## REFERENCES

1.

Martin Ligare

, “Numerical analysis of Bose–Einstein condensation in a three-dimensional harmonic oscillator potential

,” Am. J. Phys.

66

(3

), 185

–190

(1998

).2.

S.

Grossmann

and M.

Holthaus

, “λ-transition to the Bose–Einstein condensate

,” Z. Naturforsch., A: Phys. Sci.

50

, 921

–930

(1995

);K.

Kirsten

and D. J.

Toms

, “Bose–Einstein condensation of atomic gases in a general harmonic-oscillator confining potential trap

,” Phys. Rev. A

54

, 4188

–4203

(1996

);W.

Ketterle

and N. J.

van Druten

, “Bose–Einstein condensation of a finite number of particles trapped in one or three dimensions

,” Phys. Rev. A

54

, 656

–660

(1996

);N. J.

van Druten

and W.

Ketterle

, “Two-step condensation of the ideal Bose gas in highly anisotropic traps

,” Phys. Rev. Lett.

79

, 549

–552

(1997

);T.

Haugset

, H.

Haugerud

, and J. O.

Andersen

, “Bose–Einstein condensation in anisotropic harmonic traps

,” Phys. Rev. A

55

, 2922

–2929

(1997

);R. K.

Pathria

, “Bose–Einstein condensation of a finite number of particles confined to harmonic traps

,” Phys. Rev. A

58

, 1490

–1495

(1998

).3.

Consider $m$ units of energy $\u210f\omega .$ For $p=q=1$ and $r>1,$ these $m$ units must be partitioned between the two “small”-energy modes (excitations along the $x$ and $y$ axes), and the “large”-energy mode (excitation along the $z$ axis). There are $m+1$ ways to arrange the energy units with no energy in the large-energy mode, $m\u2212r+1$ ways to arrange the energy units with one excitation of the large-energy mode, etc. The largest possible number of excitations in the large-energy mode is $F(m/r),$ so the multiplicity is $n1=(m+1)+(m\u2212r+1)+\cdots +(m\u2212rF(m/r)+1).$ Summing the $F(m/r)+1$ terms in this expression gives Eq. (5). For $r=1$ and $p=q>1,$ there is only one way to arrange the energy units with no excitation in the two large-energy modes, two ways to arrange the energy units with one excitation in a large-energy mode, three ways to arrange the energy with two excitations in large-energy mode, etc. The largest possible number of excitations in the large-energy modes is $F(m/p),$ so the multiplicity is $n2=1+2+3+\cdots +(F(m/p)+1),$ which when summed gives Eq. (6).

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