Projective Hilbert space structures at exceptional points
arXiv:0704.1291v4  [math-ph]  12 Nov 2018
Projective Hilbert space structures at exceptional
points
Uwe Günthera, Ingrid Rotterb and Boris F. Samsonovc1)
a Helmholtz Center Dresden-Rossendorf, Bautzner Landstraße 400, D-01328
Dresden, Germany
b Max Planck Institute for physics of complex systems, D-01187 Dresden,
Germany
c Physics Department, Tomsk State University, 36 Lenin Avenue, 634050
Tomsk, Russia
E-mail: u.guenther@hzdr.de and rotter@pks.mpg.de
Abstract. A non-Hermitian complex symmetric 2× 2 matrix toy model is used
to study projective Hilbert space structures in the vicinity of exceptional points
(EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseux-
expanded in terms of the root vectors at the EP. It is shown that the apparent
contradiction between the two incompatible normalization conditions with finite
and singular behavior in the EP-limit can be resolved by projectively extending the
original Hilbert space. The complementary normalization conditions correspond
then to two different affine charts of this enlarged projective Hilbert space.
Geometric phase and phase jump behavior are analyzed and the usefulness of the
phase rigidity as measure for the distance to EP configurations is demonstrated.
Finally, EP-related aspects of PT −symmetrically extended Quantum Mechanics
are discussed and a conjecture concerning the quantum brachistochrone problem
is formulated.
PACS numbers: 03.65.Fd, 03.65.Vf, 03.65.Ca, 02.40.Xx
published as: J. Phys. A: Math. Theor. 40, 8815 (2007)
1. Introduction
A generic property of non-Hermitian operators is the possible occurrence of non-trivial
Jordan-blocks in their spectral decomposition [1]. For an operator H(X) depending
on a set of parameters X = (X1, . . . , Xm) ∈ M from a space M this means that, in
case of a single Jordan block, two or more spectral branches λ1(X), . . . , λk(X) may
coalesce (degenerate) at certain parameter hypersurfaces V ⊂ M under simultaneous
coalescence of the corresponding eigenvectors Φ1(X), . . . ,Φk(X): λ1(Xc) = · · · =
λk(Xc), Φ1(Xc) = · · · = Φk(Xc) ≡ Θ0(Xc) for Xc ∈ V . Spectral points of this
type are branch points of the spectral Riemann surface and are called exceptional
points (EPs) [1]. At the EPs the set of the originally k linearly independent
eigenvectors Φ1(X), . . . ,Φk(X) is replaced by the single eigenvector Θ0(Xc) and k− 1
associated vectors Θ1(Xc), . . . ,Θk−1(Xc) which form a Jordan chain. Together they
span the so called k−dimensional algebraic eigenspace (or root space) Sλ(Xc) =
span[Θ0(Xc), . . . ,Θk−1(Xc)] [1, 2] so that the total space dimension remains preserved.
1) Deceased 08 November 2012
http://arxiv.org/abs/0704.1291v4
Projective Hilbert space structures 2
The construction extends straight forwardly to the presence of several Jordan blocks
for the same eigenvalue λ(Xc). In general, the degeneration hypersurface V ⊂ M
consists of components Va of different codimension codimVa = a. Higher order Jordan
blocks require a higher degree of parameter tuning (they have a higher codimension)
and a correspondingly lower dimension of the component Va. Due to the different
dimensions of its components Va the degeneration hypersurface V =
Va itself has
the structure of a stratified manifold [3].
EPs occur naturally in quantum scattering setups [4, 5] when two or more
resonance states coalesce and higher order poles of the S-matrix form. Within the
Gamow state approach such S-matrix double poles have been considered in [6, 7, 8, 9],
whereas in the Feshbach projection operator formalism (one of the basic approaches
to analyze open quantum systems) they naturally occurred in studies of nuclei [10],
atoms [11, 12] and quantum dots [13, 14]. EP-related crossing and avoided crossing
scenarios have been studied for bound states in the continuum [11, 15, 16, 17] as well
as for phase transitions [18, 19, 20, 21]. In asymptotic analyses of quasi-stationary
systems EPs show up as hidden crossings [22]. EP-related theoretically predicted
level and width bifurcation properties have been experimentally verified in a series of
microwave resonator cavity experiments. In [23], the resonance trapping phenomenon
(width bifurcation) [10] has directly been proven. The fourfold winding around an EP
has been found experimentally [24] in full agreement with the theoretical prediction
[19, 25] and related studies [14, 26]. In [27] two-level coalescences have been associated
with chiral system behavior. The geometric phase at EPs has been discussed in
[14, 26, 27, 28, 29, 30, 31, 32].
EPs play also an important role in the recently considered PT −symmetrically
extended quantum models [33, 34, 35]. There they correspond to the phase transition
points between physical sectors of exact PT −symmetry and unphysical sectors of
spontaneously broken PT −symmetry [36, 37, 38, 39, 40].
Other, non-quantum mechanical examples where EPs play an important role are
the optics of bianisotropic crystals [41], acoustic models [42], many hydrodynamic
setups where EPs have been studied within pseudo-spectral approaches [43] as well
as a large number of mechanical models [44] where they are connected with regimes
of critical stability [45]. Recent results on magnetohydrodynamic dynamo models
indicate on a close connection between nonlinear polarity reversal mechanisms of
magnetic fields and EPs [46].
For completeness we note that the perturbation theory for systems in the vicinity
of EPs dates back to 1960 [47] (see also [2]) and that it is closely related to
singularity theory, catastrophe theory and versal deformations of Jordan structures
[48]. Supersymmetric mappings between EP configurations have been recently
considered in [49, 50].
A correct perturbative treatment of models in the vicinity of EPs has to be
built on an expansion in terms of root vectors (eigenvectors and associated vectors
Θi(Xc)) at the corresponding unperturbed eigenvalue λ(Xc) (see e.g. [2, 44]). For
X 6∈ V (away from the EP at Xc and from other EPs) the operator H(X) has a
diagonal spectral decomposition with corresponding eigenvectors Φi(X). Choosing
the normalization of these eigenvectors away from the EP and without regard to the
expansion in terms of root vectors leads to a divergence of the normalization constants
in the EP-limit X → Xc. The diagonalization break-down at Xc, the occurrence of
the Jordan block structure and the singular behavior of the eigenvector decomposition
Projective Hilbert space structures 3
are generic and were many times described in various contexts (see e.g. [51, 52]). The
natural question connected with the fitting of the root-vector based normalization and
the diagonalizable-configuration normalization (and related controversial discussions
on their physical interpretation [51, 52]) is whether and how the singular behavior
affects the projective Hilbert space structure of quantum systems.
In the present paper we answer this question by resolving the singularity with the
help of embedding the original Hilbert space H = Cn into its projective extension
n instead of projecting it down to CPn−1 as in standard Hermitian Quantum
Mechanics. Diagonal spectral decompositions and decompositions with nontrivial root
spaces live then simply in different (and complementary) affine charts of CPn similar
like monopole configurations of Hermitian systems have to be covered with two charts
(North-pole chart and South-pole chart) of the unit sphere S2 [53].
The basic construction is demonstrated on a simple complex symmetric (non-
Hermitian) 2×2−matrix toy model. The consideration of complex symmetric matrices
sets no restriction because by a similarity transformation any complex matrix can
be brought to a complex symmetric form (see, e.g. [54, 55]). The Hilbert space
notations for the 2 × 2−matrix model are fixed in section 2. In section 3, following
[32, 44] we derive the leading-order perturbative expansion in the vicinity of an EP
at Xc in terms of root vectors and fit it then explicitly with expressions of the
diagonal spectral decomposition at X 6= Xc. Combining geometric phase techniques
for non-Hermitian systems [28] with projective Hilbert space techniques from [56]
we generalize the projective geometric phase techniques of Hermitian systems to
paths around EPs (section 4). The corresponding monodromy group is identified
as parabolic Abelian subgroup of the special linear group SL(2,C) and evidence is
given that vector norm scalings are only due to complex dynamical phases whereas
geometrical phases are purely real-valued and norm preserving. In section 5 we
consider an instantaneous (stationary type) picture of the system. Within such a
picture, we resolve the singular normalization behavior by projectively embedding
the Hilbert space H = C2 →֒ CP2. We discuss the necessity for an affine multi-
chart covering of CP2 in order to accommodate diagonal-decomposition vectors and
root vectors at EPs simultaneously. The usefulness of the phase rigidity as distance
measure to EPs is discussed in section 6. In section 7 some EP-related aspects
of PT −symmetric quantum models are discussed and a conjecture concerning the
quantum brachistochrone problem [34, 57] is formulated. The Conclusions 8 are
followed by Appendix A where auxiliary results on Jordan structures of complex
symmetric matrices are listed.
2. Setup
Subject of our consideration is the behavior of a quantum system near a level crossing
point of two resonance states — supposing that for an N−level system the influence
of the other N − 2 levels is sufficiently weak. Under this assumption the setup can be
modeled by an effective complex symmetric (non-Hermitian) 2×2 matrix Hamiltonian
, H = HT . (1)
The effective energies ǫ1,2 ∈ C and the effective channel coupling ω ∈ C will in general
depend on underlying parameters X = (X1, . . . , Xk) ∈ M from a space M.
Projective Hilbert space structures 4
For nonvanishing coupling ω 6= 0 the Hamiltonian can be rewritten as
H = E0 ⊗ I2 + ω
with I2 denoting the 2× 2 unit matrix and
E0 :=
(ǫ1 + ǫ2), Z :=
ǫ1 − ǫ2
. (3)
In this representation the eigenvalues E± and eigenvectors Φ± of H take the very
simple form
E± = E0 ± ω
Z2 + 1 (4)
Z2 + 1
c± , c± ∈ C∗ := C− {0} (5)
which makes the branching behavior most transparent2). From the overlap
〈Φ+|Φ−〉 ≡ Φ∗T+ Φ− (6)
= c∗+c−
1 + |Z|2 − |Z2 + 1|+ 2Im
Z2 + 1
one reads off that 〈Φ+|Φ−〉 = 0 holds only for ImZ = 0 and that for general Z ∈ C the
two states Φ+ and Φ− are not orthogonal 〈Φ+|Φ−〉 6= 0. Following standard techniques
[44] for non-Hermitian operators, we consider a dual (left) basis Ξ± bi-orthogonal to
(H+ − E∗±)Ξ± = 0, 〈Ξk|Φl〉 ∝ δkl, k, l = ± . (7)
For complex symmetric H it holds Ξ± ∝ Φ∗± so that the most general ansatz for the
right and left basis vectors Φ± and Ξ± can be chosen as
Φ± = c±χ±, Ξ± = d
±, c±, d± ∈ C∗ (8)
χ± :=
Z2 + 1
. (9)
The bi-orthogonality
〈Ξ±|Φ∓〉 = d±c∓χT±χ∓ = 0 (10)
is ensured by the structure of χ± and holds for any value of the parameter Z ∈ C.
A normalization 〈Ξ±|Φ±〉 = 1 would set two constraints on the four free scaling
parameters c±, d± ∈ C∗
〈Ξ±|Φ±〉 = d±c±χT±χ± = 1 , (11)
so that the system would still have two free parameters which should be fixed by
additional assumptions. Subsequently, we will mainly work with the bi-orthogonality
properties of the vectors Φ±, Ξ± and fix their normalization only when explicitly
required.
Due to the arbitrary scaling parameters c±, d± ∈ C∗ of the right and left
eigenvectors Φ±,Ξ± ∈ H = C2 (8) it is natural to consider equivalence classes of such
vectors defined by corresponding lines π(Φ±), π(Ξ±). These lines form the projective
2) The fact that Φ± depends only on the single parameter Z reflects the property that after rescaling
the energy by 1/ω and shifting it by −E0/ω (these transformations do not affect the eigenvectors)
the Hamiltonian (2) depends only on Z.
Projective Hilbert space structures 5
Hilbert space P(H) = H∗/C∗ = CP1 ∋ π(Φ±), π(Ξ±) [58, 59, 60], whereH∗ := H−{0}
denotes the original Hilbert space with the point at origin {0} = (0, 0) deleted to
allow for a consistent definition of P(H). The space P(H) is covered by a single chart
of homogeneous coordinates (z0, z1)
T ∈ H and two complementary charts of affine
coordinates U0 ∋ (1, z1/z0), z0 6= 0 and U1 ∋ (z0/z1, 1), z1 6= 0. Comparison with the
structure of the auxiliary vectors χ± (9) shows that the χ± can be straight forwardly
re-interpreted as points of the projective space CP1 described by the affine coordinate
over U0: χ± ≈ π(Φ±). The vectors {Φ±,Ξ±} themselves can be understood as
sections of the natural line bundle L = {(p, v) ∈ P(H)×H| v = cp, c ∈ C∗} [53], i.e. as
Φ± = π(Φ±)⊗c±, Ξ± = π(Ξ±)⊗d∗±, where π denotes the projection π : H∗ → P(H).
The bundle structure is locally trivial π−1(U0) ≈ U0 × C∗ ∋ Φ± [61]3).
3. Jordan structure
At an EP, the two eigenvalues E± coalesce E+ = E− = E0. According to (4), this
happens for Z2 = −1 and Z = Zc := ±i and via (8) it is connected with a coalescence
of the corresponding lines π(Φ+) = π(Φ−) =: π(Φ0) encoded in
χ+ = χ− = χ0 :=
. (12)
This means that the eigenvalue E0 has algebraic multiplicity na(E0) = 2 and geometric
multiplicity ng(E0) = 1 and by definition the level crossing point is an EP of the
spectrum. The bi-orthogonality (10) of Φ± and Ξ∓ is compatible with the coalescence
of the lines due to the vanishing bi-norm χT0 χ0 = 0, i.e. the isotropy
4) of χ0, —
a generic fact holding for the (geometric) eigenvector at any EP [2, 44]. We note
that the coalescence π(Φ+) = π(Φ−) = π(Φ0) still leaves the freedom for the vectors
Φ+ = c+χ0 and Φ− = c−χ0 of being two different sections Φ+ 6= Φ− of the same fiber
π(Φ0)× C∗ over π(Φ0).
The right and left eigenvectors Φ0, Ξ0 at the EP are supplemented by
corresponding associated vectors (algebraic eigenvectors) Φ1 and Ξ1 defined by the
Jordan chain relations [44]
[H(Zc)− E0I2]Φ0 = 0, [H(Zc)− E0I2]Φ1 = Φ0 (13)
[H(Zc)− E0I2]+ Ξ0 = 0, [H(Zc)− E0I2]+ Ξ1 = Ξ0 .
From the inhomogeneity of these Jordan chain relations it follows immediately that
the root vectors Φ0 and Φ1 as well as Ξ0 and Ξ1 scale simultaneously and in a linked
way with the same single scale factor c0 and d
0, respectively. This is also visible from
their explicit representation (A.9) derived in Appendix A
Φ0 = σqc0
, Φ1 = σq
Ξ0 = σq
,Ξ1 = σq
∗−1d∗0
, q :=
2ω, Zc = ±i =: µi, c0, d0 ∈ C∗ . (15)
3) For completeness we note that the (right) eigenvectors Φ± and the dual (left) ones Ξ± could be
understood as elements of a vector bundle P(H)×F and its dual P(H)×F ∗ with pairing in the fibres
〈.|.〉 : F ∗p × Fp −→ C (see, e.g. [58]). The details of this construction will be presented elsewhere.
4) The vector χ0 behaves similar like a vector on the light cone in Minkowski space.
Projective Hilbert space structures 6
The simultaneous scaling means that the lines π(Φ0), π(Ξ0) at the EP should
be interpreted as the one-dimensional components (projections) of two-dimensional
planes which span the root space5) S(E0) [2] and which scale as a whole with a single
scale factor. Such a higher-dimensional (complex) plane-structure goes clearly beyond
the one-dimensional line structure of the projective space P(H) (mathematically one
should extend the natural line bundle of the original projective space to a more general
projective flag bundle [62, 63])6) and underlines the fact that the state at an EP itself
is not an element of the projective Hilbert space P(H) in its usual understanding.
The basis sets {Φ0,Φ1} and {Ξ0,Ξ1} satisfy the well known bi-orthogonality
conditions [44]
〈Ξ0|Φ0〉 = 〈Ξ1|Φ1〉 = 0
〈Ξ0|Φ1〉 = 〈Ξ1|Φ0〉 = d0c0 6= 0 . (16)
Again, a normalization 〈Ξ0|Φ1〉 = 〈Ξ1|Φ0〉 = 1 would only lead to a constraint
d0c0 = 1 on the scale factors, but would not fix them completely. Due to this scaling
freedom the single line π(Φ0) of a given Jordan structure, in general, still allows for
different sections Φ0,a 6= Φ0,b of the corresponding fiber π(Φ0) × C∗ ∋ Φ0,a,Φ0,b,
π(Φ0,a) = π(Φ0,b) = π(Φ0).
Let us now consider in detail what happens when the system approaches one of
the critical values Zc = ±i. For this purpose we use the well-defined (but completely
general and arbitrary) ansatz
Z = Zc + ε, |ε| ≪ 1, ε ∈ C (17)
and expand the eigenvalues (4) and the line defining vectors χ± (9) in terms of ε. This
gives the leading contributions to their Puiseux series representation [2, 44] in ε1/2 as
E± = E0 ± ε1/2∆E + o(ε1/2), (18)
∆E := ω
2Zc ,
± ε1/2
+ o(ε1/2) . (19)
Following [32, 44], we expand the eigenvectors Φ±(Z) of the diagonal spectral
decomposition in the same local ε1/2 approximation in terms of the Jordan chain
(root) vectors Φ0,1
Φ± = Φ0 + ε
1/2(b0Φ0 + b1Φ1) + o(ε
1/2), (20)
b0 = ±
∆E, b1 = ±∆E . (21)
The coefficients b0,1 are obtained by a two-step procedure. Substituting (17), (18),
(20) into the eigenvalue equation and making explicit use of the chain relations (13)
yields b1 and leaves b0 still undefined. The coefficient b0 is found by comparing the
line structures7) of Φ± with χ± in (19).
It remains to match the fiber sections — what can be done in two ways. One
may assume a single scaling coefficient c0 of the root space given and consider the
5) In the present simplest model S(E0) fills the whole Hilbert space H = C
6) Indications that all the root vectors of a Jordan chain should scale simultaneously with a single
scale factor were given, e.g., in [8] for Gamow vector setups with higher S-matrix poles.
7) The term ε1/2b0Φ0 additionally present in (20) in comparison with the corresponding result in
[32] is due to the different choice of the root (Jordan chain) vectors Φ0, Φ1. The chain relation (13)
shows that the associated vector Φ1 is defined up to additional Φ0 contributions and can be replaced
by any linear combination Φ1 + aΦ0, a ∈ C.
Projective Hilbert space structures 7
coefficients c± of the sections Φ± as derived objects. This leads to the identification
c+ = c− = σqc0. Apart from this option, one may assume the scaling coefficients
c± as primary objects and given so that they may take different values c+ 6= c−.
Correspondingly the scaling factor c0 of the root space will then be fitted to c± so
that it will take two different values
c0,± = c±/(σq) . (22)
Both constructions are possible and compatible with the smooth fitting of the line
structure encoded in the EP-limiting behavior π(Φ±) → π(Φ0).
In a way similar to the above two-step procedure with subsequent fiber fitting the
left eigenvectors can be obtained as
Ξ∗± = Ξ
0 + ε
1/2(b0Ξ
0 + b1Ξ
1) + o(ε
1/2), (23)
d± = σ
∗qZcd0,± . (24)
Here, b0 and b1 are the same as in (21) and full compatibility with the bi-orthogonality
conditions (7) as well as with (8) is easily verified by direct calculation. In case
of a single scaling factor d0 of the dual root space the coefficients d± will coincide
d+ = d− = σ
∗qZcd0.
Combining (20) and (23) one finds the limiting behavior of the inner products as
〈Ξ±|Φ±〉 = 2b1d0,±c0,±ε1/2 + o(ε1/2)
d±c±ε
1/2 + o(ε1/2) . (25)
Here, one has to distinguish two normalization schemes. If one assumes the root vector
sets {Φ0,Φ1}, {Ξ0,Ξ1} normalized, e.g., with d0,±c0,± = 1 or d0c0 = 1 in (16) then
the scalar product 〈Ξ±|Φ±〉 of the eigenvectors in the diagonal spectral decomposition
(see (25)) vanishes in the EP-limit. Starting, in contrast, from normalized eigenvector
pairs {Φ±,Ξ±} of diagonalizable Hamiltonians as in (11), i.e. with 〈Ξ±|Φ±〉 = 1,
then the scale factor products d±c± diverge as d±c± ∝ ε−1/2 for ε → 0. Both
normalization schemes are possible and compatible with the smooth limiting behavior
π(Φ±) → π(Φ0) of the lines encoded in χ±(ε → 0) → χ0 [cf. (12)]. We see that
this special behavior is only related to the fiber sections and not to the fibers (lines)
themselves. The two incompatible normalization schemes simply indicate on the need
for two complementary charts to cover the whole physical picture in the vicinity of
a 2 × 2 Jordan block J2. One of these charts (we will call it the root vector chart)
remains regular in the EP-limit, whereas the other (diagonal representation) chart
becomes singular.
The situation is similar to the two affine charts required to cover the Riemann
sphere CP1. Starting from homogeneous coordinates (x0, x1) ∈ CP1 one has the two
affine charts U0 ∋ (1, x1/x0), x0 6= 0 and U1 ∋ (x0/x1, 1), x1 6= 0. The mutually
complementary affine coordinates z := x1/x0 ∈ C1 and w := x0/x1 ∈ C1 are then
related by the well known fractional transformation w = 1/z so that the singular
|z| → ∞ limit in the z−chart corresponds simply to the regular w → 0 limit in the
w−chart. In other words, the two charts cover the North-pole region and the South-
pole region of the Riemann sphere — a construction well known, e.g., from complex
analysis and the description of magnetic monopoles [53].
Returning to the two-chart picture of the normalization we see that the original
Hilbert space H = C2 should be extended by the set of infinite vectors Φ± what can
be naturally accomplished by embedding it into a larger projective space H →֒ CP2.
Projective Hilbert space structures 8
Correspondingly the fibers π(Φ±) × C∗ should be extended as π(Φ±) × C∗ →֒
π(Φ±) × CP1. A detailed discussion of these structures will be presented in [64].
An explicit embedding construction for simplified setups with coinciding scale factors
d± = c± is given in section 5 below.
4. Geometric phase
Following earlier studies [27, 28, 29, 30, 31], geometric phases [65] of eigenvectors of
non-Hermitian complex symmetric operators have been recently considered in [32]
for paths in parameter space encircling an EP. The results showed full agreement
with the phase considerations of [27]. In this section, we combine techniques for
non-Hermitian systems [28, 32] with explicit projective space parameterizations for
Hermitian systems [56] to provide an explicit projective-space based derivation of the
phase representation for non-Hermitian systems. Such an explicit reshaping of the
results of [56] to non-Hermitian setups seems missing up to now.
Following [28, 29, 30, 32] we consider an auxiliary system with a general non-
Hermitian Hamiltonian H(t) depending on a set of non-stationary parameters X(t) =
[X1(t), . . . , Xm(t)] ∈ M, H(t) = H [X(t)] and an EP hyper-surface V ⊂ M which
is encircled by an appropriate loop Γ in parameter space M ∋ Γ = {X(t), t ∈
[0, T ] : X(0) = X(T )}. The evolution of the quantum system is governed by a usual
Schrödinger equation for the right eigenvectors
i∂tΦn(t) = H(t)Φn(t), (26)
and, due to the time invariance of the bi-orthogonal product
〈Ξm(t)|Φn(t)〉 = δmn , (27)
by a complementary evolution law for the left eigenvectors [28]
i∂tΞm(t) = H
+(t)Ξm(t) . (28)
For an adiabatic motion cycle Γ ⊂ M with Hamiltonian H [X(T )] = H [X(0)] the
resulting eigenvector Φn(t = T ) of H [X(T )] must lay on the same line as the initial
Φn(t = 0), i.e. it can only obtain an additional scaling factor which we parameterize
as complex-valued phase
Φn(T ) = e
iφn(T )Φn(0), φn(T ) ∈ C. (29)
Due to the preserved orthonormality (27) the corresponding left eigenvectors evolve
Ξm(T ) = e
(T )Ξm(0) . (30)
The complex phase φn(T ) can be split into a dynamical component [28, 56]
ǫn(T ) = −
〈Ξn(t)|H(t)|Φn(t)〉
〈Ξn(t)|Φn(t)〉
dt (31)
and the geometric phase
γn(T ) = φn(T )− ǫn(T ) . (32)
Adapting the techniques of [56] we calculate γn(T ) in terms of explicit projective space
coordinates. Setting
Φn(t) = cn(t)χn(t) =: [z0(t), z1(t)]
T = z0(t)[1, w(t)]
Ξm(t) = d
m(t)χ
m(t) =: [y0(t), y1(t)]
T = y0(t)[1, v(t)]
T (33)
Projective Hilbert space structures 9
(omitting in the projective space coordinates the mode indices m, n) one identifies
φn(T ) = −i ln [z0(T )/z0(0)] (34)
and obtains from (26), (31) and (32) the differential 1-form of the geometric phase as
dγ = − idz0
y∗0dz0 + y
y∗0z0 + y
1 + v∗w∗
. (35)
Due to the symmetry (8) between left and right eigen-lines this simplifies to
dγ = i
d ln(1 + w2) . (36)
Similar to results on Hermitian systems [56] the differential 1-form (35) is independent
of the coordinates z0 and y0 along the fibers and, hence, defines a horizontal connection
over the projective Hilbert space of the system. The mere difference in the definitions
of the projective structures is in CP1 = S3/S1 for Hermitian systems, whereas
1 = H∗/C∗ for non-Hermitian ones [61].
Let us now apply the general technique to the concrete 2× 2−matrix model (1).
Parameterizing the cycle around the EP by (17) with
ε = reiα, α ∈ [0, 2π], 0 < r ≪ 1 (37)
one reproduces the 1-forms of the geometric phases of [29]
dγ± =
d ln ε = −1
d ln r . (38)
In a similar way one obtains the same 1-forms for the corresponding left eigenvectors
Ξ±. Upon integration over a full cycle α(T ) = α(0) + 2π, r(T ) = r(0) one finds
γ±(T )− γ±(0) = −
. (39)
The relation between geometric phases γ± and the cycle phase α can be gained
also directly from the structure of the sections Φ±. These sections may be arranged
as columns of a diagonalizable 2× 2−matrix
Φ(α) := [Φ+(α),Φ−(α)] . (40)
The evolution along a cycle is then encoded in the transformation matrix W (α) =
Φ(α) [Φ(0)]
which for small ε with 0 6= |ε| ≪ 1 can be calculated from the
representation (19) as
W (α) =
2iZc sin
. (41)
The elements W (α) form an Abelian parabolic subgroup P of the special linear group
SL(2,C) ⊃ P (see, e.g., [62, 63, 66])
W (α+ β) = W (α)W (β) = W (β)W (α) (42)
corresponding to the mapping eiα ∈ S1 ≈ U(1) 7→ P ⊂ SL(2,C). For full cycles
α = 2πN, N ∈ Z they yield the monodromy transformations [67]
W0 := W (0) = I2, W1 := W (2π) =
2iZc i
W2 := W (4π) = W
2(2π) = −I2,
W3 := W (6π) = −W (2π), W (8π) = I2 = W0 . (43)
Projective Hilbert space structures 10
The geometric phase (38) and the monodromy transformations (43) show the typical
four-fold covering of the mapping α 7→ γ which was earlier described in [19, 25, 27, 32]
and experimentally demonstrated in [24]. A cycle around the EP in parameter space
M has to be passed four times in order to produce one full 2π−cycle in the geometric
phase. A (non-oriented) eigen-line π(Φ 6= Φ0) ∈ CP2 is already recovered after
two cycles π(W2Φ) = π(−Φ) = π(Φ) — similar to the eigenvalue E which for the
2×2−matrix lives on a two-sheeted Riemann surface with the same two branch points
Zc = ±i as the line bundle. For the isotropic limiting vector Φ0 at the EP it holds
(due to (12))
W (α)Φ0 = e
−iα/4Φ0 (44)
so that the parabolic subgroup P ∋ W (α) can be identified as invariance group of the
projective line at the EP
π (W (α)Φ0) = π
e−iα/4Φ0
= π(Φ0). (45)
We note that despite the non-Hermiticity of the Hamiltonian H the geometric
phase is purely real — as for Hermitian systems. Relations (38), (39) show that
possible imaginary phase contributions (which would result in a re-scaling of the
eigenvectors Φ±) are cancelled by the closed-cycle condition r(T ) = r(0). Hence,
the non-preservation of the vector norm in non-Hermitian systems is induced solely
by a complex dynamical phase ǫ and requires the presence of the bi-orthogonal basis
where a decaying behavior of the right eigenvectors8)
Φn ∝ e−iǫnt−
t, 〈Φn|Φn〉 = ||Φn||2 ∝ e−Γnt (46)
is necessarily connected with increasing vector norms of the dual left eigenvectors
Ξm ∝ e−iǫmt+
t, ||Ξm||2 ∝ eΓmt (47)
so that indeed 〈Ξm|Φn〉 = δmn. This behavior is well known from resonances and
Gamow vector theory (see, e.g. [7]).
Comparison of (8) and (46), (47) shows that the formal ansatz Ξm = Φ
m for the
eigenvectors of the complex symmetric Hamiltonian (cf. [11, 13, 14]) can be used only
for instantaneous eigenvectors at a single fixed t = t0 (which formally can be set to
t0 = 0) as well as for the subclass of real symmetric matrices (when the system becomes
Hermitian and norm-preservation of the eigenvectors holds). In contrast, for explicitly
time dependent non-Hermitian setups it only holds Ξm(t) ∝ Φ∗m(t), i.e. the dual basis
vectors necessarily live on complex conjugate lines (fibers) π[Ξm(t)] = (π[Φm(t)])
with Ξm(t) 6= Φ∗m(t) for t 6= t0.
Aspects of the parameter dependence of the phases and scalings in an
instantaneous picture with Ξm = Φ
m together with the explicit EP-limit ε → 0 are
subject of the next section.
5. Instantaneous picture
In modern quantum physics not only the properties of natural systems such as nuclei
or atoms are of interest, but rather the design and functionality of artificial quantum-
system-based devices plays an essential role. In many cases, for the understanding of
8) For simplicity, we show the relations for stationary non-Hermitian Hamiltonians H with constant
complex eigenvalues En = ǫn + i
= const ; ǫn,Γn ∈ R.
Projective Hilbert space structures 11
the dynamical features of these man-made quantum systems the time dependence is of
minor interest. The properties of these systems are mainly governed by the position
and number of EPs, i.e. the level crossing points in the complex plane, and their
dependence on external control parameters. In this context it appears natural to study
the parameter dependence of level energies and widths as well as the corresponding
eigenvectors in terms of the instantaneous picture with Ξm = Φ
m and c± = d±. This
picture is compatible with the Hermitian limit when Imǫ1,2 = Imω = 0 in (1) and the
condition Ξm = Φ
m is fulfilled by definition
We have to distinguish the two possible normalization schemes — the root-
vector based normalization (16) with d0c0 = 1 or d0,±c0,± = 1 and the diagonal-
representation based normalization (11) with d±c±χ
Tχ = 1.
In the root-vector based normalization scheme the conditions c± = d± and
d0,±c0,± = 1 together with the two relations (22) and (24) imply (in leading-order
approximation in ε) c0,± = d0,± and, hence, c0,± = κ with κ = ±1 (independently of
the signs± in the index of c0,±) as well as c± = d± = κσq. We see that in leading-order
approximation in ε the scaling factors c± = d± are rigidly fixed and independent of ε.
A geometric phase (necessarily induced via an ε−dependence) appears incompatible
with this normalization.
Let us now turn to the diagonal-representation based normalization (11). In the
EP-limit ε → 0 the normalization condition (11) for the eigenvectors (5) yields
1 = 〈Ξ±|Φ±〉 = ΦT±Φ± =
Z2 + 1
≈ ∓ 2Zc
2Zcε c
± (48)
and we find the expected divergent scaling factors as
c2± ≈ ∓2−3/2Z−3/2c ε−1/2 =⇒ c± ∼ ε−1/4 . (49)
On the one hand, (49) reproduces the local four-sheeted Riemann surface
structure connected with the geometric phase (38), (41), i.e. a fourfold winding
around the EP is needed to return to an eigenvector pointing into the same complex
direction as a starting vector. (In contrast to the root-vector normalization scheme
full compatibility with the geometric phase setup holds.)
On the other hand, it leads to divergent vector norms
||Φ±||2 = 〈Φ±|Φ±〉 ≈ 2|c±|2 ≈ |2ε|−1/2 (50)
for ε → 0. As it was indicated in section 3, the corresponding singularity can be
naturally resolved by embedding the original Hilbert space H ≈ C2 into its projective
extension H →֒ CP2 ∋ φ = (u0, u1, u2) so that the set of infinite vectors becomes well
defined. Interpreting the two components z0 and z1 of the vector (fiber section)
Φ = c(1, w) = (z0, z1) ∈ C2 (51)
as affine coordinates on the chart U2 ∋ (u0u2 ,
, 1), u2 6= 0, U2 ⊂ CP2
Φ = (c, cw) →֒ (c, cw, 1) (52)
9) When compatibility with the Hermitian limit is not required, then the bi-orthonormalization
constraints d0,±c0,± = 1 or d±c±χ
χ± = 1 fix only two of the four constants c0,±, d0,± or c±, d±
and the remaining two can be chosen arbitrarily. For instance, one may set the eigenvector scaling
factors as c0,± = C 6= 1 or c± = 1 so that d0,± = C
−1 or d± =
what would define
instantaneous pictures not compatible with the Hermitian limit.
Projective Hilbert space structures 12
we can identify Φ with the point φ ∈ CP2 with homogeneous coordinates
φ = (u0, u1, u2) = (1, w, c
−1). (53)
The singularity |c| → ∞ at the EP corresponds then simply to the point φ0 =
(1, w, 0) ∈ CP2 with u2 = 0 and we see that the affine chart U2 ∈ CP2 is no longer
appropriate for covering φ0. This is in contrast to the root-vector normalization scheme
where c is fixed and the chart U2 remains suitable for the covering. Within the present
diagonal-representation normalization, instead, φ0 should be parameterized in terms of
affine coordinates corresponding to one of the charts10) U0 or U1 with u0 6= 0 or u1 6= 0.
Most natural for our representation (51), (53) is the affine chart U0 ∋ (1, u1u0 ,
) which
can be used for a suitable projective representation of the fibre sections Φ
Φ ≈ (1, w, c−1) = (χT , c−1) ≈ (π(Φ), c−1) . (54)
Interpreting the normalization condition (48) as constraint on the affine
coordinates of Φ in the chart U2
0 = ΦTΦ− 1 = u
− 1 (55)
one immediately sees that it is equivalent to the conic (singular quadric)11)
u20 + u
1 − u22 = 0 (56)
in homogeneous coordinates which cover the whole CP2. This conic remains regular
at EPs which merely correspond to configurations with u2 = 0. In terms of
(χT , c−1)−notations it reads
χTχ− c−2 = 0 . (57)
It is clear that the conic construction is straight forwardly extendable to Hilbert space
embeddings H = Cn →֒ CPn of any dimension n. We arrive at the conclusion that
the appropriate state space for open quantum systems in an instantaneous setting
will be related to the projective extension CPn of the original Hilbert space H = Cn
with states identified with conics
k=0 u
k − u2n = 0. This is in contrast to Hermitian
systems where it is sufficient to project the Hilbert space H∗ = Cn − {0} down to
the base space CPn−1, i.e. π : H∗ → P(H∗) ≈ CPn−1. In non-Hermitian setups
each fiber π(Φ) × C∗ should be supplemented by ∞. This suggests to extend them
to π(Φ) × CP1. From the above construction we see that the singular behavior with
regard to the two affine charts is only related to the scale factors c ∈ CP1, whereas
π(Φ) behaves smoothly and regular. On its turn, this suggests to reconsider the model
dependent physical interpretation of the eigenvector self-orthogonality (isotropy) and
the corresponding diverging or non-diverging sensitivity in perturbation expansions
like in [51, 52] as result of divergent or non-divergent normalization constants.
The Hilbert space extension H = C2 →֒ CP2 together with the observed
simultaneous scaling of the whole root space Sλ obtained in section 3, the upper
and lower triangular (parabolic subgroup type) structure of the Sλ−related matrices
in (A.7), (A.8) and the parabolic subgroup structure (41) at EPs provides one
more indication that the natural structure at EPs is connected with projective flags
10) A projective space CPn ∋ (z0, z1, . . . , zn) is covered by n + 1 affine charts Uk ∋
, . . . ,
, . . . ,
) with zk 6= 0 (see, e.g., [59, 61]) in straight forward dimensional
extension of the two-chart covering of the Riemann sphere CP1 mentioned in section 3.
11) For conics and quadrics in projective spaces see, e.g. [59, 62, 68].
Projective Hilbert space structures 13
[63]. A study of Jordan chain related flag bundles and the mappings between their
complementary affine charts will be presented in [64].
Returning to the ε → 0 limit in (49) we see that
→ −1 =⇒ c+
=⇒ Φ+ → ±iΦ− , (58)
i.e. the two eigenvectors (fiber sections) Φ+, Φ− are phase-shifted one relative
to the other by ±i when tending to their common coalescence line at ε = 0:
π(Φ+) = π(Φ−) = π(Φ0). We note that this relative ±i phase-shift of the vectors
Φ+, Φ− is generic for models in their instantaneous picture and with d± = c± and
normalization 〈Ξ±|Φ±〉 = 1.
A further result which immediately follows from (49) is the typical distance-
dependent phase jump behavior in the vicinity of the EP. In a sufficiently close
vicinity of an EP (|ε| ≪ 1) any sufficiently smooth trajectory in an underlying
parameter space can be roughly approximated by a straight line segment with an
effective parametrization of the type
ε = eiα0(ρ+ is), s ∈ [−s0, s0] ⊂ R (59)
where α0 =const fixes the direction orthogonal to the effective trajectory in the
complex ε−plane and ρ is the minimal distance ρ = |ε(s = 0)| of this trajectory
to the EP. The parameter along the path is s ∈ [−s0, s0] ⊂ R. This parametrization
gives:
[ε(s)]
= e−i
−iθ(s)|ε(s)|−1/4
|ε(s)| =
ρ2 + s2
θ(s) =
arctan(s/ρ) ∈ (−π/8, π/8) (60)
and we observe that the minimal distance ρ between the parameter trajectory and the
EP defines the smoothness of the phase changes. The closer the path approaches the
EP the more it will take the form of a Heaviside step function with jump height π/4:
θ(s; ρ → 0) → π
Θ(s)− 1
. (61)
The phase jump behavior can be used as implicit indicator of a possible close location
of an EP — a fact especially useful in numerical studies of systems with complicated
parameter dependence, but where phases of eigenvectors can be easily extracted.
Jumps ±π/4 of wave function phases have been observed numerically in [69] for the
model Hamiltonian (1) and in [14] for the special case of a small quantum billiard.
According to these results, the phases of the components change smoothly (as a
function of a certain control parameter) in approaching the EP and jump by π/4
at the smallest distance from this point. Other phase jump values are possible, but
require especially tuned paths.
6. Phase rigidity
In numerical studies of man-made open quantum systems depending in a complicated
way on several parameters X = (X1, . . . , Xm) ∈ M it is usually important to know
how close a given configuration is located to an EP. EPs dominate the system behavior
Projective Hilbert space structures 14
also in their vicinities, spectral bands may merge at EPs [51] or the transmission
properties of quantum dots (QDs) may become optimal at EPs [70]. A measure for
the distance between a given point in parameter space and a closely located EP would
provide a convenient tool for adjusting and tuning parameters so that a system may
be ’moved’ in parameter space toward to or away from this EP.
In [70] it has been shown numerically that within the instantaneous picture
(Φ = Ξ∗) an appropriate measure for the detection of EP vicinities is the fraction
〈Φ|Φ〉
. (62)
We note that originally similar fractions have been introduced in [71] to describe the
transitions between Hamiltonian ensembles with orthogonal and unitary symmetry
in Hermitian quantum chaotic systems. There the square modulus |r|2 was dubbed
”phase rigidity”. In our considerations of non-Hermitian systems we use this term in
loose analogy for r itself.
Decomposing Φ into real and complex components Φ = Φr + iΦi we find from
the normalization that
ΦTΦ = 1 = ΦTr Φr − ΦTi Φi , ΦTr Φi = 0 (63)
and12) hence that the norm is bounded below
||Φ||2 = 〈Φ|Φ〉 = ΦTr Φr +ΦTi Φi = 2ΦTi Φi + 1 ≥ 1 . (64)
The phase rigidity can be expressed as
||Φ||2
∈ [0, 1] (65)
where according to (50) for the EP-limit ε → 0 holds
r ≈ |2ε|1/2 → 0 . (66)
The opposite limit r → 1 is reached when the channel coupling ω in the Hamiltonian
(1) vanishes, i.e. when the interaction between the two decaying resonance states
tends to zero and any eigenvector can be taken purely real-valued in the instantaneous
picture.
Finally, we note that for certain quantum dot systems the phase rigidity r is
closely related to the transmission properties of these systems. The capability of
corresponding numerical studies (including the visualizations of transmission and
phase rigidity ’landscapes’ over parameter space) has been recently demonstrated in
[70].
7. PT −symmetric models
Toy model Hamiltonians of 2× 2−matrix type have been often used as test ground in
PT −symmetrically extended Quantum Mechanics (PTSQM) [33, 34, 35]. They can
be obtained from non-Hermitian complex symmetric 2 × 2−matrix Hamiltonians by
12) In equation (63) it can be set ΦTr Φr =: cosh
2 β and ΦTi Φi =: sinh
2 β. This hyperbolic structure
shows analogies with the mass shell condition E2 − p2 = m2 of special relativity. The EP-limit
ΦTr Φr,Φ
i Φi → ∞ corresponds, e.g., to the light-cone limit where the vectors become isotropic — a
fact which seems to play an important role in connection with the conjectured Hilbert space worm
holes [34] related to the brachistochrone problem of PT −symmetric Quantum Mechanics (PTSQM).
Projective Hilbert space structures 15
imposing a PT −symmetry constraint. In a suitable parametrization they have the
reiθ s
s re−iθ
, r, s, θ ∈ R (67)
and commute with the operator PT
[PT , H ] = 0, P =
. (68)
Here, P is the parity reflection operator and T — the time inversion (acting as complex
conjugation). The eigenvalues of H are
E± = r cos(θ)±
s2 − r2 sin2(θ) (69)
and the corresponding eigenvectors can be represented as [35]
|E+〉 =
eiα/2
2 cos(α)
=: c+χ+
|E−〉 =
ie−iα/2
2 cos(α)
=: c−χ− (70)
where
sin(α) =
sin(θ) . (71)
With regard to the indefinite (Krein space type [38]) PT inner product (u, v) = PT u·v
the vectors are normalized as
(E±, E±) = ±1, (E±, E∓) = 0 . (72)
The indefinite PT inner product is then mapped by the dynamical operator C with
[C, H ] = 0 and
C = 1
cos(α)
i sin(α) 1
1 −i sin(α)
(see, e.g., [35]) into the positive definite (Hilbert space type) CPT inner product
((u, v)) = CPT u · v which yields
((E±, E±)) = 1, ((E±, E∓)) = 0 . (74)
Let us now reshape the model in terms of the EP-relevant notations of section 2.
A simple comparison of (1), (3) with (67), (71) shows that
Z = i sin(α) (75)
and, hence, that
C = 1
cos(α)
and that the model is actually one-parametric with essential parameter Z. Together
with (2) the parametrization (76) leads to a representation of the Hamiltonian (67) as
H = E0I2 + s cos(α)C , E0 = r cos(θ) (77)
and [C, H ] = 0 is fulfilled trivially.
The compatibility of the PT and the CPT inner products (72), (74) with the bi-
orthogonality relations (7) is ensured by the fact that for an eigenvector Φ = c(1, b)T
Projective Hilbert space structures 16
exact PT symmetry requires PT Φ ∝ Φ and, hence, c∗b∗(1, 1/b∗)T ∝ c(1, b)T so that
|b|2 = 1. For such vectors it holds PT Φ ∝ Ξ∗ and due to the dynamically tuned
C also CPT Φ ∝ Ξ∗. As result one finds CPT Φk · Φl ∝ PT Φk · Φl ∝ Ξ+k Φl and
full compatibility of the bi-orthogonality with the PT and CPT inner products is
established.
From (75) we see that possible EPs are solely defined by the value of α. From
Zc = ±i we find the corresponding critical αc as
αc = ±π/2 + 2Nπ, N ∈ Z . (78)
Furthermore, it follows from (71) that a purely Hermitian model with θ = nπ,
n ∈ Z corresponds to α = Nπ, N ∈ Z. Exact PT −symmetry is preserved for
α ∈ R− {π/2+ πZ}, and the corresponding models are parameterized by elements Z
belonging to the purely imaginary straight line segment connecting the two EPs, i.e.
by Z ∈ (−i, i), ReZ = 0.
According to (70), at the EPs the eigenvectors lay on the same line π(|E+〉) =
π(|E−〉) ≈ χ0 = (1, Zc)T and their norms diverge for α → αc like
|||E±〉||2 = 〈E±|E±〉 ≈
| cos(α)|
→ ∞ . (79)
The operator C in (73) shows the same singular behavior, i.e. the C−induced mapping
between the Krein space and the Hilbert space breaks down at the EPs. In analogy to
the singularity resolution presented in section 5 we may map the vectors |E±〉 ∈ C2
into elements from the affine chart U2 ⊂ CP2 corresponding to points e± ∈ CP2 with
homogeneous coordinates
|E±〉 7→ e± =
χT±, c
. (80)
The original normalization via PT inner product PT |E±〉 · |E±〉 = 1 acts then as
generalized conic
PT χ± · χ± −
T c−1±
c−1± = 0 (81)
which remains regular in the EP-limit α → αc, but shows the typical EP-related self-
orthogonality (isotropy) of the lines PT χ± ·χ± → 0. Again we arrive at the conclusion
that the original Hilbert space H = C2 should be projectively embedded into CP2 in
order to accommodate EP-related singularities.
Finally, we note that the recently uncovered solutions of the PT −symmetric
brachistochrone problem with vanishing optimal passage time [34] occurs for α = π/2
what according to (78) can be identified as an EP-regime13). This fact appears
compatible with the results of [57] where a vanishing passage time was reported for
arbitrary non-Hermitian Hamiltonians. In this regard it is natural to conjecture that a
vanishing optimal passage time might be a generic EP-related feature of non-Hermitian
systems not necessarily restricted to PTSQM models.
8. Conclusion
In the present paper we considered projective Hilbert space structures in the
vicinity of EPs. Starting from a leading-order Puiseux-expansion of the bi-
orthogonal eigenvectors of a non-Hermitian (complex symmetric) diagonalizable
2×2−matrix Hamiltonian in terms of root vectors (algebraic eigenvectors) at an EP the
13) The corresponding state vector alinement without link to EPs was observed also in [72].
Projective Hilbert space structures 17
normalization divergency of the eigenvectors in the EP-limit has been parameterized.
It has been shown that the natural projective line structure related to the eigenvectors
of the diagonal Hamiltonian has to be replaced at an EP by a higher dimensional
projective structure in which all the root vectors of a Jordan block scale simultaneously
with the same single factor. For a simplified setup with left eigenvectors equated to
their complex conjugate right counterparts, the normalization divergency has been
resolved by embedding the original Hilbert space H = C2 into its projective extension
H →֒ CP2. Eigenvectors normalized according to the diagonalizable Hamiltonian and
eigenvectors with a normalization inherited from the root vector normalization live
then merely in different (complementary) affine charts of CP2. The states themselves
can be interpreted as conics in CP2. The line structure of the states behaves smoothly
and independently of these charts and their possibly singular transition functions. This
indicates on the possibility of a technically efficient description of the global behavior
of the non-Hermitian system by factoring the eigenvectors in globally smoothly varying
non-singular projective line components π(Φk) and possibly diverging scale factors
With the help of the Puiseux expanded eigenvectors it has been shown that the
geometric phase obtained on circles around EPs of complex symmetric Hamiltonians
is purely real-valued and that the corresponding monodromy transformations are
induced by an Abelian parabolic subgroup of SL(2,C). Furthermore the Puiseux
expansion has been used to explain phase jumps which in prior work had been
numerically observed in the vicinity of EPs. An analytical foundation for the usefulness
of the phase rigidity as distance measure to EPs has been provided. Finally, a
PT −symmetric model has been studied. It has been shown that the EP-related
singularities show up not only in the normalization conditions of the eigenvectors
but also in the dynamical symmetry operator C. The normalization singularity has
been resolved via a projective extension of the original Hilbert space. From the
singularity structure it has been conjectured that the zero passage time effect in
the brachistochrone problem of non-Hermitian Hamiltonians might be a generic EP-
related artifact.
Acknowledgement.
We thank Hugh Jones and Andreas Fring for useful comments on [34, 57]. This
work has been supported by the German Research Foundation DFG, grant GE 682/12-
3 (U.G.) and by the grants RFBR-06-02-16719, SS-5103.2006.2 (B.F.S.).
Appendix A. Jordan normal forms for complex symmetric 2× 2 matrices
At the EPs with Zc = ±i =: µi the matrix
H(Zc)− E0I2 = ω
1 −Zc
=: M (A.1)
is related to its Jordan normal form J2(0) =
by a similarity transformation
M = PRJ2(0)R
−1P−1 . (A.2)
14) The question concerning the physical interpretation of diverging or non-diverging normalizations
and the corresponding diverging or non-diverging sensitivity in perturbation expansions is highly
model dependent (see e.g. [51, 52]) and still requires a detailed investigation.
Projective Hilbert space structures 18
From the symmetry properties
M = MT , J2(0) = S2J
2 (0)S2 (A.3)
with S2 =
and P 2 = S2 one finds
1 −iµ
−iµ 1
, P = PT = (P−1)+ (A.4)
0 q−1
, q :=
2ω . (A.5)
The elementary Jordan block J2(0) has right and left root vectors Θ0,Θ1 and Ψ0,Ψ1
satisfying
J2(0)Θ0 = 0, J2(0)Θ1 = Θ0
JT2 (0)Ψ0 = 0, J
2 (0)Ψ1 = Ψ0 . (A.6)
The explicit solutions of these Jordan chains can be arranged as Toeplitz and Hankel
matrices
Θ = [Θ0,Θ1] =
c0 c1
, Ψ = [Ψ0,Ψ1] =
0 d∗0
d∗0 d
(A.7)
Ψ̃ := ΨS2 =
d∗0 0
d∗1 d
. (A.8)
From the simplest realization of the bi-orthonormality condition Ψ+Θ = S2, Ψ̃
I2 one finds the parameters c1 = d1 = 0, d0c0 = 1. Via similarity transformations
Θ0,1 7→ Φ0,1 = PRΘ0,1 and Ψ0,1 7→ Ξ0,1 = P
Ψ0,1 one arrives at the root
vectors of M
Φ0 = σqc0
, Φ1 = σq
Ξ0 = σq
, Ξ1 = σq
∗−1d∗0
. (A.9)
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