# Continous spcetrum

1. Aug 26, 2007

### Sangoku

$$\mathcal L [y(x)]=-\lambda_{n} y(x)$$

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 continous n and m then the scalar product

$$<y_{n} |y_{m} > =\delta (n-m)$$ (Dirac delta --> continous Kronecker delta --> discrete case )

am i right ?? ... for the problem we have a continous set of eigenvalues $$\lambda _{n}=h(n)$$ where n >0 is any real and positive number

2. Aug 26, 2007

### AiRAVATA

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