# Convergence of self-adjoint operators with converging spectra

Let $$A=\int_{\sigma(A)} \lambda dP(\lambda)$$ be an unbounded self-adjoint operator on a Hilbert space $$H$$ with dense domain $$D\subset H$$.

Let $$A_n=\int_{\sigma(A)\cap[-n,n]} \lambda dP(\lambda)$$ (see motivation), which then defines a sequence of bounded (by $$n$$, from spectral radius) self-adjoint operators $$\{A_n\}$$. I want to show strong convergence $$A_n \rightarrow A$$, e.g. $$||A_n\psi-A\psi||$$ converges for all $$\psi \in D$$. Since $$A$$ is unbounded and $$\psi$$ has infinite support in general, I am struggling to come up with a convergence argument, not least because I think the operator $$A$$ (smeared field operator in QFT, but could be position operator in QM also) should have a spectrum that is $$\mathbb{R}$$, so $$A_n-A$$ has a spectrum $$\sim (-\infty,-n)\cap(n,\infty)$$, and I am not hugely familiar with these operators outside of the usual countable dimension assumptions in QM. If A were bounded, I would make an argument based on existence of an $$n >$$spectral radius, but of course that does not help here. Any hints would be much appreciated.

Just use $$\|(A-A_n)\psi\|^2=\int_{\sigma(A)\cap(-\infty, -n)} \lambda^2\;\mathrm{d}\|P(\lambda)\psi\|^2+\int_{\sigma(A)\cap(n ,\infty)} \lambda^2\;\mathrm{d}\|P(\lambda)\psi\|^2, \quad \psi \in D$$ and apply monotne convergence theorem (the integrals above are just standard integrals in $$\mathbb{R}$$).