Differential operators-eigenvalues and eigenfunctions

Question

Problem:

Differential Operators : Eigenvalues and Eigenfunctions

Let L = - d²/dx² with boundary conditions u(0) = 0, u'(O) = u(1) ,so that the domain of L is S = {u:Lu is square integrable; u(0) = 0, u'(O) = u(1)}.

(a) For the above differential operator FIND S* for the adjoint with respect to

                            = ∫¹₀ v^-u dx.

and compare S with S*.

(b) COMPARE the eigenvalues λ_n

                     Lu_n = λ_nu_n n = 0,1,2,...

With the eigenvalues λ^*_n of

                       L*v_{n =} λ_n^*u_n n = 0,1,2,...

If the two sequences of eigenvalues are different, point out the distinction; if you find they are the same, justify that result.

(c) EXHIBIT the corresponding eigenfunctions.

(d) WHAT is the eigenvalue with the smallest modulus ("absolute value")? What is the corresponding eigenfunction?

(e) VERIFY that  ∫¹₀ v^-u dx = 0 for n≠m.