1. Consider the eigenvalue problem - u" (x) = Au(x) for - T

1. Consider the eigenvalue problem (a) (b) (c) (d) u" for < T, (1) u(T) + 0, u' (T) + u' 0. Let U {U e T]) u(T) + 0, u' (T) + u' 0} be equipped with the usual L2-inner product on T], that is, (u, v ) u(a;) v(x) dat. Prove that the operator S[u] u" is self-adjoint on the domain U. Explain why all eigenvalues of (1) are i) real and ii) positive. Find all eigenvalues of (1) and the corresponding eigenfunctions. Determine the multiplicity of the eigenvalues.