Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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2.5. Exact diagonalization <strong>of</strong> the <strong>Bogoliubov</strong> problem<br />
The eigenvalues λm and the (right) eigenvectors am <strong>of</strong> U are defined by<br />
�<br />
[Uij − λmδij] (am)j = 0 ⇔ det (Uij − λmδij) = 0 . (2.70)<br />
j<br />
Analogously the eigenvectors <strong>of</strong> the adjoint operator U † = �<br />
j,k ejU ∗ kj e† k (or<br />
left eigenvectors <strong>of</strong> U) can be defined<br />
�<br />
j<br />
� � ∗<br />
Uji − µnδi,j (bn)j = 0 ⇔ det � U ∗ �<br />
ji − µnδi,j = 0 . (2.71)<br />
Taking the complex conjugate and interchanging the indices, we see, that µ ∗ n<br />
and λm satisfy the same equation. Thus, the eigenvalues can be enumerated<br />
such that µ ∗ n = λn. Finally, we compute the scalar product b † nU am and let<br />
U act both to the right and to the left<br />
b † nU am = �<br />
(b † n)iUij(am)j<br />
i,j<br />
(2.70)<br />
�<br />
= (b † n)iλm(am)i<br />
From this, it follows the bi-orthogonality relation<br />
i<br />
(2.71)<br />
�<br />
= µ ∗ n(b † n)j(am)j .<br />
(λm − λn) (bn, am) = 0 . (2.72)<br />
Thus, the eigenvector am is orthogonal to the adjoints <strong>of</strong> eigenvectors with<br />
different eigenvalues bm. This, however, does not imply that the eigenvectors<br />
themselves are pairwisely orthogonal. Only in the case <strong>of</strong> a Hermitian<br />
operator with Uij = U ∗ ji , the left equations <strong>of</strong> (2.70) and (2.71) imply that<br />
an = bn, which then restores the usual orthogonality <strong>of</strong> eigenvectors.<br />
2.5.5. <strong>Bogoliubov</strong> eigenstates with non-zero frequency<br />
Let us express the bi-orthogonality relation derived above in terms <strong>of</strong> the<br />
<strong>Bogoliubov</strong> eigenstates. Let aν = ( uν<br />
vν ) be an eigenvector <strong>of</strong> ηQ with eigenvalue<br />
ων: ηQ aν = �ων aν. With η2 = 1, it can be seen that the matrix η<br />
relates the eigenvectors to those <strong>of</strong> the adjoint operator Qη<br />
Qη(ηaν) = η(ηQ)aν = �ων ηaν ⇒ bν = ηaν. (2.73)<br />
With this, the bi-orthogonality relation (2.72) reads<br />
0 = (ων − ωµ)b † �<br />
νaµ = (ων − ωµ) d d r [u ∗ ν(r)uµ(r) − v ∗ ν(r)vµ(r)] (2.74)<br />
j<br />
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