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--- hamburgminicourse2017:stone_s_theorem [2017/04/02 15:17] – created markus
+++ hamburgminicourse2017:stone_s_theorem [2017/04/02 15:26] (current) – markus
@@ Line 28: / Line 28: @@
 Finally we show that the new-formed evolution operator is unitary, which also yields uniqueness of the so-formed solution as well as conservation of propabilities. $H_\lambda$ is not even self-adjoint but the related $H_{(\lambda)} = \frac{1}{2}(H_\lambda + H_{-\lambda})$ that converges to $H$ just as well is. This means
 \[
-\langle \exp(-i H t)\varphi,\exp(-i H t)\psi \rangle = \lim_{|\lambda| \rightarrow \infty} \langle \exp(-i H_{(\lambda)} t)\varphi,\exp(-i H_{(\lambda)} t)\psi \rangle = \langle \varphi,\psi \rangle. \Box
+\langle \exp(-i H t)\varphi,\exp(-i H t)\psi \rangle = \lim_{\lambda,\mu \rightarrow \infty} \langle \exp(-i H_{(\lambda)} t)\varphi,\exp(-i H_{(\mu)} t)\psi \rangle = \langle \varphi,\psi \rangle. \Box
 \]
+Stone's theorem can also be taken as a one-to-one correspondence between self-adjoint operators and unitary one-parameter groups. For any given unitary one-parameter group on a Hilbert space there exists a unique self-adjoint generator in the role of the Hamiltonian. This operator is then defined as a limit as [[the_abstract_cauchy_problem_in_banach_space|before]] that does not necessarily converge for all $\psi \in \mathcal{H}$ which gives the respective domain of the Hamiltonian.
+\[
+H\psi = \lim_{t \rightarrow 0} \frac{\exp(-i H t)\psi - \psi}{t}
+\]
+This amounts to an interesting shift in perspective, making the whole evolution operation the fundamental ingredient defining the dynamics of a system with the Hamiltonian remaining in the position of a derived object.
+Note that the same result as Stone's theorem is amenable via spectral theory of operators where a resolution of identity yields a projection valued measure $d E_H$. The unitary evolution operator is then given as the exponential applied to the eigenvalues in the spectral representation.
+\[
+\exp(-i H t) = \int \exp(-i \varepsilon t) \,d E_{H}(\varepsilon)
+\]
+ >> [[Classes of static potentials]]