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?

 

[tacman]

, and h(n) is any continuous function.

Yes, you are correct. The fact that the eigenvalues are continuous means that we have a continuous spectrum, and in this case, the Dirac delta function is replaced by the continuous Kronecker delta function. This function still satisfies the properties of the Dirac delta function, such as being zero everywhere except at the point of interest, and integrating to 1 over the entire domain. Therefore, the scalar product <y_n |y_m> is indeed equal to the continuous Kronecker delta function, which is defined as 1 when n=m and 0 otherwise. This leads to the integral given in the problem, which shows that the eigenfunctions y_n and y_m are orthogonal.