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I'm puzzling out the way one can determine $exp(iA)$ operator for an unbounded $A$. More precisely I would like to know, how could I deal with the momentum operator $p_x = -i\hbar\partial_x$.

I'm trying to follow the instruction which I learned from Frederic Schullers lectures (https://www.youtube.com/watch?v=GbqA9Xn_iM0 lecture 10 and 11)

The instruction is as following:

1) Construct the positive real-valued measure $\mu_{\psi}$ using Stieltjes inversion formula:

$\mu_{\psi}((-\infty,\ \lambda]) = \lim_{\delta\to 0^{+}}\lim_{\epsilon\to 0^{+}}\frac{1}{\pi} \int\limits_{-\infty}^{\lambda+\delta}dt \ Im<\psi, R_p(t + i\epsilon)\psi> $, where $R_p(z)$ is a resolvent operator.

2)Construct complex-valued measure $\mu_{\psi, \phi}$ by polarization formula:

$\mu_{\psi, \phi}(\Omega) = \frac{1}{4} [\mu_{\psi+\phi}(\Omega) - \mu_{\psi-\phi}(\Omega)+i\mu_{\psi-i\phi}(\Omega)-i\mu_{\psi+i\phi}(\Omega)], \Omega$ is a Borel set in $\mathbb{R}$

3) Construct projection-valued measure $P$ as following:

$<\psi, P(\Omega)\phi> := \int \chi_\Omega d\mu_{\psi, \phi}$

4) Calculate the integral:

$exp(i\hbar\partial_x) := \int_{\mathbb{R}} e^{i\lambda}\ P(d\lambda)$

My achievments are really poor. Actually, I've just calculated the resolvent for the momentum operator: $R_p(z) = \frac{i}{\hbar} \int\limits_{0}^{\infty}dt\ e^{izt\hbar^{-1}}u(t)$, where $u(t)\psi(x) = \psi(x-t)$. But I got in trouble even at the first step while calculating Stieltjes integral. I've achieved such thing to calculate:

$\frac{i}{h}\int\limits_{0}^{\infty}da\ e^{iza\hbar^{-1}}\ \int_{\mathbb{R}} dx\ \psi^{*}(x)u(a)\psi(x)$ and I don't know how to deal with it.

Maybe someone tried this way and can give me some advices? However, I bet it is not the shortest way to rigorously construct exponential of unbounded operator. Are they any other ideas to define exp? If they are, what is the main purpose of the spectral theorem and all of equastions I mentioned before. It seems to be not constructive in this case.

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    $\begingroup$ would SE Mathematics be better? $\endgroup$ – ZeroTheHero Sep 10 '17 at 21:44
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    $\begingroup$ the definition of the exponential of an unbounded operator cannot be given through the power series, at least not for the whole of the space. $\endgroup$ – QCrypt Sep 10 '17 at 22:53
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    $\begingroup$ one should consider the exponential mapping in the differential geometric sense instead $\endgroup$ – QCrypt Sep 10 '17 at 22:59
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    $\begingroup$ However, usually in physics, one deals with subspaces of the whole hilbert space, where the power-series expressions are valid. I was wondering, what kind of problem do you have such that the power series expressions are not valid ? it would be useful to add some context on this point. $\endgroup$ – QCrypt Sep 10 '17 at 23:08
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    $\begingroup$ OP should check the standard references: Reed and Simon (volumes I and II), Hall (Quantum Theory for Mathematicians), etc. $\endgroup$ – Ryan Unger Sep 11 '17 at 1:46

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