Under what condition on the constants c and c' are the boundary conditions
\(\displaystyle f(b) = cf(a) \)
\(\displaystyle f'(b) = c'f'(a) \)
self-adjoint for the operator \(\displaystyle L(f) = (rf')' + pf \) on [a,b]? Assume that r and p are real.
Boundary conditions are self-adjoint if
\(\displaystyle [r(f'\bar g - f\bar g')]^b_a = 0\)
Using this evaluation and boundary conditions, I end up with
\(\displaystyle f'(a)\bar g [c'r(b)-r(a)] + f(a)\bar g'[r(a) - cr(b)] = 0\)
which implies that \(\displaystyle c' = c = r(a)/r(b) \)
However, the back of the book has \(\displaystyle c \bar c'= r(a)/r(b).\) What am I doing wrong?
\(\displaystyle f(b) = cf(a) \)
\(\displaystyle f'(b) = c'f'(a) \)
self-adjoint for the operator \(\displaystyle L(f) = (rf')' + pf \) on [a,b]? Assume that r and p are real.
Boundary conditions are self-adjoint if
\(\displaystyle [r(f'\bar g - f\bar g')]^b_a = 0\)
Using this evaluation and boundary conditions, I end up with
\(\displaystyle f'(a)\bar g [c'r(b)-r(a)] + f(a)\bar g'[r(a) - cr(b)] = 0\)
which implies that \(\displaystyle c' = c = r(a)/r(b) \)
However, the back of the book has \(\displaystyle c \bar c'= r(a)/r(b).\) What am I doing wrong?