inverse Bogolyubov transformation.

A general formula for operators $U$, $V$ for a given Bogolyubov transformation is known (see e.g. formula (5.15) from Berezin’s book [@Berezin]). Using it we find the following explicit formulas for the two blocks of $\hat D_{m}^{-1}$: D\_[m]{}\^[-1]{} = (

[cc]{} D\^[NS]{} & 0\
0& D\^[R]{}

) , \[IONS\] D\^[NS]{} = [N]{}()|00|\_[NS]{} \_[k&gt;0]{}( i(a\^\_[k]{}|a\^\_[k]{} - b\_[-k]{}b\_[k]{}) + f\^[-1]{}(k)(a\^\_[k]{}b\_[-k]{} + |a\^\_[k]{}b\_[k]{}) ) , \[IOR\] D\^[R]{} = ()\_[n=1]{}\^( i(a\^\_[n]{}|a\^\_[n]{} - b\_[-n]{}b\_[n]{}) + f\^[-1]{}(n)(a\^\_[n]{}b\_[-n]{} + |a\^\_[n]{}b\_[n]{}) ) where = {

[l@l]{} |0|\_[R]{} + |0|\_[R]{}b\_[0]{} , m&gt;0\
|0|\_[R]{} b\_[0]{} + |0|\_[R]{} , m&lt;0

. and all creation operators $a_{n}^{\dagger}, \bar a_{m}^{\dagger}$ are understood to act on $|0\rangle$ and $|\sigma\rangle$ from the left. The normalisation factors ${\cal N}(\nu) $, $ {\tilde {\cal N}}(\nu)$ are given in terms of infinite products () = \_[n=0]{}\^ f(n+1/2) , () = \_[n=1]{}\^ f(n) . where we denoted $\nu = |m|R$ - the dimensionless mass.

Taking the logarithms of ${\cal N}(\nu) $, $ {\tilde {\cal N}}(\nu)$ and applying Abel-Plana summation formula we obtain the following integral expressions \[N1\] [N]{}() = , \[N2\] () = 2\^[1/4]{} ,

The ratio of the normalisation factors $\tilde {\cal N}/{\cal N}$ interpolates between $1$ at $\nu=0$ and $2^{1/4}$ at $\nu=\infty$. Below is 