The complete eigenspectrum will contain all the information needed to determine
the expectation values and fluctuations of the individual spins in the system,
either for a single spinwave mode, or, for the system in thermal equilibrium.
To determine expectation values of the spin components or their squares, we
need to know 
and 
in terms of
the normal modes 
and 
. This means we need to invert
the defining relations (2.12). First of all, the overall normalization
of 
and 
must be chosen so that their commutator is
unity, 
.
From the definitions, we must require
where 
was used. We assume in what follows
that the 
and 
coefficients are now
rescaled to give the unit normalization and unit commutator of 
with
in Eq. (3.1). Then, the following inverse expressions
are assumed,
To determine the new coefficients 
and 
in terms of the 
and 
coefficients,
one can form the commutator of 
or 
[Eqs. (2.12)] with
and with 
, giving
On the other hand, forming the commutator of 
and 
with 
and 
[Eq. (3.2)]
leads to equivalent results,
Thus there is the conversion between the coefficients;
As an application of these results, we can determine the
local magnetization for a site by finding an expectation value of the
original lab frame spin components, to quadratic order in the creation
and annihilation operators. In order to do this, we first need
expectation values of the spin components in the tilde coordinate system.
From their definitions, 
, because these are linear in 
and
. However, the 
component will be reduced slightly
below S due to spin fluctuations of the modes. In order to preserve the
overall spin length and the commutation relations of 
and 
with 
, it is necessary to
use the following expression for 
(as in the standard
Holstein-Primakoff [26] transformation):
where the latter terms are the spin lowering and raising operators. Using Eq. (3.2) and Eq. (3.5), the expectation value of this expression is
where terms linear in the 
and 
operators and terms
like 
and 
are zero in the
unperturbed (single-vortex) state and in single-quantum states and therefore
do not appear here.
The expectation values of the 
and 
operators
will be determined by the type of state, whether it be a state with one
mode excited or a thermodynamic ensemble of states (equilibrium state).
For example, if the state that we are perturbing from
(single vortex) is denoted |0>, then single-quantum
excited states are denoted, |k>,
The fundamental expectation values are 
,
and 
. On the other hand, if the
interest is in a thermal ensemble, then the expectation value required
will be the Bose-Einstein occupation, 
, where 
is the inverse
temperature. It is clear that the expectation values of the
components will be less than S as a result of fluctuations,
while the expectation values of 
and 
will be zero. As a result, it is straightforward to use Eq. (3.7) in the
coordinate transformation Eq. (2.6) to obtain the expectation values in the
original lab coordinates, i.e., the spin is just reduced in effective length,
We also want to know the spin fluctuations associated with some state. The spin fluctuations will be defined in terms of squares of Cartesian spin components, relative to the vortex state. For instance, the in-plane and out-of-plane spin fluctuations are described by
Using the definitions of the tilde coordinates, Eq. (2.6), these are equivalent to
Making use of the expansion of spin components in the operators
and 
, Eq. (3.2), together with Eq. (3.5),
one can write the fluctuations in the tilde coordinates,
Finally, the resulting in-plane and out-of-plane fluctuations are:
