Partial Differential Equations Separation of Variables and F

Partial Differential Equations: Separation of Variables and Fourier Series

Consider the eigenvalue problem (6.15) with Robin boundary conditions. Suppose lambda = 0 is an eigenvalue. Find the eigenfunction. Find a necessary condition on the coefficients a_0, a_l. Prove that this condition is also sufficient to guarantee that lambda = 0 is an eigenvalue.

Solution

Part b:

If we have a zero eigenvalue, then we have V = 0 or V(x) = ax + b.

Plugging in the boundary conditions gives

a a₀b = 0,

a + a_L(aL + b) = 0.

We want to find a nontrivial solution for a, b in this equation, or, as above, we need the matrix

1 -a₀

   _{1 +} a_LL a_{L (matrix)}

to have determinant zero, or a_L + a₀(1 + a_LL) = 0.(This is the necessary condition for the coefficients of a₀ and a_L. This gives the solution for part b).

Part a:

If this determinant is zero, we know that the two equations we have are redundant, so we can solve either. The simpler to solve is the first, which gives a = a₀b, and of course we will have one free choice for b.

So,the eigenfunction is V(x) = a₀x + 1, and we can of course choose any scalar multiple of this. Hence proved part a.

Part c:

From the above proofs, it is sufficient that   = 0 is an eigenvalue.