Abstract
The exact density of states for one particle in the field of an attractive $\ensuremath{\delta}$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The Bose-Einstein phase is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature ${T}_{c}$. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, ${E}_{b}$. The transition temperature and the discontinuity of the specific heat at ${T}_{c}$ are also obtained as a function of ${E}_{b}$.
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