aa r X i v : . [ qu a n t - ph ] A p r Complex modes in unstable quadratic bosonic forms
R. Rossignoli, A.M. Kowalski
Departamento de F´ısica, Universidad Nacional de La Plata, C.C.67, La Plata (1900), Argentina
We discuss the necessity of using non-standard boson operators for diagonalizing quadratic bosonicforms which are not positive definite and its convenience for describing the temporal evolutionof the system. Such operators correspond to non-hermitian coordinates and momenta and areassociated with complex frequencies. As application, we examine a bosonic version of a BCS-likepairing Hamiltonian, which, in contrast with the fermionic case, is stable just for limited values ofthe gap parameter and requires the use of the present extended treatment for a general diagonalrepresentation. The dynamical stability of such forms and the occurrence of non-diagonalizablecases are also discussed.
Quadratic bosonic forms arise naturally in many areasof physics at different levels of approximation. Startingfrom the basic example of coupled harmonic oscillators,their ubiquity is testified by their appearance in stan-dard treatments of quantum optics [1], disordered sys-tems [2], Bose-Einstein condensates [3–6] and other in-teracting many-body boson and fermion systems [7, 8].In the latter they constitute the core of the random-phase approximation (RPA), which arises as a first or-der treatment in a bosonized description of the systemexcitations, or alternatively, from the linearization of thetime-dependent mean field equations of motion (time de-pendent Hartree, Hartree-Fock (HF) or HF-Bogoliubov(HFB) [7, 8]). The ensuing forms are quite general andmay contain all types of mixing terms ( q i p j , q i q j and p i p j ) when expressed in terms of coordinates and mo-menta. Although the standard situation, i.e., that wherethe RPA is constructed upon a stable mean field (theHartree, HF or HFB vacuum), corresponds to a posi-tive form, in more general treatments the RPA can alsobe made on top of unstable mean fields, as occurs inthe study of instabilities in binary Bose-Einstein con-densates [3–6], and even around non-stationary runningmean fields, as in the case of the static path + RPAtreatment of the partition function [9, 10], derived fromits path integral representation. In these cases the ensu-ing forms may not be positive and may lead, as is wellknown, to complex frequencies. Quadratic bosonic formsare also relevant in the study of dynamical systems [11–13], providing a basic framework for investigating diverseaspects such as integrals of motion and semiclassical lim-its.Now, a basic problem with such forms is that whilein the fermionic case they can always be diagonalized bymeans of a standard Bogoliubov transformation [7], inthe bosonic case they may not admit a similar diagonalrepresentation in terms of standard boson operators, norin terms of usual hermitian coordinates and momenta.These cases can of course only arise in unstable formswhich are not positive definite. The aim of this workis to discuss the diagonal representation of such formsin terms of non-standard boson-like quasiparticle opera-tors (or equivalently, non-hermitian coordinates and mo-menta), associated with complex normal modes. This re-quires the use of generalized Bogoliubov transformations since the usual one leads to a vanishing norm in the caseof complex frequencies. The present treatment allowsthen to identify the operators characterized by an expo-nentially increasing or decreasing evolution, providing aprecise description of the dynamics and of the quadraticinvariants in the presence of instabilities. It will alsobecome apparent that an analysis of the dynamical sta-bility based just on the Hamiltonian positivity may notbe sufficient.As application, we will discuss a bosonic version of aBCS-type pairing Hamiltonian, which, in contrast withthe fermionic case, exhibits a complex behavior, loosingits positive definite character above a certain thresholdvalue of the gap parameter, and becoming dynamicallyunstable above a second higher threshold. In the presenceof a perturbation it may even lead to a reentry of dynam-ical stability after an initial breakdown. This exampleillustrates the existence of simple quadratic forms whichcannot be written in diagonal form in terms of standardboson operators or coordinates and momenta. Moreover,it also shows the existence of non-diagonalizable caseswhich do not correspond to a zero frequency (and henceto a free particle term, in contrast with standard Gold-stone or zero frequency RPA modes arising from meanfields with broken symmetries [7]), and which are char-acterized by equations of motions which cannot be fullydecoupled.A general hermitian quadratic form in boson annihi-lation and creation operators b i , b † i , can be written as H = X i,j A ij ( b † i b j + δ ij ) + ( B ij b † i b † j + B ∗ ij b i b j ) (1a)= Z † H Z , H = (cid:18) A BB ∗ A t (cid:19) , Z = (cid:18) bb † (cid:19) , (1b)where A is an hermitian matrix, B is symmetric and Z † = ( b † , b ), with b , b † arrays of components b i , b † i . Theextended matrix H is hermitian and satisfies in addition¯ H ≡ T H t T = H , T = (cid:18) (cid:19) . (2)The boson commutation relations [ b i , b j ] = [ b † i , b † j ] = 0,[ b i , b † j ] = δ ij , can be succinctly expressed as Z Z † − ( Z † t Z t ) t = M , M = (cid:18) − (cid:19) . (3)It is well known that if the matrix H possesses onlystrictly positive eigenvalues, the quadratic form (1) canbe diagonalized by means of a standard linear Bogoliubovtransformation for bosons preserving Eqs. (3) [7]. Thisis the standard situation where (1) represents a stablesystem with a discrete positive spectrum, such as a sys-tem of coupled harmonic oscillators. In general, however,and in contrast with the fermionic case, it is not alwayspossible to represent Eq. (1) as a diagonal form in stan-dard boson operators. The physical reason is obvious.If H is not strictly positive, Eq. (1) may represent theHamiltonian of systems like a free particle or a particlein a repulsive quadratic potential ( H ∝ p − q ) when ex-pressed in terms of coordinates and momenta, which donot possess a discrete spectrum. Nonetheless, one maystill attempt to write (1) as a convenient diagonal formin suitable operators, such that the ensuing equations ofmotion become decoupled and trivial to solve.Let us consider for this aim a general linear transfor-mation [7, 8] Z = W Z ′ , Z ′ = (cid:18) b ′ ¯ b ′ (cid:19) , (4)where ¯ b ′ i is not necessarily the adjoint of b ′ i , although b ′ i ,¯ b ′ j are still assumed to satisfy the same boson commuta-tion relations, i.e., Z ′ ¯ Z ′ − ( ¯ Z ′ t Z ′ t ) t = M , where ¯ Z ′ ≡ (¯ b ′ , b ′ ) = Z ′ t T . Since Z † = ¯ Z ′ ¯ W , with ¯ W ≡ T W t T , thematrix W should then fulfill WM ¯ W = M , (5)implying W − = M ¯ WM . No conjugation is involved in(5). Note that ¯ Z ≡ Z t T = Z † while in general ¯ Z ′ = Z ′† = ¯ Z ′ ¯ W ( W † ) − . If ¯ b ′ = b ′† then ¯ W = W † (andviceversa) and Eq. (4) reduces to a standard Bogoliubovtransformation for bosons [7, 8]. Eq. (4) allows to rewrite H as H = ¯ Z ′ H ′ Z ′ , H ′ = ¯ WHW = (cid:18) A ′ B ′ ¯ B ′ A ′ t (cid:19) , (6)where the relation (2) is preserved ( ¯ H ′ = H ′ , implying B ′ , ¯ B ′ symmetric), although in general H ′† = H ′ . Find-ing a representation where H ′ is diagonal implies thenan eigenvalue equation with “metric” M , i.e., HW = MWMH ′ , which can be recast as a standard eigenvalueequation for a non-hermitian matrix ˜ H :˜ H W = W ˜ H ′ , ˜ H ≡ M H = (cid:18) A B − B ∗ − A t (cid:19) . (7)This matrix is precisely that which determines the tem-poral evolution of the system when H is the Hamiltonian,as the Heisenberg equation of motion for b , b † is i dZdt = − [ H, Z ] = ˜ H Z . (8) Its solution for a time independent ˜ H is therefore Z ( t ) = U ( t ) Z (0) , U ( t ) = exp[ − i ˜ H t ] , (9)(or in general U ( t ) = T exp[ − i R t ˜ H ( t ′ ) dt ′ ], where T de-notes time ordering). The eigenvalues of ˜ H characterizethen the temporal evolution and can be complex in un-stable systems. Nevertheless, since ˜ H † = HM = M ˜ HM and (Eq. (2)) T ˜ H t T = −M ˜ HM , (10)it is easily verified that the commutation relations (3)are always preserved ∀ t ∈ ℜ , as ¯ U ( t ) ≡ T U t T = U † ( t )and U ( t ) M ¯ U ( t ) = M . Moreover, the last identity re-mains valid also for complex times (although in this case¯ U ( t ) = U † ( t )), so that Eq. (9) is a particular exampleof the general transformation (4), becoming a standardBogoliubov transformation for bosons for t ∈ ℜ .Eq. (10) implies that Det[ ˜ H t − λ ] = Det[ ˜ H + λ ], sothat the eigenvalues of ˜ H (the same as those of ˜ H t ) al-ways come in pairs ( λ i , λ ¯ i ) of opposite sign ( λ ¯ i = − λ i ).Eq. (10) also entails that the corresponding eigenvec-tors W i (columns of W ) satisfy the orthogonality rela-tions ¯ W j M W i = − ¯ W i M W j = 0 if λ i = − λ j , with¯ W i ≡ W ti T , which are those required by Eq. (5) (therequired norm is ¯ W ¯ i M W i = 1). In addition, for H her-mitian, Det[ ˜ H − λ ] ∗ = Det[ ˜ H † − λ ∗ ] = Det[ ˜ H − λ ∗ ], sothat if λ is an eigenvalue, so is λ ∗ . Combined with (10)this implies that if W i is eigenvector with eigenvalue λ i , W ¯ i ∗ ≡ T W ∗ i is eigenvector with eigenvalue − λ ∗ i . For λ i real , the required norm can then be reduced to the usualone for bosons [7], W † i M W i = 1. However, for λ i com-plex , the usual norm vanishes ( W † i M W i = ¯ W ¯ i ∗ M W i = 0as λ i = − λ ¯ i ∗ = λ ∗ i ) while the present one does not in gen-eral. Note finally that the eigenvalues of ˜ H are the sameas those of ˜ H s ≡ √HM√H . When those of H are all non-negative , √H and hence ˜ H s are hermitian , so thatall eigenvalues of ˜ H are real .Let us assume now that the matrix ˜ H is diagonaliz-able , such that a non-singular matrix W of eigenvectorsexists. Then ¯ WMW will be non-singular, and due to theorthogonality relations can be set equal to M if eigen-vectors are ordered and chosen such that ¯ W ¯ j M W i = δ ij .The ensuing W will then satisfy Eqs. (5) and (7) with ˜ H ′ diagonal. Through the relation H ′ = M ˜ H ′ and Eq. (6)we obtain finally the diagonal representation H = X i λ i (¯ b ′ i b ′ i + ) , (11)where b ′ i = ¯ W ¯ i M Z , ¯ b ′ i = Z † M W i , with W i , W ¯ i theeigenvectors with eigenvalues λ i and − λ i satisfying thepresent norm ( ¯ W ¯ i M W i = 1). If λ i is real, we may choose W ¯ i = T W ∗ i such that ¯ W ¯ i = W † i (with W † i M W i = 1) andhence ¯ b ′ i = b ′ i † . Nonetheless, for complex λ i , ¯ b ′ i = b ′ i † .Eq. (11) remains, however, physically meaningful, as theeigenvalues λ i determine the temporal evolution. We im-mediately obtain from (11) and (9) the decoupled evolu-tion b ′ i ( t ) = e − iλ i t b ′ i (0) , ¯ b ′ i ( t ) = e iλ i t ¯ b ′ i (0) , (12)in all cases, together with the quadratic invariants ¯ b ′ i b ′ i = Z † M W i ¯ W ¯ i M Z . If all eigenvalues λ i are real and pos-itive (with ¯ b ′ i = b ′ i † ), we have the standard case of apositive definite quadratic form. If all λ i are real butsome of them are negative (with ¯ b ′ i = b ′ i † ), the system isunstable in the sense that H is no longer positive anddoes not possess a minimum energy, but the spectrum isstill discrete and the temporal evolution (9) remains sta-ble. Finally, when some of the λ i are complex, the evo-lution becomes unbounded, with b ′ i ( t ) ( ¯ b ′ i ( t )) increasing(decreasing) exponentially for Im( λ i ) > t . In these cases the sign of λ i in (12) depends on thechoice of operators and can be changed with the trans-formation b ′ i → − ¯ b ′ i , ¯ b ′ i → b ′ i (which preserves the com-mutation relations) such that ¯ b ′ i b ′ i + → − (¯ b ′ i b ′ i + ) (for λ i real the sign can be fixed by the additional condition¯ b ′ i = b ′ i † ). Cases where ˜ H is not diagonalizable (whichmay arise when its eigenvalues are not all different) arealso dynamically unbounded as the temporal evolutiondetermined by Eq. (9) will contain terms proportional tosome power of t (times some exponential; see example).We may also express (1) in terms of hermitian coordi-nates q = ( b + b † ) / √ p = ( b − b † ) / ( √ i ),satisfying [ p i , p j ] = [ q i , q j ] = 0, [ q i , p j ] = iδ ij , as H = X i,j T ij p i p j + V ij q i q j + U ij q i p j + U tij p i q j (13a)= R t H c R, H c = (cid:18) V UU t T (cid:19) , R = (cid:18) qp (cid:19) , (13b)where V, T = Re( A ± B ) and U = Im( B − A ), with T, V and H c symmetric . The corresponding transformation is Z = S R, H c = S † HS , (14)where S = √ ( i − i ) is unitary and satisfies S † = S t T .The commutation relation for R reads RR t − ( RR t ) t = M c , M c = S † MS = (cid:18) i − i (cid:19) , (15)and the transformation (4) becomes R = W c R ′ , W c M c W tc = M c , (16)where W c = S † WS and R ′ = ( q ′ p ′ ) satisfies Eq. (15). Notethat q ′ , p ′ will not be hermitian if W c is complex. Stan-dard linear canonical transformations among hermitiancoordinates and momenta correspond to W c real , whichis equivalent to the condition ¯ W = W † in (5).We may now rewrite (13) as H = R ′ t H ′ c R ′ , where H ′ c = W tc H c W c is symmetric although not necessarily real. Finding a representation with H ′ c diagonal impliesthen the non-standard eigenvalue problem˜ H c W c = W c ˜ H ′ c , ˜ H c = M c H c = i (cid:18) U t T − V − U (cid:19) , (17)with U ′ = 0 and V ′ , T ′ diagonal in ˜ H ′ c = M c H ′ c ,which leads to the coupled equations ˜ H c W ci = − iV ′ i W c ¯ i ,˜ H c W c ¯ i = iT ′ i W ci , for the columns of W c . The requirednorm (Eq. (16)) is again ¯ W c ¯ i M W ci = 1. The matrix ˜ H c determines the evolution of q, p , as idR/dt = ˜ H c R , andits eigenvalues are of course the same as those of ˜ H , as˜ H c = S † ˜ HS . If a matrix W c (real or complex) satisfying(16)–(17) exists, we obtain the diagonal form H = X i ( T ′ i p ′ i + V ′ i q ′ i ) , T ′ i V ′ i = λ i , (18)where p ′ i = − ¯ W ci M R , q ′ i = ¯ W c ¯ i M R and λ i are theeigenvalues of ˜ H or ˜ H c . For λ i = 0 we may always set T ′ i = V ′ i = λ i by a scaling p ′ i → s i p ′ i , q ′ i → q ′ i /s i , where s i = p V ′ i /T ′ i can be complex, in which case we maychoose W ci = S † ( W i + W ¯ i ) / √ W c ¯ i = i S † ( W i − W ¯ i ) / √ W i , W ¯ i the eigenvectors of ˜ H with eigenvalues ± λ i satisfying ¯ W ¯ i MW i = 1, such that p ′ i + q ′ i = 2¯ b ′ i b ′ i + 1.The ensuing operators p ′ i , q ′ i will not be hermitian when λ i is complex, but their evolution will still be given by theusual expressions q ′ i ( t ) = q ′ i (0) cos( λ i t ) + p ′ i (0) sin( λ i t ), p ′ i ( t ) = p ′ i (0) cos( λ i t ) − q ′ i (0) sin( λ i t ).When ˜ H is diagonalizable , Eq. (18) is obviously equiv-alent to (11) (with Z ′ = S R ′ for T ′ = V ′ ). However, Eq.(18) is more general since it may also contain free par-ticle terms T ′ i p ′ i when λ i = 0, which cannot be writ-ten in the form (11). In these cases the matrix ˜ H isnot diagonalizable , as easily recognized from the ensuinglinear evolution p ′ i ( t ) = p ′ (0), q ′ i ( t ) = q ′ i (0) + tT ′ i p ′ i (0),having a degenerate eigenvalue 0. Nonetheless, it shouldbe emphasized that it is not always possible to representEq. (13) in the diagonal form (18), as non-diagonalizablecases where no eigenvalue of ˜ H vanishes, also exist (seeexample). Let us also remark that if one considers just hermitian q ′ i , p ′ i in (18), with T ′ i , V ′ i real , the eigenvalues λ i of ˜ H are either real ( T ′ i V ′ i ≥
0) or purely imaginary( T ′ i V ′ i < H pos-sesses full complex eigenvalues (see example) cannot bewritten in the diagonal form (18) unless non-hermitiancoordinates and momenta q ′ , p ′ are admitted .The following example clearly illustrates the previoussituations. Let us consider the Hamiltonian H = X ν = ± ε ν ( b † ν b ν + ) + ∆( b + b − + b † + b †− ) (19a)= X ν = ± ε ν ( p ν + q ν ) + ∆( q + q − − p + p − ) , (19b)which represents two boson modes interacting through aBCS-like pairing term. We assume ε + > ε − >
0, andwrite ε ± = ε ± γ , with ε >
0, 0 < γ < ε . The eigenvaluesof the ensuing matrix H (or H c ), two-fold degenerate, are σ ± = ε ± p γ + ∆ , (20)which are both positive only for | ∆ | < p ε − γ = √ ε + ε − (the condition for a positive mass and poten-tial tensor in (19b)). However, the four eigenvalues of˜ H = MH are λ ± ν = ± [ νγ + p ε − ∆ ] , ν = ± , (21)which are real for | ∆ | ≤ ε = ( ε + + ε − ) /
2. Thus, if p ε − γ < | ∆ | < ε , H is no longer positive defi-nite ( σ − < λ ± ν remain real (anddistinct) implying that the temporal evolution is still bounded (quasiperiodic). However, for | ∆ | > ε , all eigen-values are complex (with non-zero real part if γ = 0) andthe evolution becomes unbounded.Let us obtain now the diagonal representation of H . Itis sufficient to consider in (5) a BCS-like transformationfor bosons of the form b ν = ub ′ ν − v ¯ b ′− ν , b † ν = u ¯ b ′ ν − vb ′− ν , (22)which corresponds to q ν = uq ′ ν − vq ′− ν , p ν = up ′ ν + vp ′− ν .The commutation relations are preserved if u − v = 1( WM ¯ W = M ) and the inverse transformation ( M ¯ WM )is obtained for v → − v ( b ′ ν = ub ν + vb †− ν , ¯ b ′ ν = ub † ν + vb − ν ). Now, for (cid:18) uv (cid:19) = r ε ± α α , α = p ε − ∆ , (23)where we assume α = 0 ( | ∆ | 6 = ε ) and signs in squareroots are to be chosen such that 2 αuv = ∆, we mayexpress H as a sum of two independent modes, H = X ν = ± λ ν (¯ b ′ ν b ′ ν + ) = X ν = ± λ ν ( p ′ ν + q ′ ν ) , (24)where λ ν ≡ λ + ν . If | ∆ | < ε , u, v are both real, so that¯ b ′ ν = b ′† ν , with q ′ ν , p ′ ν , hermitian, while if | ∆ | > ε , u, v are complex , implying ¯ b ′ i = b ′† i and q ′ i , p ′ i no longer hermitian.Instead, ( λ ν ) ∗ = − λ − ν and u ∗ = iv (with Im( α ) > > b ′ ν † = ib ′− ν , ¯ b ′ ν † = i ¯ b ′− ν and q ′ ν † = iq ′− ν , p ′† ν = − ip ′− ν . Note that in this case the usual normvanishes ( | u | − | v | = 0) but the present one remainsunchanged ( u − v = 1 still holds).If | ∆ | < p ε − γ , λ ± >
0, so that both modes havea discrete positive spectrum. However, if p ε − γ < | ∆ | < ε , λ + > λ − <
0, so that the spectrum ofthe lowest mode, though still discrete, becomes negative ,implying that H has no longer a minimum energy. Care-ful should be taken here to select the correct eigenvaluein (21), as ˜ H still has two positive eigenvalues ( λ −− > | ∆ | = p ε − γ , λ ±− = 0, reflectingthe onset of the instability, but ˜ H is still diagonalizable , as u, v remain finite. The lowest mode in (24) has here asingle degenerate eigenvalue 0. Finally, for | ∆ | > ε , theoperators b ′ ν , ¯ b ′ ν represent complex modes with an expo-nentially increasing or decreasing evolution. The evolu-tion of the original operators b ν , b † ν for any | ∆ | 6 = ε canbe immediately obtained from (12) and (22) and is givenby b ν ( t ) = e − iλ ν t [ b ν + v (1 − e iαt )( vb ν + ub †− ν )] , (25)where b ν ≡ b ν (0), b † ν ≡ b † ν (0), with b † ν ( t ) = [ b ν ( t )] † . Itbecomes clearly unbounded for | ∆ | > ε .For | ∆ | = ε , ˜ H is not diagonalizable , even though itseigenvalues λ ± ν are in this case all real and non-zero (butdegenerate), and H cannot be written in the form (24).However, the time evolution can still be obtained from(25) taking the limit α →
0, which leads to b ν ( t ) = e − iνγt [(1 − itε ) b ν − it ∆ b †− ν ] . (26)The factor t confirms that the evolution equations can-not be fully decoupled in this case, while the exponentialmultiplying this factor shows that they do not arise froma free particle term either. We may, however, rewrite H in this case (assuming for instance ∆ = ε ) as H = γ (¯ b s + b s + − ¯ b s − b s − ) + 2∆¯ b s − ¯ b s + , (27)where b ν = ( b sν + ¯ b s − ν ) / √ b † ν = (¯ b sν − b s − ν ) / √
2, with b sν † = − b s − ν , ¯ b sν † = ¯ b s − ν , also satisfy boson commutationrelations. In the form (27) H is “maximally decoupled”,in the sense that the evolution equations for ¯ b sν are fully decoupled , while those of b sν are coupled just to ¯ b s − ν . Thisleads to ¯ b sν ( t ) = e iνγt ¯ b sν , b sν ( t ) = e − iνγt ( b sν − it ∆¯ b s − ν ).Eq. (26) can also be obtained from these expressions. Theassociated invariants in this case are ¯ b s − ¯ b s + and ¯ b s + b s + − ¯ b s − b s − , i.e., the two terms in (27), which are mutuallycommuting.If b ν , b † ν were fermion operators, Eq. (19a) would rep-resent essentially a generic term of the standard BCSapproximation to a pairing Hamiltonian [7] [ H BCS = P k,ν ε kν b † kν b kν + P k ∆ k ( b k + b k − + b † k − b † k + ), where k ± denote time reversed states, ∆ k the BCS gap, b kν , b † kν fermion operators and the splitting between ε k ± may rep-resent the effect of a Zeeman coupling to a magnetic field].In the fermionic case, Eq. (19a) (with → − ) can bewritten as P ν λ ν ( b ′† ν b ′ ν − ) ∀ ∆, where λ ν = νγ + α ,with α = √ ε + ∆ , are the quasiparticle energies and b ′ ν , b ′† ν quasiparticle fermion operators defined by b ν = ub ′ ν + νvb ′†− ν , with u, v = p ( α ± ε ) / α . The analogousboson problem is, in contrast, stable just for limited val-ues of ∆, as the latter decreases (rather than increases)the “quasiparticle energies” λ ν . The onset of complexfrequencies occurs finally when λ − = − λ + .Let us also mention that in general, when H is notpositive regions of dynamical stability may also arise be-tween fully unstable regions. For instance, if a pertur-bation κ ( b † + b − + b †− b + ) is added to (19), the eigenval-ues of H and ˜ H become σ ± ν = ε + ν p γ + (∆ ± κ ) and λ ± ν = ± q ˜ λ ν − κ ( ε /γ − λ ν = νγ + p ∆ c − ∆ and ∆ c = ε (1 + κ /γ ). Those of H are split, and as-suming κ small such that H is positive at ∆ = 0, thetwo lowest ones σ ±− become negative at different values∆ c ± = p ε − γ ± | κ | . In such a case λ ±− becomes imag-inary for ∆ c − < | ∆ | < ∆ c + , but returns again to realvalues for ∆ c + < | ∆ | < ∆ c if | κ | < γ / p ε − γ , ex-hibiting a reentry of dynamical stability. Finally, both λ ± become full complex for | ∆ | > ∆ c . A diagonal rep-resentation of the general form (24) is feasible except atthe critical values ∆ c ± and ∆ c .In summary, we have extended the standard method-ology employed for diagonalizing an hermitian quadraticbosonic form, employing generalized quasiparticle boson-like operators for describing unstable cases with arbitrarycomplex frequencies. In this way the operators exhibitingan exponentially increasing or decreasing temporal evolu-tion are explicitly identified, together with the associated quadratic invariants, allowing for a precise characteriza-tion of the system evolution in the presence of generalinstabilities. While positive definite forms can be con-sidered completely stable, those which are not positivebut whose matrix ˜ H is diagonalizable and has only real eigenvalues, can still be considered dynamically stable,as the temporal evolution remains quasiperiodic, in con-trast with the case where ˜ H has complex eigenvalues oris non-diagonalizable. Finally, we have seen that a BCS-like hamiltonian for bosons can be completely stable, justdynamically stable, or unstable depending on the valuesof the gap parameter, and requires the present general-ized approach for a diagonal representation valid for largegaps. Moreover, it also shows that cases where ˜ H is non-diagonalizable are not necessarily associated with zerofrequencies or free particle terms, and may arise even ifall its eigenvalues are non-zero. For such cases the evo-lution equations cannot be fully decoupled.RR and AMK are supported by CIC of Argentina. [1] P. Meystre, M. Sargent, Elements of Quantum Optics(Springer, NY, 1991).[2] V. Gurarie, J.T. Chalker, Phys. Rev. Lett. 89, 136801(2002).[3] E.V. Goldstein and P . Meystre, Phys. Rev. A , 2935(1997).[4] C.K. Law, H. Pu, N.P. Bigelow, J.H. Eberly, Phys. Rev.Lett. , 3105 (1997).[5] H. Pu and N.P. Bigelow, Phys. Rev. Lett. , 1134(1998).[6] S. Alexandrov, V.V. Kavanov, J.Phys. Condens. Matter14, L327 (2002); V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 1, 50 (2004); V.I. Yukalov, Laser Phys. Lett.1, 435 (2004).[7] P. Ring and P. Schuck, The Nuclear Many-Body Problem ,(Springer, NY, 1980).[8] J.P. Blaizot and G. Ripka,