# Question on computation in the paper "Objective interpretations of quantum mechanics and the possibility of a deterministic limit" from A. Sudbery

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!