• Incand

## Homework Statement

Under what condition on the constant ##c## and ##c'## are the boundary conditions ##f(b) = cf(a)## and ##f'(b)=c'f'(a)## self-adjoint for the operator ##L(f) = (rf')'+pf## on ##[a,b]##? (Assume that ##r,p## are real.)

## Homework Equations

The boundary conditions are self-adjoint (relative L) if
##\left[r(f'\bar g-f\bar g ')\right]_a^b=0##
for all ##f,g## satisfying the boundary conditions.
(Assuming that ##L## is formally self-adjoint as it is in this case.)

## The Attempt at a Solution

##\left[r(f'\bar g-f\bar g ')\right]_a^b = r(b)\left( f'(b)\bar g (b) - f(b)\bar g '(b)\right) -r(a)\left( f'(a)\bar g(a)-f(a)\bar g '(a)\right) = f'(a)\bar g (a)\left(c'\bar c r(b)-r(a)\right) + f(a)\bar g(a)\left(-r(b)c\bar c'+r(a)\right)##

So a solution would be ##c'\bar c = c\bar c' = \frac{r(a)}{r(b)}##.

The answer to the question says only ##c\bar c' = \frac{r(a)}{r(b)}## but I got that ##c'\bar c## also has to have the same value. Am I correct here or did i make a mistake?

## Homework Statement

Under what condition on the constant ##c## and ##c'## are the boundary conditions ##f(b) = cf(a)## and ##f'(b)=c'f'(a)## self-adjoint for the operator ##L(f) = (rf')'+pf## on ##[a,b]##? (Assume that ##r,p## are real.)

## Homework Equations

The boundary conditions are self-adjoint (relative L) if
##\left[r(f'\bar g-f\bar g ')\right]_a^b=0##
for all ##f,g## satisfying the boundary conditions.
(Assuming that ##L## is formally self-adjoint as it is in this case.)

## The Attempt at a Solution

##\left[r(f'\bar g-f\bar g ')\right]_a^b = r(b)\left( f'(b)\bar g (b) - f(b)\bar g '(b)\right) -r(a)\left( f'(a)\bar g(a)-f(a)\bar g '(a)\right) = f'(a)\bar g (a)\left(c'\bar c r(b)-r(a)\right) + f(a)\bar g(a)\left(-r(b)c\bar c'+r(a)\right)##

So a solution would be ##c'\bar c = c\bar c' = \frac{r(a)}{r(b)}##.

The answer to the question says only ##c\bar c' = \frac{r(a)}{r(b)}## but I got that ##c'\bar c## also has to have the same value. Am I correct here or did i make a mistake?

$c'\bar c$ is the complex conjugate of $\bar c' c$. The two are equal as $\frac{r(a)}{r(b)}$ is real and therefore equal to its complex conjugate.

• Incand
$c'\bar c$ is the complex conjugate of $\bar c' c$. The two are equal as $\frac{r(a)}{r(b)}$ is real and therefore equal to its complex conjugate.
Thanks!