Thank you Professor Farid for your insight in the question.

1. My problem is defined on a plane/disc (with possible out-of-plane deformations)

2. Both U and V vanish at the edges of the plane/disc

3. The coefficients are real, continuous and differentiable over the whole plane; no discontinuities. The functions are not simple though, but some how complicated.

Please, I can send you the full equations if you don't mind.

Thank you

Dear professor Farid, thank you so much for the thorough discussion of my problem.

Yes, it is true that I should have specified both Driclet and Neuman boundary conditions. Sorry for mentioning the the out-of-plane d.o.f, it is actually captured through a condition like z^2=x^2+y^2.

In fact, I have no doubt your answer will make an impact on my ongoing research. More expecially the last part on cylindrical coordinates. Unfortunately seperation of variables will be difficult to achieve (analytically) in my case. for instance, the coefficients contain such factors as cos[r sin( φ)]

I have read Griff's Quantum book before but with ur advice I need to look at it again.

Please, may I get your lecture notes on this topic.

Thank you.

Dear Professor Farid, I have put my equations in a pdf file which is attached here.

I am sorry for the delay in putting the equations here, I had to travel from my country to ICTP-Italy for a program. Could you let me make further clarification if the need arises.

I thank you in advance.

Dear Professor Farid, thank you once more for your very comprehensive and detail insight. Please,

1. In your N-by-N matrix equation for the coeffients, you left out "Phi" term. Did you assume its fourier coefficients are orthogonal to the Ψ(r,φ) coefficients?

2. Sorry for my redundancies in the boundary conditions (BCs). Trully, I was initially working with explicit time-dependence in the equations. I converted them to Omega and ε and forgot to remove them in the BCs, particularly the Ψ(0,φ) = ψ(φ). The zero was meant for time. So this condition should be ignored. I also understand that I was very wrong with the BC: Ψ(r,π) = Ψ(r,-π), since this gives discrete BCs. The right one to employ should be Ψ(r,φ) = Ψ(r,φ+2π) or Ψ(r,0) = Ψ(r, 2π).

3. Eventhough I am familiar with Dirac Bra-Ket notation, I am not very sure how I will rewrite my PDEs in the notation. Or you meant writing them like;

Del^2 Ψ(r,φ) = L Ψ(r,φ)  as  < r, φ|Del^2|Ψ> = L < r, φ|Ψ>.

Please, I  don't understand your last statement!!!

Thank you

Thank you for your prompt reply. At least, I can continue from here now.

I am most grateful for all your contributions.

Thank you professor Farid.

In fact, I am looking at hydrogen problem now.

Dear professor,

Now, when do we use Bessel or Lagurre Solutions? I have seen these two types of solutions for the same system.