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Bose-condensed atomic systems with nonlocal interaction potentials

The general approach for describing systems with Bose-Einstein condensate, where atoms interact through nonlocal pair potentials, is presented. A special attention is paid to nonintegrable potentials, such as the dipolar interaction potential. The potentials that are not absolutely integrable can have not well defined Fourier transforms. Using formally such not defined Fourier transforms leads to unphysical conclusions. For making the Fourier transform well defined, the interaction potential has to be regularized. This is illustrated by the example of dipolar interactions.


Introduction
Atomic systems exhibiting Bose-Einstein condensation are widely studied both theoretically and experimentally, as can be inferred from the books [1][2][3] and review articles [4][5][6][7][8][9][10][11][12][13]. Trapped dilute atomic gases are often characterized by local interaction potentials of the delta-function type. But recently, trapped atoms interacting through dipolar forces have been condensed (see reviews [14][15][16]). Dipolar interactions are nonlocal and long-range and their theoretical description is more complicated than that of locally interacting atoms, because of which one usually considers only the Bogolubov approximation that is valid at temperatures close to zero and asymptotically weak interactions.
Moreover, the dipolar interaction potential does not possesses a well defined Fourier transform. Formally taking this transform and using it leads to unphysical consequences.
The aim of the present paper is to suggest a self-consistent approach for atomic systems with nonlocal interaction potentials, valid for finite temperatures and for interactions of any strength. A special attention is paid to nonintegrable potentials, whose Fourier transforms are not well defined. As a typical example of this kind, dipolar interactions are treated. We show that such interactions need to be regularized in order to get a correct description of atomic systems. Such a regularization is necessary for atoms of any statistics, whether Bose or Fermi. For concreteness, we consider here atomic systems of Bose-Einstein statistics.
Throughout the paper, we use the notation where the Planck and Boltzmann constants are set to unity, = 1, k B = 1.

Self-consistent approach
A self-consistent approach for describing Bose-condensed systems has been developed in Refs. [17][18][19]. This approach resolves the Hohenberg-Martin dilemma [20] of conserving versus gapless theories and provides a theory that is conserving as well as gapless.
The general form of the energy Hamiltonian iŝ where the field operators satisfy the Bose commutation relations, U = U(r) is an external potential, if any, and Φ(r) is a nonlocal interaction potential. The field operators depend on time t, which, for simplicity, is not explicitly shown. For the occurrence of Bose-Einstein condensation, the necessary and sufficient condition is the global gauge symmetry breaking [1,12,13,21]. A convenient way of breaking the gauge symmetry is by employing the Bogolubov shift [22,23] representing the field operators aŝ ψ(r) = η(r) + ψ 1 (r) , where η(r) is the condensate function and ψ 1 (r) is the operator of uncondensed atoms. Note that the Bogolubov shift is an exact canonical transformation, but not an approximation, as one sometimes writes.
To avoid double counting, the condensate function and the field operator of uncondensed atoms are assumed to be orthogonal, η * (r)ψ 1 (r) dr = 0 .
The condensate function plays the role of the functional order parameter, such that η(r) = ψ (r) .
The condensate function is normalized to the number of condensed atoms While the number of uncondensed atoms is the statistical average of the operatorN The evolution equation for the condensate function can be written as i ∂ ∂t η(r, t) = δH δη * (r, t) .
And the equation of motion for the operator of uncondensed atoms reads as i ∂ ∂t ψ 1 (r, t) = δH δψ † 1 (r, t) .
The grand Hamiltonian generating these equations is where the first term is the energy operator (1), the Lagrange multipliers µ 0 and µ 1 guarantee the validity of the normalization conditions (6) and (7), while the last term, having the form preserves the validity of condition (5). It has been proved [13,24] that the variational equation (9) is equivalent to the Heisenberg equation of motion.
In systems with broken gauge symmetry, in addition to the single-particle density matrix there exists the anomalous matrix The condensate function defines the density of condensed atoms The diagonal elements of the single-particle density matrix give the density of uncondensed atoms And the diagonal elements of the anomalous matrix define the density of pair-correlated atoms. The total atomic density is the sum yielding the total number of atoms The partial atomic ratios give the fractions of condensed, n 0 , and uncondensed, n 1 atoms, respectively, For an equilibrium system, the statistical operator has the form with β = 1/T being the inverse temperature, and with the same grand Hamiltonian (10). The superfluid fraction is given by the expression where d is the real-space dimensionality, the operator of momentum iŝ and the variance of an arbitrary operatorÂ is defined as For a self-adjoint operator, this yields In equilibrium, the average momentum is zero, which, for dimensionality d = 3, leads to

Uniform systems
When there is no external potential, U = 0, or when the trap is sufficiently large, the system can be treated as uniform. Then it is possible to Fourier transform the field operators, Usually, one assumes that the interaction potential also enjoys the Fourier transformation If the Fourier transform is well defined, there exists the limit Note that if the trapping potential U is not zero, but sufficiently smooth, it is possible to resort to the local-density approximation, for which the similar Fourier transforms are also assumed.
The single-particle density matrix becomes with the momentum distribution n k = a + k a k .
And the anomalous average is The condensate function and atomic densities do not depend in the spatial variable. The total average density reads as with the condensate density ρ 0 = |η| 2 (29) and the density of uncondensed atoms The diagonal anomalous average becomes In the Hartree-Fock-Bogolubov approximation, we obtain [12,13] the momentum distribution and the anomalous average (27) reads as Here we use the notations and defining the spectrum of collective excitations In equilibrium, which is equivalent to the variational condition From the latter, we get the condensate chemical potential The condition of the condensate existence [12,13] lim requires a gapless spectrum, such that in agreement with the Bogolubov [22,23] and Hugenholtz-Pines [25] theorems. Condition (41) yields the chemical potential of uncondensed atoms Then expression (34) takes the form In that way, for the spectrum of collective excitations (36) we find In order to simplify the formulas, we notice that expressions (32) and (33) strongly increase as k → 0. Therefore the main contribution in the summations, containing n p and σ p , comes from the region of small p. This suggests the possibility of using the approximations p =0 Then the chemical potentials (39) and (42) become and, respectively, Equation (43) reduces to while expression (35) becomes The collective spectrum (44) takes the form

Thermodynamic characteristics
Having spectrum (50), it is possible to calculate the grand potential where With approximation (45), we have Using here the fractions of condensed and uncondensed atoms and the notation we get The average of the grand Hamiltonian can be represented as which defines the system chemical potential For the latter, we find which, in approximation (45), reads as With the given grand potential, it is straightforward to calculate any required thermodynamic characteristics. It is necessary to notice that all thermodynamic characteristics depend on the quantity Φ 0 that, hence, has to be well defined. Even in the simplest Bogolubov approximation, when so that it is admissible to neglect ρ 1 and σ 1 , as compared to the condensate density ρ 0 → ρ, Anyway, thermodynamic characteristics do depend on Φ 0 . Thus the limit Φ 0 has to be well defined. It must be a scalar in order that thermodynamic characteristics be scalar quantities. This implies that the Fourier transform of the interaction potential has to be correctly defined, giving an unambiguous limit (23).

Fourier transform
It is useful to remember the conditions when Fourier transforms can be correctly defined, since not each function enjoys a well defined Fourier transform. For this purpose, let us remind some known mathematical facts. For simplicity, we recall these facts for the case of one variable. The generalization to several variables is straightforward.
, is of bounded variation in a finite interval if in that interval it has only a finite number of extrema and a finite number of finite discontinuities.
Dirichlet theorem. If a function f (x) is of bounded variation in any finite interval and is absolutely integrable, such that exists and the inverse Fourier transform gives The details can be found in Ref. [26].
For the case of several variables, the sufficient condition for the existence of Fourier transform of a potential Φ(r) is its absolute integrability: We also may notice that the correct definition of the limit (23) requires the validity of interchanging the limit and integration.
Lebesgue theorem. A sufficient condition for the interchange of a limiting procedure and integration is the absolute integrability of the considered function.
For the case of the interaction potential, the limit (23) exists, so that the change of the orders lim is valid, provided that the interaction potential is absolutely integrable, according to condition (61). A good illustration of nonlocal potentials is the potential of dipole-dipole interactions. Dipoles can be electric, with the dipolar moment d i , or magnetic, with the magnetic moment µ i , where the index i enumerates particles. We shall consider dipoles with the moment d i , keeping in mind that the same problems concern dipoles with the moment µ i . Dipolar interactions are widespread for different kinds of condensed matter [27], for polymers [28], biological systems [29,30], and for many magnetic nanoclusters and nanomolecules [31,32].
Two dipoles at the distance r from each other interact through the dipolar potential where One often considers polarized dipoles, all directed along the same unit vector e d , so that Then the dipolar potential reduces to where ϑ is the angle between the direction of a dipole and the spatial vector n = r/r, If the dipoles are directed along the axis z, then cos ϑ = z/r. One formally defines the Fourier transform Then in the general case (62), one gets with k = |k|. And for the case of polarized dipoles, one has with ϑ k being the angle between the dipole direction and the vector k, If the dipole direction is along the axis z, then ϑ k = k z /k. However, there are problems with the dipolar interaction potential. Thus, it is easy to see that the integral is not well defined. For an infinite system, it has to be understood as an improper integral But if one integrates, first, over spherical angles, one gets zero. While, if one integrates, first, over the radius, one gets infinity. Hence contrary to condition (23). Moreover, the limit k → 0 for D k is not defined at all, as is seen from the above forms of D k . Such a limit explicitly depends on the angle (68). Usually, one alleges that this anisotropy is appropriate for a system with dipolar interactions. However, this is absolutely wrong. The quantity D 0 , similar to Φ 0 , enters many thermodynamic characteristics, as is explained in Sec. 4. The thermodynamic quantities, such as chemical potential, energy, grand potential, by their meaning are scalars and principally cannot be anisotropic. Since the limit D 0 is not defined, then all thermodynamic characteristics are not defined, which has no sense.
This problem arises because the dipolar potential (62) or (63) is not absolutely integrable. Really, since This integral tends to infinity for any of the limits, either R → ∞, or b → 0. Therefore the Fourier transform for an absolutely nonintegrable function may not exist, which is the case for the dipolar potential whose Fourier transform is defined neither for k → 0 nor for k → ∞. The problem arises not due to the anisotropy of the dipolar potential, but due to its nonintegrability, which makes the formally introduced Fourier transform senseless.

Cutoff regularization
Actually, the difficulties with the dipolar interaction potential are well known in physics of condensed matter [27] and have been discussed in many publications. Thus the divergence of the dipolar potential at short distance, resulting in infinite molecular polarizability, has been called the polarization catastrophe [33,34]. The way out of this catastroph is physically transparent, requiring to consider dipolar particles not as point-like objects, but as finitesize particles described by spatial distributions. This way leads to the smearing of dipolar interactions at short distances, which can be characterized by different smearing functions [35][36][37][38][39][40]. The simplest is the short-range cutoff regularization removing the 1/r 3 singularity. This implies the use of the short-range regularized potential where Θ(r) is a unit-step function and b is the sum of two atomic radii. In the case of identical atoms, b is the effective atomic diameter. For the Fourier transform Considering the polarized potential (63), directing the axis z along the wave vector k, integrating out the angle ϕ, and using the notation Integrating over x gives with where y = kr. Taking the last integral results in Integral (73) has the property The presence of the short-range cutoff makes it possible to define the short-wave limit, that is, the large-k limit, Recall that without the short-range cutoff regularization the large-k limit is not defined, as is clear from equation (67). Respectively, in the limiting expression (74), it is impossible to set b to zero. However, the long-wave limit, when k → 0, is not defined, since and we return to the problem discussed in the previous section. This is connected with the fact that potential (71) is not absolutely integrable, being divergent at large r.

Screening regularization
When one considers separate molecules or clusters, such finite small systems require only short-range regularization. But for large systems, a long-range regularization, taking into account long-range correlations, is also necessary. The necessity of regularizing the dipolar potential by long-range screening was emphasized by Jonscher [41][42][43]. Such a screening is usually described by an exponential function [39,[41][42][43][44][45].
The dipolar potential that is regularized both for short-range as well as for long-range interactions can be written in the form where κ is a screening wave vector, hence 1/κ is a screening radius. This potential is absolutely integrable, so that it enjoys a well defined Fourier transform For the case of polarized dipoles, we have with the integral in which y = kr. With the change of the variable x = r/b, giving y = kbx, we get Removing the screening returns us back to the case of the previous section, with a not well defined Fourier transform. It is clear that the limits b → 0 and κ → 0 do not commute with the limit k → 0, It is possible to notice that integral (79) depends, actually, on two variables Therefore integral (79) can be represented as with the integral The latter integral can be expressed through the exponential integral function in which the integral is defined in the sense of the principal value, with a branch cut along the negative real axis [46]. Then we find To show that this expression is real valued, we can employ the series representation for the exponential integral function where γ = 0.57721 is the Euler-Mascheroni constant. Using the relation Function (84) possesses the following properties. When q → 0, under a finite c, we have This defines the long-wave limit of the Fourier transform (77) in the form Note that this limit is principally dependent on a finite value of the screening parameter κ that cannot be set to zero. Thus in the long-wave limit, we get that agrees with the integral and which is in agreement with condition (23). In the short-wave limit, when k → ∞, and c being finite, we get which gives for the Fourier transform Here the finiteness of the short-range cutoff b is important.
In the limit of a small screening parameter, under finite q, we find where the coefficient functions are , a 2 (q) = 3 2q 3 (3 sin q − q cos q) , and the notation for the sine integral is used. When q → 0, we get If we try to find here the long-wave limit, we get the Fourier transform This expression is divergent at k → 0, demonstrating that the expansion in powers of the screening parameter (c → 0 is not defined. The function J q (c) is not analytical at c = 0, hence, the Fourier transform D k (b, κ) is not analytical at κ = 0. For a very large screening parameter, we have with d 1 (q) = 3 q 3 3 − q 2 sin q − 3q cos q , d 2 (q) = 3 q 3 12 − q 2 q cos q + 5q 2 − 12 sin q .
Here, the long-wave limit of the Fourier transform is defined, yielding In this way, in order that the dipolar interaction potential would enjoy a well-defined Fourier transform, it is necessary to regularize this potential both, for short-range as well as for long-range interactions. The behavior of the function J q (c), defining the Fourier transform of the dipolar potential, is illustrated in Fig. 1.
The regularized dipolar potential is an effective potential taking into account short-range and long-range particle correlations. One may ask whether the use of an effective potential, instead of the bare interaction potential, is admissible. The answer is yes. The study of any many-particle system can be started with a self-consistent mean-field approximation, containing an effective potential, which is called the correlated mean-field approximation [47]. The higher approximations, beyond the correlated mean-field approximation, can be obtained by means of an iterative procedure for Green functions [48][49][50].

Excitation spectrum
After the interaction potential is properly regularized, so that it enjoys a well defined Fourier transform, it is possible to study the properties of the system. It is necessary to emphasize that without the regularization the formal investigation of the system properties in a meanfield approximation is not correct and would lead to wrong conclusions. It is admissible to deal with a singular bare interaction potential only in higher-order approximations, taking into account particle correlations smearing the singularities in the bare potential. However, in a simple mean-field approximation, the use of a bare potential having no well defined Fourier transform is inadmissible. But a correlated mean-field approximation involving an effective regularized potential is justified.
Let us consider the excitation spectrum of a Bose-condensed system having the interaction potential consisting of two terms, Φ(r) = 4π a s m δ(r) + D(r, b, κ) .
The first term describes local atomic interactions, with a s being the scattering length. And the second term is the regularized dipolar interaction potential. Strictly speaking, the scattering length depends on the strength of the dipolar interactions, but in a wide range of the scattering-length values, it can be varied independently of the dipolar part [14][15][16]. The Fourier transform of potential (96) is Keeping in mind that lim contains only the local interaction part. For arbitrary k, the Fourier transform (97) can be represented as where The dipolar term can be written as For the spectrum of collective excitations (50), we have In the long-wave limit, the spectrum is of phonon type, The sound velocity here is obtained by taking account of the properties of the function so that Thus for the sound velocity, we find The sound velocity is anisotropic because of D k . Although it is necessary to stress that the anisotropy appears only if ρ 1 and σ 1 are not zero, but are defined by the expressions in which the functions n k and σ k are defined in equations (32) and (33). The condensate density ρ 0 = ρ − ρ 1 is expressed through the density of uncondensed atoms ρ 1 . The functions n k and σ k are connected with each other, It is important to emphasize that the anomalous average σ 1 is of order or even larger than ρ 1 . Because of this, they are to be taken into account together or both omitted. But neglecting the anomalous average σ 1 , while keeping the normal density ρ 1 , is principally wrong [12,13,51].

Bogolubov approximation
The Bogolubov approximation is applicable for temperature close to zero and asymptotically weak interactions, such that both ρ 1 and σ 1 are much smaller than the condensate density ρ 0 → ρ. Neglecting ρ 1 and σ 1 in spectrum (102) yields the Bogolubov spectrum In the long-wave limit, the spectrum is of phonon type, with the sound velocity As is evident, in the Bogolubov approximation, the sound velocity is isotropic. Anisotropy arises only in the higher-order approximation, as is seen in the sound velocity (104). But at finite k, spectrum (106) is anisotropic. We can consider two opposite cases, the so-called parallel geometry, with k parallel to the dipole direction, and the perpendicular geometry, when It is convenient to introduce the dimensionless spectrum Also, we define the correlation length Then spectrum (106) leads to the expression Let us denote the spectrum for the parallel geometry as and for the perpendicular geometry, as The quantity a D ≡ md 2 0 (118) is called the dipolar length. And the ratio characterizes the relative strength of the dipolar interactions with respect to the local interactions. With these notations, for spectrum (116) we get And spectrum (117) becomes Assuming that the screening parameter is inversely proportional to the correlation length, we have Then for spectra (120) and (121), we obtain ε (q) = q 1 + 2αJ q (c) + q 2 4c 2 (123) and, respectively, One also considers the relative difference between the parallel and perpendicular geometries, defined by the quantity As examples of atoms with magnetic dipoles [14][15][16], it is possible to mention 52 Cr, with the magnetic dipole µ 0 = 6µ B and the dipolar length a D = 2.4 × 10 −7 cm, 168 Er, with µ 0 = 7µ B and a D = 1.05 × 10 −6 cm, and 164 Dy, with µ 0 = 10µ B and a D = 2.09 × 10 −6 cm. Some molecules can have the dipolar lengths as large as a D ∼ 10 −4 cm.
The typical behavior of the excitation spectra (123) and (124) are shown in Figs. 2, 3, and 4. Increasing the dipolar interactions leads to the appearance of a roton minimum and then to the roton instability. The relative difference (125) is shown in Figs. 5 and 6. The minimal relative interaction strength α min corresponds to the appearance of the roton minimum, which is defined as the occurrence of a zero derivative of the spectrum, with respect to the wave vector. The maximal α max is the relative dipolar strength at which the roton minimum touches zero, thus displaying the roton instability.
The origin of the roton minimum in systems with dipolar interactions is rather clear, being caused by the attractive part of these anisotropic interactions, which results in the appearance of the negative term in the spectrum (124). The possibility of the roton minimum in the spectrum of trapped Bose gases with dipolar forces has been theoretically considered for quasi-one-dimensional [52][53][54][55] and quasi-two-dimensional [56][57][58] cases. For sufficiently strong dipolar forces, Bose gas can become unstable [59][60][61], which also depends on the trap shape. This instability is due to the roton minimum touching zero [62].
The most intensive experimental studies have been done for 52 Cr, whose Bose-Einstein condensation has been experimentally observed [63,64]. The scattering length for these atoms is a s = 105a B = 0.555 × 10 −6 cm. With the peak density ρ = 3 × 10 14 cm −3 , the correlation length becomes ξ c = 2.186 × 10 −5 cm. For the dipolar length a D = 2.4 × 10 −7 cm, the relative dipolar parameter α = 0.144. If the short-range cutoff b is of order of the scattering length a s , then the screening parameter c is of order 0.025.
By employing Feshbach resonance, the scattering length of 52 Cr can be varied in a wide range, producing pure dipolar trapped gas and reaching its collapse [65,66].
The collective spectrum of dipolar chromium gas was measured by Bismut et al. [67]. Because of the small relative dipolar strength α = 0.144, the spectrum does not show roton minimum. Qualitatively, the measured spectrum [67] as well as the relative difference ∆(q) are in agreement with our results for small α.

Local-density approximation
The theory of the previous sections, developed for uniform systems, can be straightforwardly generalized to trapped atoms, when the trapping potential is sufficiently smooth, such that the local-density approximation be valid. Then the off-diagonal parts of correlation functions are treated as fast, while their diagonal parts, as slow. As a result, the spatial dependence, induced by the trapping potential, enters only through the atomic densities and density distributions that are treated as slow functions of the spatial variables [13].
In the presence of an external trapping potential U(r), using the local-density approximation, we have the following representation for the single-particle density matrix and for the anomalous average where V is a quantization volume, needed only at the intermediate stage, and where Here, we use the notations and defining the local spectrum of collective excitations The density of uncondensed atoms and the anomalous average become The total atomic density is The condensate function, defined by condition (8), satisfies the equation In the Bogolubov approximation, when almost all atoms are condensed, so that the quantities related to uncondensed atoms and the anomalous average can be neglected, the condensatefunction equation (134) simplifies to the nonlinear Schrödinger equation The condition of the condensate existence (41), which now reads as yields Substituting this into equation (129) gives Then for the spectrum of collective excitations (131), we have the equation Resorting to approximation (45), we get, instead of (137), instead of (138), and for expression (130), we have The equation for the spectrum (139) becomes Keeping in mind dipolar interactions, we use the Fourier transform (99), which gives the spectrum equation In the Bogolubov approximation, the spectrum equation reduces to the form In the long-wave limit, the spectrum is phononic, with the isotropic sound velocity A special case is when the scattering length can be varied by means of the Feshbach resonance [65,66] and even made zero. In an isotropic harmonic trap, the Bose condensate of 52 Cr becomes unstable and collapses at a s ≈ 15a B = 0.794 × 10 −7 cm. This corresponds to α ≈ 1.
When the scattering length is zero, a s = 0, hence Φ 0 = 0, the spectrum in the Bogolubov approximation reads as In the long-wave limit, using the behaviour of f k from Sec. 8, we find Although the roton instability here occurs when the inequality is valid, but such a quadratic spectrum does not satisfy the Landau criterion for superfluidity, which means that the condensate is not stable at all. This is in agreement with the fact that in a uniform system, Bose condensate with purely dipolar forces is not stable. This is also true for a large isotropic trap. But if the trap is of pancake shape, the condensate can be stabilized [65,66]. However, for a strongly anisotropic trap, the local-density approximation can be not appropriate.
Recall that at nonzero temperature and finite interactions, the condensate density does not coincide with the superfluid density, although superfluidity and condensation arise together at the Bose condensation point. In the self-consistent mean-field approach [12,13,[17][18][19], the superfluid density writes as with the dissipated heat Employing the equality where β = 1/T , we find

Conclusion
The main message of this paper is that it is necessary to be cautious dealing with nonlocal long-range interactions. Such long-range interactions can exist in different finite quantum systems [68]. Before considering Fourier transforms of such long-range interactions, it is necessary to check whether the considered interaction potential is absolutely integrable. The absolute integrability is a sufficient condition for the existence of the corresponding Fourier transform. If the interaction potential is not absolutely integrable, it may have no well defined Fourier transform. Then, formally calculating a Fourier transform of a nonintegrable potential, one can get an incorrect expression leading to senseless unphysical results. For instance, one can come to a conclusion that thermodynamic characteristics, such as chemical potentials and energies, are not scalars, which certainly has no meaning. The consideration is specified by the example of a Bose-condensed system with nonlocal interactions. The self-consistent mean-field approach is used [12,13,[17][18][19]. This approach enjoys the unique properties, as compared to all other mean-field approximations: It is the sole mean-field theory guaranteeing the correct second order of the Bose-Einstein condensation transition [69]. Also, it is the sole mean-field approach yielding the values of the condensate fraction in very good agreement with Monte Carlo calculations for arbitrary interaction strength [70].
As a particular case, atoms with dipolar interactions are treated. The bare dipolar interaction potential, for a three-dimensional system, is not absolutely integrable, and does not have a correctly defined Fourier transform. A formally calculated Fourier transform is not well defined and leads to wrong conclusions. To get a correctly defined Fourier transform of the dipolar potential, it is necessary to regularize it, making it absolutely integrable. The correctly defined Fourier transform of the regularized potential yields the results essentially differing from the ill-defined transform of the bare potential. Thus, all thermodynamic characteristics are scalars, as they should be. The spectrum of collective excitations is anisotropic. But the sound velocity in the Bogolubov approximation is isotropic. The sound velocity becomes anisotropic only in higher approximations, e.g., in the Hartree-Fock-Bogolubov approximation taking into account the necessary self-consistency conditions [12,13,[17][18][19].
The spectrum of collective excitations and the relative spectrum difference for the parallel and perpendicular geometries, at a small relative dipolar strength, are in qualitative agreement with those experimentally observed [67] for the atoms of 52 Cr. However a detailed comparison with particular experiments has not been the aim of the present paper. This requires a separate publication. The goal of this paper has been to suggest a general approach for describing systems with nonlocal long-range interactions and to explain the necessity of regularizing the related interaction potentials before taking their Fourier transforms.               Figure 6: Relative difference between the collective spectra in the parallel and perpendicular geometries for the screening parameter c = 1 and different relative dipolar interaction strengths.