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Three examples of spatial Bose-Einstein condensations in which the particles macroscopically occupy the lowest localized state of an inhomogeneous external potential are analyzed. The three cases are (a) a small potential well in a large box which causes a spectrum with a gap, (b) a harmonic oscillator potential, and (c) randomly sized trapping potentials caused by ’’impurities.’’ All three cases are two dimensional so that no Bose condensation occurs without the inhomogeneous potentials. An attempt is made to keep the treatments as mathematically simple as possible. A review of much of the literature of spatial Bose condensations is provided. The special problem of the form of the thermodynamic limit in an inhomogeneous potential is discussed for case (b). Numerical examples applying to monolayer 4He adsorbed on a surface are treated.
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Masut et al. (1979) studied this question.
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