# Computing integrals with differential operator

I've to compute this expression

$$\hat{H} = \frac{1}{4}g_2\int d^3R\int d^3r\ \bar{\Psi}(\vec{R}+\frac{\vec{r}}{2})\bar{\Psi}(\vec{R}-\frac{\vec{r}}{2})\left[ \delta(\vec{r})\nabla_{\vec{r}}^2 +\nabla_{\vec{r}}^2\delta(\vec{r}) \right]\Psi(\vec{R}+\frac{\vec{r}}{2})\Psi(\vec{R}-\frac{\vec{r}}{2})$$

where $\bar{\Psi}$ is the conjugate of $\Psi$.

How can i calculate it with Mathematica? I don't know how to represent the differential operator and doing this kind of integral with Mathematica.

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Instead to give me negative vote, you can explain me what i've to add to my question.

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I will venture to guess that the downvote was because there is no Mathematica code shown, just an image. It's considerable work for readers to try to transcribe an image into code. As for representing the operator, maybe write it out as a sum of derivatives in Cartesian or spherical coordinates,then apply to the functions to itsright. – Daniel Lichtblau Jun 21 '14 at 19:19
This is just a mathematical integral that i'd like to calculate in Mathematica. I don't know how to write it with the code. Or even, i know how to integrate a function, but i haven't try to evaluate an integral with distributions. – user13653 Jun 21 '14 at 19:23
Probably premature to post questions to this site if you are unfamiliar with Mathematica notation. It's not really a site suitable for learning the basics, and there are other resources for that; an internet search should bring up some useful hits. – Daniel Lichtblau Jun 21 '14 at 19:31
In addition to things written above the question is formulated incorrectly. It is unclear, what the author wants to do with this integral. This integral is well-known. Integrals of such kind, for example, enter the potential giving rise to Ginzburg-Landau equation. They cannot be simply calculated in the case of arbitrary function Psi. May be, rewritten in a different form? I think, the author should better think about the statement of his question. – Alexei Boulbitch Jun 23 '14 at 9:44