Does a Continuous Spectrum Confirm Orthogonality in Self-Adjoint Operators?

[Sangoku]

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

 

[AiRAVATA]

I believe you should use the inner product $<L[y_n],y_m>$ and use the fact that L is self-adjoint, but I am not sure. Why don't you see the discrete proof for orthogonal functions and try to extend it?