates, expanding in eigenfunctions of angular momentum and then solving the radial equation, which will be an ODE. In 2D, the "spherical harmonics" are just the functions $e^{im \phi}$ where m is an integer. So try writing $f_m(r,\phi) = u_m(r) e^{i m \phi}$, which should give you an ordinary differential equation for $u_m(r)$. Then the general solution will be a linear combination of the $f_m$, ie, you should expect a linearly independent solution for each integer m.
This is a little messier in 2D than in 3D, and the equation will have a first derivative term that doesn't appear in the ordinary schrodinger equation. In fact, the solutions $u_m(r,\phi)$ are probably going to be Bessel functions.