A discussion of Landau damping for the collective rigid dipole oscillations
in presence of two-dimensional betatron tune spread can be found in Ref. [5].
The problem of the stability is expressed by the dispersion integral
where 
is the coherent horizontal tune shift induced
by the machine impedance in the absence of tune spread and Q the coherent horizontal tune.
A similar dispersion relation holds for vertical betatron
oscillations.
The amplitude-dependent incoherent horizontal tune in presence of
octupoles (see Eq. (8)) is written
For a Gaussian distribution 
in the
two transverse planes, the double integral has infinite
tails either in one or in both tune directions (depending on the relative sign of the
detuning coefficients) and it is impossible to obtain quantitative results concerning
the loss of Landau damping for a given real coherent tune shift caused by the broad-band
impedance. Since the case of a truncated Gaussian distribution gives rise to complicated
pathologies of the stability limit [5], we assume that the transverse beam
distribution function in 2D is `quasi-parabolic' with the same r.m.s. normalised
betatron spread 
in both planes
This distribution is normalised to unity and its projection on the horizontal betatron
plane is
The normalised physical beam profile in x, given by
extends over 
and is shown in Fig. 1:
it is a bell-shaped curve smoothly going to zero, together with its first and second
derivative, at about 
.
Since our distribution is zero beyond the boundary
, it is convenient to
perform the integral in Eq. (10) by parts.
Then, normalising the action variables by 
,
introducing the `full horizontal tune spread'
and measuring the tune shifts in units of S, with
the dispersion relation (10) can be written
where the complex beam transfer function 
is given by
For a given ratio c=b/a between the octupole cross and direct term,
the dispersion relation Eq. (16)
can be viewed as a mapping from the complex q-plane to the complex
-plane.
When a<0, the spread S used to normalise the tune shifts is positive and
the stability limit is then obtained by plotting
versus 
for values of q with vanishing, positive imaginary part.
Reversing the sign of the detuning coefficients a and b, the stability limit corresponds
to complex conjugate values of q and thus of ; it is the mirror symmetric
curve in the -plane obtained by reflection about the real axis.
The integral in the beam transfer function can be done analytically and we obtain
The beam transfer functions and the corresponding
stability limits for a<0 (solid curves) are shown in
Figs. 2-I, 2-II and 2-III for c=0, 1 and -1, respectively.
The dashed and dotted lines in these plots correspond to instability growth rates
of 5% and 10%, respectively, of the full horizontal tune spread S, i.e., to
and 
.
Figure 2: Beam transfer function (a) and stability limit (b) for
relative cross-anharmonicities c=0, c=1, c=-1.
The transfer function 
is proportional to the coherent
beam response observed in the limit of vanishing intensity while exciting
the beam at frequency q by a shaker: the zero-intensity beam tune corresponds
to the maximum of 
and, for finite beam intensity,
Landau damping is guaranteed provided the real coherent tune shift
induced by the broad-band impedance
is roughly within the incoherent tune spread around this peak.
It is important to realize that, as a consequence of our normalisation by the
full horizontal tune spread S,
the coherent tune shifts induced by the machine impedance
in the complex -plane have an opposite sign for a>0.
For example, an inductive impedance gives rise to negative real coherent tune shifts
.
If the detuning coefficient a is chosen negative, so as to produce
an amplitude detuning with the same sign as ,
the corresponding normalised tune shift
is as well negative. In the one-dimensional case,
this situation provides the largest Landau damping (up to 
)
as can be seen in Fig. 2-Ib. The analysis of the two-dimensional problem shows
that this situation can be reversed (Fig. 2-II) and that the
stability threshold can be increased by a factor of two
(Fig. 2-III) with a proper sign of the relative detuning
coefficient c.
