  This paper treats a quantum network from a physical approach, explicitly
finds the physical eigenstates and compares them to the quantum-graph
description. The basic building block of a quantum network is an X-shaped
potential well made by crossing two quantum wires, and we consider a massive
particle in such an X well. The system is analyzed using a variational method
based on an expansion into modes with fast convergence and it provides a very
clear intuition for the physics of the problem. The particle is found to have a
ground state that is exponentially localized to the center of the X well, and
the other symmetric solutions are formed so to be orthogonal to the ground
state. This is in contrast to the predictions of the conventionally used
so-called Kirchoff boundary conditions in quantum graph theory that predict a
different sequence of symmetric solutions that cannot be physically realized.
Numerical methods have previously been the only source of information on the
ground-state wave function and our results provide a different perspective with
strong analytical insights. The ground-state wave function has the shape of a
solitonic solution to the non-linear Schr{\"o}dinger equation, enabling an
analytical prediction of the wave number. When combining multiple X wells into
a network or grid, each site supports a solitonic localized state. The solitons
only couple to each other and are able to jump from one site to another as if
they were trapped in a discrete lattice.
