Proof of $-\frac{1}{4}$ upper bound to the spectrum of $\Delta$

Question

Picture below is from this paper. I don't know how to get the inequation 1. Because in the inequation 2, $f'$ is the derivative of $f(y)$, there is not $\partial_{xx}f$. Even though integrate inequation 2 , I can't get inequation 1. How to get it ?

Answer 1

In equation $(1)$ the function $f$ is assumed to be smooth and compactly supported. Thus $$ \int_H \partial_x^2 f(x,y) dx dy = \int_0^\infty \int_{-\infty}^\infty \partial_x^2 f(x,y) dx dy =0 $$ since for a fixed $y >0$ $$ \int_{-\infty}^\infty \partial_x^2 f(x,y) dx = \lim_{T \to \infty} \partial_x f(x,y) \Big\vert^{x=T}_{x=-T} =0 $$ due to the fact that the support of $f$ is compact. Hence for such $f$ $$ \int_H \Delta f(x,y) dx dy = \int_H \partial_y^2 f(x,y) dx dy, $$ and so you can use equation $(2)$.