  We study two interacting quantum particles forming a bound state in
$d$-dimensional free space, and constrain the particles in $k$ directions to
$(0,\infty)^k \times \mathbb{R}^{d-k}$, with Neumann boundary conditions.
First, we prove that the ground state energy strictly decreases upon going from
$k$ to $k+1$. This shows that the particles stick to the corner where all
boundary planes intersect. Second, we show that for all $k$ the resulting
Hamiltonian, after removing the free part of the kinetic energy, has only
finitely many eigenvalues below the essential spectrum. This paper generalizes
the work of Egger, Kerner and Pankrashkin (J. Spectr. Theory 10(4):1413--1444,
2020) to dimensions $d>1$.
