# Theorems on orthonormal bases and spectrum

Are there theorems similar to the following: If $T$ is symmetric and $D(T)$ contains an ONB of eigenvectors of $T$, then $T$ is essentially self adjoint and the spectrum of $\bar{T}$ is the closure of the point spectrum $\sigma_p(T)$?

I am interested in these theorems as in physics they do the following to deduce that the spectrum of $A:=-(1/2)(d^2/dx^2+x^2)$ is $1/2,1,3/2,\ldots$ find some of the eigenvalues and show that the associated eigenvectors are an ONB. Why does it follow that $\sigma(A)=\sigma_p(A)$?

Everything happens on Schwarz space

As a related question: Reason for Continuous Spectrum of Laplacian " has a discrete spectrum consisting of the eigenvalues ... as can be seen from the eigenfunction basis" how come?

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