- #1
Sangoku
- 20
- 0
given a self-adjoint operator
\mathcal L [y(x)]=-\lambda_{n} y(x)
where the index 'n' can be any positive real number (continous spectrum) then my question is if
\int_{c}^{d}\int_{a}^{b}dn y_{n}(x)y_{m} (x)w(x)dx = 1
deduced from the fact that for continuous n and m then the scalar product
<y_{n} |y_{m} > =\delta (n-m) (Dirac delta --> continuous Kronecker delta --> discrete case )
am i right ?? ... for the problem we have a continuous set of eigenvalues \lambda _{n}=h(n) where n >0 is any real and positive number
\mathcal L [y(x)]=-\lambda_{n} y(x)
where the index 'n' can be any positive real number (continous spectrum) then my question is if
\int_{c}^{d}\int_{a}^{b}dn y_{n}(x)y_{m} (x)w(x)dx = 1
deduced from the fact that for continuous n and m then the scalar product
<y_{n} |y_{m} > =\delta (n-m) (Dirac delta --> continuous Kronecker delta --> discrete case )
am i right ?? ... for the problem we have a continuous set of eigenvalues \lambda _{n}=h(n) where n >0 is any real and positive number
