At the moment I am reading the paper: "Objective interpretations of quantum mechanics and the possibility of a deterministic limit", from A. Sudbery, and I am struggling to understand a certain computation he does in his work.

I hope my problem is understandable without knowing the details about the paper.

So, we have intervals $I_k=[x_k-\delta, x_{k+1}+\delta]$, which overlap by a small amount of $2\delta$ and the projection operators, $\Pi_k$, on those intervals are given by:

$$\Pi_k\phi(x)= \Theta(x-(x_k-\delta))\Theta((x_{k+1}+\delta)-x) $$

Where, $\phi(x)$ is a wavefunction of a continuous variable $x$. For $I_k, I_l$ being adjacent, i.e. $l=k-1$, he computes the following:

$$ \langle\phi\mid \Pi_{k-1}H\Pi_k \mid \phi\rangle= \int_{x_{k-1}-\delta}^{x_k+\delta}\overline{\phi(x)}\left(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V \right) \left[\phi(x)\Theta(x_{k+1}+\delta-x)\right] \, \text{d} x$$

This step seems clear to me, since $H$ denotes the normal Hamiltonian and we use the continuous inner product. But the next step is what I don't get. He claims that the above equation equals:

$$\langle\phi\mid \Pi_{k-1}H\Pi_k \mid \phi\rangle = -\frac{\hbar^2}{2m}\left[\overline{\phi}\frac{d\phi}{dx} -\frac{d\overline{\phi}}{dx}\phi\right]_{x=x_k-\delta}+\int_{x_k-\delta}^{x_k+\delta}\overline{\phi(x)}H\phi(x)\, \text{d}x $$

Where does the term:

$$-\frac{\hbar^2}{2m}\left[\overline{\phi}\frac{d\phi}{dx} -\frac{d\overline{\phi}}{dx}\phi\right]_{x=x_k-\delta}$$

come from and how does one compute such an integral?

I would be very greatful for any help! Thank you so much!



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