By the spectral theorem, the operator in parentheses acts at spectral value $t\in\mathbb R$ by
$$
\frac1{2\pi i}\int_a^b
\left[\frac1{t-x-i\epsilon}
-\frac1{t-x+i\epsilon}\right]dx
=\frac1\pi\int_a^b
\frac\epsilon{(t-x)^2+\epsilon^2}\,dx.
$$
The Poisson kernel converges to $1$ for $t\in(a,b)$, to $1/2$ at $t=a,b$, and to $0$ outside $[a,b]$. It is uniformly bounded, so dominated convergence in the <spectral measure of a normal operator> gives the strong limit
$$
E((a,b))+\frac12E(\{a\})+\frac12E(\{b\})
=\boxed{\frac12[E((a,b))+E([a,b])]}.
$$
Applying this operator to $v$ proves the claimed <Stone formula>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 358 / 3 / b / Solution

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Stone's formula recovers spectral projections of a self-adjoint operator from the jump of its resolvent across the real axis, with half weight at interval endpoints.


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 Incoming links: Stone formula