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# Differential Operators : Eigenvalues and Eigenfunctions

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Let L = 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
(v,u) =S 1-->0 v-bar u dx
and compare S with S*.
(b) COMPARE the eigenvalues of
Lu =...
with the eigenvalues of
L*v n = ...
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 .... for n does not equal m.