The MHD α2-dynamo, ZZ2-graded pseudo-Hermiticity,level crossings and exceptional points of branching type ∗)

Uwe Gunther, Frank Stefani, Gunter Gerbeth

Research Center Rossendorf, Department of Magnetohydrodynamics,P.O. Box 510119, D-01314 Dresden, Germany

Received 30 July 2004

The spectral branching behavior of the 2 × 2 operator matrix of the magneto-hydro-dynamic α2-dynamo is analyzed numerically. Some qualitative aspects of level crossingsare briefly discussed with the help of a simple toy model which is based on a Z2-graded-pseudo-Hermitian 2× 2 matrix. The considered issues comprise: the underlying SU(1, 1)symmetry and the Krein space structure of the system, exceptional points of branchingtype and diabolic points, as well as the algebraic and geometric multiplicity of correspond-ing degenerate eigenvalues.

PACS : 91.25.Cw, 02.30.Tb, 02.40.Xx, 11.30.Er, 11.30.–jKey words: MHD dynamo, non-Hermitian operators, discrete symmetries, Krein space,

level crossings, branching points, singularities

1 The 2 × 2-operator matrix of the MHD α2-dynamo

The magnetic fields of planets, stars and galaxies are maintained by homoge-neous dynamo effects, which can be successfully described within magnetohydrody-namics (MHD) [1]. This is achieved by appropriately combining the Maxwell equa-tions of electrodynamics with the Navier–Stokes equations of hydrodynamics — forcertain topologically non-trivial, helical flow fields. The resulting highly complicatedequation systems are subject of large and cost-intensive computer simulations (see,e.g., Ref. [2]). Recently, the homogeneous dynamo effect has been demonstrated inlarge scale liquid sodium experiments in Riga/Latvia [3] and Karlsruhe/Germany[4]. Next generation experiments are currently planned at 7 sites around the world.

One of the simplest dynamo toy models, which can be regarded as importantfor MHD dynamo theory as the harmonic oscillator for quantum mechanics, is thespherically symmetric α2-dynamo [1] in its kinematic regime. Its operator matrixhas the form [5]

Hl[α] =(−Q[1] αQ[α] −Q[1]

)(1)

and consists of formally selfadjoint blocks

Q[α] := pαp + αl(l + 1)

r2. (2)

(p = −i(∂r + 1/r) is the radial momentum operator). It describes the coupledl-modes of the poloidal and toroidal magnetic field components in a mean-field

∗) Presented at the 2nd International Workshop “Pseudo-Hermitian Hamiltonians in QuantumPhysics”, Prague, Czech Republic, June 14-16, 2004

Czechoslovak Journal of Physics, Vol. 54 (2004), No. 10 1075

U. Gunther, F. Stefani, and G. Gerbeth

dynamo model with helical turbulence function (α-profile) α(r). The differentialexpression (1) has the symmetry property [5]

Hl[α] = JH†l [α]J, J =

(0 II 0

)(3)

and shows that — depending on its domain D(Hl[α]) and the chosen boundaryconditions for the two-component eigenfunctions ψ — the corresponding operatorcan be pseudo-Hermitian (J-selfadjoint). Modulo J-selfadjoint extensions, this willhold for functions ψ with idealized boundary conditions at r = 1:

D(Hl[α]) : =ψ =

(ψ1

ψ2

): ψ ∈ H ≡ H ⊕H, H = L2(Ω, r2dr),

Ω = [0, 1], ψ(1) = 0, rψ(r)|r→0 → 0, (4)

whereas the physically realistic boundary conditions

Blψ∣∣∣r=1

= 0, Bl = diag[∂r + (l + 1)/r, 1] (5)

lead to a non-J-Hermitian operator.The dynamo effect starts when the first eigenvalue λ of Hl[α] enters the right

half-plane (λ) > 0. This leads to an exponential growth ∼ eλt of the solutions ofthe time dependent kinematic dynamo problem — and a corresponding growth ofthe magnetic field components. In Ref. [6] it was demonstrated numerically thatthis can happen even on a complex-valued branch of the spectrum, so that thedynamo can start in an oscillating regime.

In Figs. 1–5 some typical spectral branches are shown for an α2-dynamo operatorwith physically realistic boundary conditions (5). The real and imaginary compo-nents of the eigenvalues for modes with angular mode number l = 1 and radialmode numbers n = 1, . . . , 9 are depicted over a scaling parameter C of an α(r)-profile, which is chosen as quartic polynomial α(r) = C× [a0 +a2r

2 +a3r3 +a4r

4].Of special physical interest are the critical value Cc, where the spectrum entersthe right half-plane (λ) > 0, as well as the location of those level crossing points,where two real-valued branches (non-oscillating regime) of the spectrum meet andcontinue to evolve as two complex conjugate branches (oscillating dynamo regime).Within the considered range of C such transitions occur locally only pairwise —there are always only two spectral branches which are locally involved in such tran-sitions. As shown in Fig. 3, globally, there are more branches involved in mutualtransitions. A detailed and rigorous non-numerical study is still missing. Below, wecollect some few building blocks for such an analysis.

Some first and rough qualitative aspects of the behavior of the dynamo system ata level crossing point can be easily understood, e.g., by passing from the eigenvalueproblem for the linear pencil

Ll[α, λ]ψ :=(Hl[α] − λ

)ψ = 0 (6)

1076 Czech. J. Phys. 54 (2004)

The MHD α2-dynamo . . .

-200

-100

0

100

200

300

400

500

0 10 20 30 40 50 60 70 80 90

Re(

λ)

C

n=1n=2n=3n=4n=5n=6n=7n=8n=9

-60

-40

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90

Im(λ

)

C

n=1n=2n=3n=4n=5n=6n=7n=8n=9

Fig. 1. Large-scale behavior of real and imaginary parts (λ), (λ) of the α2-dynamospectrum (for physically realistic boundary conditions (5)) as function of the scaling pa-rameter C in the concrete α-profile α(r) = C×

1− 26.09 × r2 + 53.64 × r3 − 28.22 × r4

.

Czech. J. Phys. 54 (2004) 1077

U. Gunther, F. Stefani, and G. Gerbeth

-60

-40

-20

0

20

40

60

10 15 20 25 30 35

Re(

λ)

C

n=1n=2n=3n=4n=5n=6

-20

-15

-10

-5

0

5

10

15

20

10 15 20 25 30 35

Im(λ

)

C

n=1n=2n=3n=4n=5n=6

Fig. 2. Zooming into a scaling region with multiple level-crossings and branches which“mutually influence” each other.

1078 Czech. J. Phys. 54 (2004)

The MHD α2-dynamo . . .

-50

0

50

100

55 60 65 70 75

Re(

λ)

C

n=5n=6n=7n=8n=9

-60

-40

-20

0

20

40

60

55 60 65 70 75

Im(λ

)

C

n=5n=6n=7n=8n=9

Fig. 3. Zooming into another scaling region with multiple level crossings. Interestingly,although locally only two branches undergo a transition from pairwise real-valued eigen-values to pairwise complex-conjugate ones, such transitions globally occur also for morebranches. In the depicted scaling region three branches participate in mutual transitions.This is a natural indication of the underlying Riemann surface structure of the operator

spectrum (see, e.g., [7, 8]).

Czech. J. Phys. 54 (2004) 1079

U. Gunther, F. Stefani, and G. Gerbeth

100

120

140

160

180

200

52 53 54 55 56 57 58

Re(

λ)

C

n=2n=3n=4

-10

-5

0

5

10

52 53 54 55 56 57 58

Im(λ

)

C

n=2n=3n=4

Fig. 4. The real and imaginary parts of crossing spectral branches without transition fromreal to complex eigenvalues “feel” each other.

1080 Czech. J. Phys. 54 (2004)

The MHD α2-dynamo . . .

420

430

440

450

460

470

480

490

500

84 84.5 85 85.5 86 86.5 87 87.5 88

Re(

λ)

C

n=3n=4n=7

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

84 84.5 85 85.5 86 86.5 87 87.5 88

Im(λ

)

C

n=3n=4n=7

Fig. 5. The long-term parallel location (repelling) of different branches without real-to-complex transition is not a numerical artefact. The branches can be faithfully identified

as different ones.

Czech. J. Phys. 54 (2004) 1081

U. Gunther, F. Stefani, and G. Gerbeth

of the 2 × 2-operator matrix via substitution

ψ =

(ψ1

1α

[Q(1) + λ]ψ1

), α(r) = 0 , (7)

to the equivalent eigenvalue problem of the associated quadratic operator pencil

Ll[α, λ]ψ1 ≡

[Q[1] + λ]1α

[Q[1] + λ] −Q[α]ψ1 = 0

= (A2λ2 + A1λ + A0)ψ1 = 0. (8)

Solving the quadratic (functional) equation

Ml[α, λ] := (Ll[α, λ]ψ1, ψ1) = a2λ2 + a1λ + a0 = 0, aj := (Ajψ1, ψ1) , (9)

for λ we see from the solutions

λ± =1

2a2

(−a1 ±

√a21 − 4a0a2

)(10)

that a level crossing occurs when the discriminant ∆ = a21 − 4a0a2 of Eq. (9)

vanishes. At the level crossing points it holds

Ml[α, λ] = 0, ∂λMl[α, λ] = 0 (11)

and the operator pencil has a Jordan–Keldysh chain [9–11] ψ1, χ1, φ1, consist-ing of the eigenfunction ψ1 and the associated functions χ1, φ1 which satisfy therelations

Ll[α, λ0]ψ1 = 0, (12)Ll[α, λ0]χ1 + ∂λLl[α, λ]|λ0

ψ1 = 0, (13)

Ll[α, λ0]φ1 + ∂λLl[α, λ]|λ0χ1 + 1

2 ∂2λLl[α, λ]

∣∣λ0

ψ1 = 0. (14)

Due to the lack of rigorous analytical and non-numerical operator theoretic studiesof the α2-dynamo, we will collect in the next section some few qualitative factsabout a highly simplified 2× 2 matrix model with some rough structural analogiesto the operator matrix (1).

2 ZZ2-graded pseudo-Hermiticity

In this section1) we start from the simple model of an α2-dynamo with idealizedboundary conditions (4) so that the corresponding operator Hl[α] is J-selfadjoint(J-pseudo-Hermitian) [see relation (3)]. We will try to get a rough intuitive insight

1) The few issues presented in this and the next section are part of a detailed study given in:U. Gunther, Z2-graded pseudo-Hermitian systems: exceptional points, pseudo-unitary fibrationsand nontrivial holonomy, in preparation.

1082 Czech. J. Phys. 54 (2004)

The MHD α2-dynamo . . .

into some of the basic properties of such a system in the vicinity of a level crossingpoint.2)

General pseudo-Hermitian operators H were defined in Refs. [14] as operatorswhich satisfy a relation

H = ηH†η−1. (15)

Because of the involution property of J , i.e. J2 = I, J = J−1, the operator Hl[α]is part of a narrow subclass of operators with H = µH†µ, µ2 = I, µ−1 = µ. To thesame subclass belong the PT -symmetric Hamiltonians of Refs. [15–18], which are P-pseudo-Hermitian and for which necessarily holds P2 = I, as well as the SchrodingerHamiltonian associated to a Wheeler–DeWitt equation of minisuperspace quantumcosmology considered in Ref. [14]. A general feature of systems with involutivepseudo-Hermiticity operators µ is an underlying Z2-graded structure of their Hilbertspace: Every state (vector) of the Hilbert space can be naturally split into (projectedonto) µ-even and µ-odd components (for PT -symmetric Hamiltonians into parity-even and parity-odd states, see, e.g., [14, 19])

x = P+x + P−x = x+ + x−, P± := 12 (I ± µ) , (16)

withµx± = ±x±. (17)

Explicitly, this leads to a natural Z2-grading of the Hilbert space H → H+⊕H− anda Krein space structure3) [21]: Beside the usual inner product (., .) of the Hilbertspace H, which induces a non-negative norm (x, x) = (x+, x+) + (x−, x−) ≥ 0 forx± ∈ H±, one can consider a Krein space (Kµ, [., .]µ) with indefinite inner product[x, y]µ = (µx, y) = (x+, y+) − (x−, y−) (and a corresponding indefinite “norm”).One naturally distinguishes states (vectors) of positive type [x, x]µ > 0, of negativetype [x, x]µ < 0, and isotropic states [x, x]µ = 0. (In rough analogy to Minkowskispace this corresponds to time-like, space-like, and light-like vectors.) Because ofµH = H†µ a µ-selfadjoint (µ-pseudo-Hermitian) operator is selfadjoint in the Kreinspace Kµ:

[Hx, y]µ = (µHx, y) = (x,H†µy) = (x, µHy) = [x,Hy]µ . (18)

A natural representation of a Z2-graded system can be given in terms of 2-componentvectors and 2 × 2-operator matrices

x =(x+

x−

), H =

(H++ H+−H−+ H−−

),

µ =(I 00 −I

), P+ =

(I 00 0

), P− =

(0 00 I

),

(19)

2) After finishing this proceedings contribution we became aware that some aspects of levelcrossings in Z2-graded pseudo-Hermitian systems had been briefly discussed earlier — with thehelp of different techniques — in Refs. [12, 13]. The present analysis overlaps only marginally withthose results and provides a different view on the subject.

3) For a detailed introduction into the operator theory over Krein spaces see, e.g., Refs. [11, 20].

Czech. J. Phys. 54 (2004) 1083

U. Gunther, F. Stefani, and G. Gerbeth

where H++ = P+HP+, etc. The µ-pseudo-Hermiticity implies

H++ = H†++, H−− = H†

−−, H+− = −H†−+. (20)

The operator Hl[α] of the α2-dynamo with idealized boundary conditions can betransformed into this representation by first diagonalizing the involution operatorJ

J → µ = S−1JS, S =1√2

(I −II I

). (21)

Applying then the same transformation S to Hl[α] and the elements of the Hilbertspace H one obtains the equivalent operator [5]

Hl[α] = S−1Hl[α]S =12

(Q[α− 2] + α −Q[α] + αQ[α] − α Q[−α− 2] − α

)(22)

which acts on 2-vectors

ψ =(ψ+

ψ−

)=

1√2

(ψ2 + ψ1

ψ2 − ψ1

). (23)

We leave a detailed analysis of this highly non-trivial operator to future studies.Instead we try to get a rough qualitative understanding of the level-crossing in

a Z2-graded system. For this purpose, we consider its simplest example4) — theeigenvalue crossing of a 2 × 2 matrix with the basic symmetry properties (20):

H =(

a b−b∗ d

)=(e0 00 e0

)+ h, h :=

(f b

−b∗ −f

),

e0 = 12 (a + d), f = 1

2 (a− d), b = b1 + ib2, a, d, b1, b2 ∈ R.(24)

For the eigenvalues holds

det (H −EI) = 0 =⇒ E = e0 ±√

f2 − b21 − b22 (25)

and we see that a level crossing occurs at

∆(f, b1, b2) := f2 − b21 − b22 = 0. (26)

∆(f, b1, b2) = 0 defines a two-dimensional variety (a double-cone) in the threedimensional parameter space M (f, b1, b2) so that for f = 0 = |b| the degeneracyhas co-dimension one. This is in obvious contrast to a Hermitian 2×2 (spin) matrixproblem with

hs =(

f bb∗ −f

)(27)

4) This gives a level crossing with transition from a pair of real-valued eigenvalues to a pair ofcomplex conjugate eigenvalues. Apart from such level crossing with real-to-complex transitions,there occur level crossings without such transitions. They are present in all Figs. 1–5 and can beeasily understood in a matrix setup as crossings of spectral branches which belong to differentZ2-blocks.

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The MHD α2-dynamo . . .

and eigenvalues

det (Hs −EsI) = 0 =⇒ Es = e0 ±√

f2 + b21 + b22. (28)

Here a degeneracy ∆s(f, b1, b2) := f2+b21+b22 = 0 occurs only in the single (diabolic)point f = b1 = b2 = 0 and the crossing has co-dimension three [22, 23].

Another difference between the two models is the underlying symmetry of “iso-energetic” (adiabatic) deformations. Obviously, the eigenvalues E and Es in (25)and (28) are invariant, respectively, under SO(1, 2) and SO(3) transformations inthe parameters f, b1, b2 [24]. This is in natural correspondence with the generatorsof “iso-energetic” (adiabatic) transformations which are defined by the matricesH and Hs themselves. Modulo the Abelean U(1) transformations induced by the

elements ie0

(1 00 1

)these generators are Lie algebra elements of the type

ih = −b2σ1 − b1σ2 + ifσ3 ∈ su(1, 1) ∼ so(1, 2) ∼ sl(2,R), (29)ihs = i (b1σ1 + b2σ2 + b3σ3) ∈ su(2) ∼ so(3). (30)

(σi are the Pauli matrices.) Of course, this interlinks5) also nicely with the invari-ance transformations of the two-dimensional Krein space6) (Kµ, [., .]µ), µ = σ3 andthe 2-vector Hilbert space Hs of the spin system. The inner product [., .]µ is in-variant under pseudo-unitary U(1, 1) ∼ U(1) × SU(1, 1) transformations, whereasthe inner product (., .) in Hs is invariant under unitary U(2) ∼ U(1) × SU(2)transformations.

Further insight into the structure of the simple pseudo-Hermitian 2 × 2 matrixeigenvalue problem can be gained by diagonalizing the matrix H. Before we do thisexplicitly, we note that any 2 × 2 involution matrix η:

η2 = I, η = η−1 = η† (31)

belongs to one of the following classes η+ or η−

det(η+) = 1, η+ ∈ I,−I, (32)det(η−) = −1, η− = a1σ1 + a2σ2 + a3σ3, a2

1 + a22 + a2

3 = 1. (33)

This means that two involution matrices η1−, η2− are connected by an SU(2) ∼Z2 × SO(3) rotation.

5) Because of its underlying pseudo-unitary symmetry pseudo-Hermitian (PT -symmetric) quan-tum mechanics is not a special variant of quaternionic quantum mechanics [25]. One sees thisimmediately from the simplest models with H and Hs as Hamiltonians. Quaternionic structuresare naturally connected with SU(2) symmetries [24, 25] and Hs, whereas the pseudo-Hermitianmodel with Hamiltonian H has invariants which are related to SU(1, 1) symmetry transforma-tions. Additionally, we note that SU(2) is a compact group whereas SU(1, 1) is a non-compactone.

6) Because of dim(H±) = 1 < ∞ for the considered 2 × 2 matrix model, this Krein space is aso-called Pontryagin space [11].

Czech. J. Phys. 54 (2004) 1085

U. Gunther, F. Stefani, and G. Gerbeth

For non-degenerate eigenvalues E the diagonalization is done as

H = SDS−1, D =(e0 − ε 0

0 e0 + ε

), S =

( −f + ε

b∗γ1

−f − ε

b∗γ2

γ1 γ2

),

ε := ∆1/2, γ1,2 ∈ C.(34)

The complex constants γ1,2 are still arbitrary and we can fix them by requiringthat the diagonalization preserves the pseudo-Hermitian structure, i.e., that thereexists an involution matrix η so that

D = ηD†η. (35)

From the subtitution chain

H = µH†µ =⇒ SDS−1 = µ(S−1

)†D†S†µ

=⇒ D = S−1µ(S−1

)†D†S†µS (36)

and relation (35) one identifies

η = S†µS = S−1µ(S−1

)†= µ−1 = µ† (37)

so thatµ = SS†µSS†, η = S†SηS†S. (38)

From the latter equations one concludes that |det(S)|2 = 1. Explicit calculations7)show that one can set det(S) = 1 and that S is in general neither unitary, S† = S−1,nor pseudo-unitary. Complementary information can be gained from the explicitstructure of the matrix D. It holds

∆ > 0 : =⇒ ε = ∆1/2 > 0, D =(e0 − ε 0

0 e0 + ε

)= D† , (39)

∆ < 0 : =⇒ ε = i|ε|, D =(e0 − i|ε| 0

0 e0 + i|ε|

),

D† =(e0 + i|ε| 0

0 e0 − i|ε|

) (40)

and one obtains from (35) and the explicit form of S in (34) (by appropriatelytuning the constants γ1,2) that

∆ > 0 : f > 0 : η =(−1 00 1

), (41)

f < 0 : η =(

1 00 −1

), (42)

∆ < 0 : η =(

0 11 0

). (43)

7) See footnote 1.

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The MHD α2-dynamo . . .

We see that smooth changes of the starting µ-pseudo-Hermitian matrix H lead toqualitative “switchings” (discontinuities in the mapping µ → η which correspond torotations in the space of 2×2 involution matrices η− — see Eq. (33)) in the pseudo-Hermitian structure of the diagonal matrix D. These “switchings” occur when thesystem intersects critical surfaces in the parameter space: (1) the surface f = 0and (2) the degeneration double cone ∆ = 0. In the latter case, the “switching”in the pseudo-Hermiticity matrices corresponds to a “switching” from a Hermitiandiagonal matrix D = D† for ∆ > 0 to a complex-valued diagonal matrix D = D†

for ∆ < 0. A “switching” occurs also in the properties of the eigenvectors of thediagonal matrix D. Choosing these eigenvectors in the simplest form as

D|±〉 = (e0 ± ε) |±〉, |+〉 =(

01

), |−〉 =

(10

),

〈±|±〉 = 1, 〈+| = (0, 1), 〈−| = (1, 0), (44)

one finds for the Krein space inner products

∆ > 0 : [±,±]η = ±sign(f), [±,∓]η = 0 , (45)∆ < 0 : [±,±]η = 0, [±,∓]η = 1. (46)

This means that the eigenvectors |±〉 are of positive or negative type for real-valuedeigenvalues, ∆ > 0, and of isotropic type for pair-wise complex conjugate eigenval-ues, ∆ < 0. We explicitly reproduced a basic result of Krein space theory which isdiscussed in [20] and which is also implicitly present, e.g., in Refs. [14, 17, 18].

3 Exceptional points

From the structure of S in (34) one sees that in the degeneration limit ε → 0the determinant det(S) vanishes, det(S) → 0, S becomes singular and the diago-nalization (34) breaks down. Instead the matrix D turns into a Jordan block. Forε = 0, f2 = |b|2 = 0, f = ±|b|, a = d± 2|b| one finds the explicit relation

H = SDS−1, D =(E 10 E

), S =

(∓|b|b∗

1b∗

1 0

), E = d± |b|. (47)

Additionally, one observes8) that only |−〉 survives as an (geometric) eigenvectorof D, whereas |+〉 is now an associated vector (algebraic eigenvector)

(D −EI)|−〉 = 0, (D −EI)|+〉 = |−〉, (D −EI)2|+〉 = 0. (48)

Hence, the degenerate eigenvalues on the double cone have algebraic multiplicitytwo and geometric multiplicity one. The degeneracy is a branching point degeneracy

8) The existence of a Jordan block structure with a single geometric eigenvector is a genericfeature of systems at level-crossing points and was observed for an 1D pseudo-Hermitian Hamil-tonian, e.g., in [26].

Czech. J. Phys. 54 (2004) 1087

U. Gunther, F. Stefani, and G. Gerbeth

[27] — a double cone of exceptional points of branching type [28] in the sense ofKato [29].

A singularity of higher order occurs at the center of the double cone, i.e., in thediabolic point [30] at the origin f = |b| = 0 of the parameter space M (f, b1, b2).There the transformation matrix S in (47) becomes singular and instead of theJordan block (47) the matrix H has the form

H =(E 00 E

), E = a = d. (49)

This is the same type of co-dimension three degeneracy as in the case of the spinmatrix Hs — with two (geometric) eigenvectors, i.e., the algebraic and the geometricmultiplicity of this eigenvalue coincide and equal two.

The question of whether nontrivial holonomy (geometric phases [23]) and chiral-ity properties of eigenfunctions [31] in the vicinity of level-crossing points can playa role in the dynamics of unstable magnetic field configurations with field reversals[2] remains as one of the interesting open issues.

We thank the Professors M.V. Berry, W.D. Heiss, H. Langer, C. Tretter, and M. Znojil

for useful comments at different stages of the work. U.G. acknowledges support from DFG

grant GE 682/12-2.

References

[1] H.K. Moffatt: Magnetic field generation in electrically conducting fluids. CambridgeUniversity Press, Cambridge, 1978.F. Krause and K.-H. Radler: Mean-field magnetohydrodynamics and dynamo theory.Akademie-Verlag, Berlin and Pergamon Press, Oxford, 1980.Ya.B. Zeldovich, A.A. Ruzmaikin, and D.D. Sokoloff: Magnetic fields in astrophysics.Gordon & Breach Science Publishers, New York, 1983.

[2] G.A. Glatzmaier and P.H. Roberts: Nature 377 (1995) 203.

[3] A. Gailitis et al.: Phys. Rev. Lett. 84 (2000) 4365; 86 (2001) 3024; Rev. Mod. Phys.74 (2002) 973.

[4] U. Muller and R. Stieglitz: Phys. Fluids 13 (2001) 561.

[5] U. Gunther and F. Stefani: J. Math. Phys. 44 (2003) 3097; math-ph/0208012.

[6] F. Stefani and G. Gerbeth: Phys. Rev. E 67 (2003) 027302; astro-ph/0210412.

[7] W.D. Heiss and W.H. Steeb: J. Math. Phys. 32 (1991) 3003.

[8] A. Marshakov: Seiberg-Witten theory and integrable systems. World Scientific, Singa-pore, 1999.

[9] A. Markus: Introduction to the spectral theory of polynomial operator pencils. Transla-tions of Mathematical Monographs, Vol. 71. Am. Math. Soc., Providence (RI), 1988.

[10] H. Baumgartel: Analytic perturbation theory for matrices and operators. Akademie-Verlag, Berlin, 1984; Operator Theory, Adv. Appl., Vol. 15, Birkhauser Verlag, Basel,1985.

1088 Czech. J. Phys. 54 (2004)

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[11] A. Dijksma and H. Langer: in Lectures on operator theory and its applications, FieldsInstitute Monographs, (Eds. A. Bottcher et al.), Am. Math. Soc., Providence (RI),1996, p. 75.

[12] A. Mostafazadeh: Nucl. Phys. B 640 (2002) 419; math-ph/0203041.

[13] A. Mostafazadeh: J. Math. Phys. 43 (2002) 6343; Erratum 44 (2003) 943;math-ph/0207009.

[14] A. Mostafazadeh: J. Math. Phys. 43 (2002) 205; math-ph/0107001.

[15] C.M. Bender and S. Boettcher: Phys. Rev. Lett. 80 (1998) 5243; physics/9712001.

[16] C.M. Bender, S. Boettcher, and P.N. Meisinger: J. Math. Phys. 40 (1999) 2201;quant-ph/9809072.

[17] C.M. Bender, D.C. Brody, and H.F. Jones: Phys. Rev. Lett. 89 (2002) 270401; Erra-tum 92 (2004) 119902; quant-ph/0208076.

[18] C.M. Bender, P.N. Meisinger, and Q. Wang: J. Phys. A 36 (2003) 6791;quant-ph/0303174.

[19] M. Znojil: in Quantum Theory and Symmetries, (Eds. E. Kapuscik and A. Horzela),Word Scientific, Singapore, 2002, p. 626; math-ph/0106021.

[20] T.Ya. Azizov and I.S. Iokhvidov: Linear operators in spaces with an indefinite metric.Wiley-Interscience, New York, 1989.

[21] H. Langer: Czech J. Phys. 54 (2004) 1113 (this issue).

[22] J.v. Neumann and E. Wigner: Phys. Z. 30 (1929) 467.

[23] M.V. Berry: Proc. R. Soc. Lond. A 392 (1984) 45.

[24] B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov: Modern geometry – methods andapplications. Part I. The geometry of surfaces, transformation groups, and fields.Springer, New York, 1992.

[25] S.L. Adler: Quaternionic quantum mechanics and quantum fields. Oxford UniversityPress, Oxford, 1995.

[26] P. Dorey, C. Dunning, and R. Tateo, J. Phys. A34 (2001) L391; hep-th/0104119.

[27] M.V. Berry: Czech J. Phys. 54 (2004) 1039 (this issue).

[28] W.D. Heiss: Czech J. Phys. 54 (2004) 1091 (this issue).

[29] T. Kato: Perturbation theory for linear operators. Springer, Berlin, 1966.

[30] M.V. Berry and M. Wilkinson: Proc. R. Soc. Lond. A 392 (1984) 15.

[31] W.D. Heiss and H.L. Harney: Eur. Phys. J. D 17 (2001) 149; quant-ph/0012093.

Czech. J. Phys. 54 (2004) 1089