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Thank you very much for the interesting responses in my previous question The Quotient exponential operator. I have another complicated formula related to the previous one in the following $$ \exp\left[-x\bigg(\frac{d}{dx}\bigg)^{2}+\frac{d}{dx}\right]\big(f(x)\big). $$
I would like to develop this equation and decompose it using a reference or a Book: I have found this kind of operators in G. Dattoli and A. Torre's works. I inform everyone that the two researchers produce many papers and Books in the fields of Physics and mathematics, so that is very difficult to search and find the required response within their large numbers of papers. If someone can help me I will be very grateful. Respectful regards.

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    $\begingroup$ Depends how explicit an answer you're expecting. With higher derivatives like that in the exponent, you're generally going to get forms where the argument of $f$ contains derivatives, which may or may not be very useful (depends - if $f$ is itself exponential, that may then reduce to something simpler again). Alternatively, you might get answers in the form of series. $\endgroup$ Jun 23, 2021 at 0:00
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    $\begingroup$ It's worth to notice that $$\left[-x(\frac{d}{dx})^{2}+\frac{d}{dx}\right]^2 = x^2(\frac{d}{dx})^{4}.$$ $\endgroup$ Jun 23, 2021 at 2:23
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    $\begingroup$ It's also worth noting that, if we denote $A=d/dx$ and $B=-x(d/dx)^2 $, then we have the specific algebra $[A,B]=-A^2 $. $\endgroup$ Jun 23, 2021 at 7:45
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    $\begingroup$ Correct the question, then, perhaps? It saves the reader from reading the comments. You can certainly adapt the remark of @MichaelEngelhardt to this case, and find how the algebra works out. $\endgroup$
    – username
    Jun 23, 2021 at 12:39
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    $\begingroup$ Formally, this would be $u(x,1)$, where $u(x,t)$ solves the equation $u_t=-xu_{xx}+u_x$ with initial condition $u(x,0)=f(x)$. However, this initial value problem is not well-posed, at least not for $x$ on the real line. $\endgroup$ Jun 23, 2021 at 15:20

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