Fluctuation-induced potential for an impurity in a semi-infinite one-dimensional Bose gas

Understand

We consider an impurity in a semi-infinite one-dimensional system of weakly-interacting bosons.

• We calculate the interaction potential for the impurity due to the end of the system, i.e., the wall.
• For local repulsive (attractive) interaction between the impurity and the Bose gas, the interaction potential is attractive (repulsive).
• At short distances from the wall it decays exponentially crossing over into a universal $1/r^2$ behavior at separations $r$ above the healing length.