Eigenvalue problem-transcendental equation-positive-definite

Question

Problems:

Eigenvalue Problem : Transcendental Equation, Positive-Definite and Orthonormal

Solve the eigenvalue problem

                          -u"(x) = λu(x)
                           u'(1) = λu(1)   0                           u(0) = 0

as follows:

Let U = (^u(x)_u1)be a two-component vector whose first component is a twice differentiable function u(x), and whose second component is a real number u₁.Consider the corresponding vector space H with inner product.

                               Ξ ∫¹₀ u(x)v(x)dx + u₁v₁

Let S⊂H be the subspace

                                S = {U:U = (^u(x)_u1) ; u(0) = 0}

and let

                                 LU = (^-u"(x)_u'(1)).

The above eigenvalue problem can now be rewritten in standard form

                                      LUλU with U∈S.

(a) PROVE or DISPROVE that L is self adjoint, i.e. that = .

(b) PROVE or DISPROVE that L is positive-definite, i.e. that > 0 for U ≠ 0^→.

(c) FIND the (transcendental) equation for the eigenvalues of L.

(d) Denoting these eigenvalues by λ1, λ2, λ3, . . . , EXHIBIT the orthonormalized eigenvectors U_n=1,2,3,. .., associated with these eigenvalues.