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

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SUMMARY

The discussion centers on the orthogonality of self-adjoint operators represented by the equation \(\mathcal{L}[y(x)]=-\lambda_{n} y(x)\), where \(n\) is a continuous spectrum of positive real numbers. The participants confirm that the scalar product \(\langle y_{n} | y_{m} \rangle = \delta(n-m)\) holds true, indicating orthogonality through the Dirac delta function. The continuous set of eigenvalues \(\lambda_{n}=h(n)\) reinforces the need to utilize the inner product \(\langle L[y_n], y_m \rangle\) to validate the self-adjoint nature of the operator. The suggestion to extend discrete proofs of orthogonality to the continuous case is also highlighted.

PREREQUISITES
  • Understanding of self-adjoint operators in functional analysis
  • Familiarity with Dirac delta functions and their properties
  • Knowledge of eigenvalues and eigenfunctions in quantum mechanics
  • Basic concepts of inner product spaces
NEXT STEPS
  • Study the properties of self-adjoint operators in Hilbert spaces
  • Explore the application of Dirac delta functions in quantum mechanics
  • Learn about the spectral theorem for continuous spectra
  • Investigate the extension of discrete orthogonality proofs to continuous cases
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Mathematicians, physicists, and graduate students specializing in quantum mechanics, functional analysis, or operator theory will benefit from this discussion.

Sangoku
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given a self-adjoint operator

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

where the index 'n' can be any positive real number (continous spectrum) then my question is if

[tex]\int_{c}^{d}\int_{a}^{b}dn y_{n}(x)y_{m} (x)w(x)dx = 1[/tex]

deduced from the fact that for continuous n and m then the scalar product

[tex]<y_{n} |y_{m} > =\delta (n-m)[/tex] (Dirac delta --> continuous Kronecker delta --> discrete case )

am i right ?? ... for the problem we have a continuous set of eigenvalues [tex]\lambda _{n}=h(n)[/tex] where n >0 is any real and positive number
 
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I believe you should use the inner product [itex]<L[y_n],y_m>[/itex] 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?
 

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