Spectral theorem/ exchanging limit of series and operator

Question

I am currently learning quantum mechanics and there is one typical scenario i encounter in my physics books:

Suppose $\mathcal{H}$ is a Hilbert space and $A: \operatorname{Dom}(A)\to \mathcal{H}$ is a (linear) operator, $Dom(A)\subseteq\mathcal{H}$. In addition, let $\mathcal{B}:=\{v_n: n \in\mathbb{N}\}$ be a set of eigenvectors of $A$: $A(v_n)=\lambda_nv_n$.

If there is a $v \in \operatorname{Dom}(A)$ and a sequence of coefficients $(c_n)_{n \in\mathbb{N}}$ with $v=\sum\limits_{n=1}^{\infty}c_nv_n$, then the authors write $A(v)=A(\sum\limits_{n=1}^{\infty}c_nv_n)=\sum\limits_{n=1}^{\infty}c_nA(v_n)=\sum\limits_{n=1}^{\infty}(c_n\lambda_n)v_n$.

From what i know, A is not continuous in general, but this is a special case of the spectral theorem. I think that a physicist would write $A=\sum\limits_{n=1}^{\infty}|v_n\rangle\langle v_n|$ and call this the spectral theorem for the discrete case. The problem is that when looking into a math book about the spectral theorem, i see a lot of integrals and things get very complicated. So it would be nice if someone could explain this special case of the spectral theorem (its requirements and statements), or suggest a good source.

Answer 1

The operator $A$ is typically not continuous, but since it is self-adjoint, it is closed (see below). One of the conclusions of the spectral theorem is that $v\in \mathrm{Dom}(A)$ is equivalent to the convergence of the series of nonnegative numbers $$ \sum_n \lambda_n^2\lvert c_n\rvert^2. $$ This implies that the sequence $ S_N:= \sum_{n=1}^N \lambda_n c_n v_n $ is convergent, because $$\lVert S_M-S_N\rVert=\left\lVert \sum_{n=N}^M \lambda_n c_n v_n\right\rVert=\sqrt{\sum_{n=N}^M \lvert c_n\rvert^2\lambda_n^2} ,$$ so $S_N$ satisfies the Cauchy condition.

The formula $$\tag{1} A\sum_n c_n v_n= \sum_n \lambda_n c_n v_n$$ is now proven by using the fact that $A$ is closed$^{[1]}$. Indeed, $A(\sum_1^N c_n v_n)=\sum_{1}^N \lambda_n c_n v_n$, and we just saw that the right-hand side converges. Since $\sum_n \lvert c_n\rvert^2<\infty$, by reasoning as above we see that $\sum_1^N c_n v_n$ converges, so by the closedness property we can pass to the limit and prove (1).

Remark. Here we supposed that $A$ has a discrete spectrum. This means that every spectral value is an eigenvalue and that there exists an orthonormal basis of $\mathcal{H}$ made of eigenvectors. If this is not the case, the series $\sum_n$ have to be replaced by integrals. This is why mathematics books explain the spectral theorem in terms of spectral integrals. The basic idea, however, is already fully contained in the discrete spectrum case.

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$[1]$. This means that if $f_n\in \mathrm{Dom}(A)$ is such that $f_n\to f$ and $Af_n \to g$, then $f\in \mathrm{Dom}(A)$ and $g=Af$.