Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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