Krein Collisions in Hamiltonian Systems

• Krein collisions are spectral interactions in Hamiltonian systems where neutral modes with opposite signatures meet, often triggering instability as eigenvalues depart from the imaginary axis.
• They are studied using frameworks ranging from discrete-spectrum Hamiltonian models to PT-symmetric and Vlasov–Poisson systems, each offering unique diagnostic tools.
• Applications in plasma physics and warm two-stream problems illustrate how resonance between positive- and negative-action modes defines stability boundaries and bifurcation outcomes.

Krein collisions are spectral collisions of neutral modes distinguished by a signature, and they occupy a central role in stability theory for Hamiltonian and Hamiltonian-like systems. In the discrete-spectrum Hamiltonian setting, the relevant modes lie on the imaginary axis and carry a Krein signature, i.e. the sign of the quadratic Hamiltonian restricted to the corresponding eigenvector, equivalently the sign of its energy; a collision of opposite-sign modes is the necessary precursor for eigenvalues to leave the imaginary axis and form unstable quartets (Hagstrom et al., 2010). The same qualitative mechanism persists, with substantial reformulation, in noncanonical Hamiltonian systems with continuous spectrum, in complex $G$-Hamiltonian reductions of plasma models, and in non-Hermitian $\mathcal{PT}$-symmetric spectral problems where the signature must be defined through adjoint eigenvectors rather than a Hamiltonian quadratic form (Chernyavsky et al., 2017, Zhang et al., 2016, Hagstrom et al., 2010).

1. Discrete-spectrum Hamiltonian mechanism

For a linear Hamiltonian system with a quadratic Hamiltonian, spectral stability means all eigenvalues lie on the imaginary axis, or equivalently in frequency variables all frequencies are real. In that case each eigenmode carries a Krein signature, and the classical Krein–Moser theorem quoted for this setting states that a stable finite-dimensional linear Hamiltonian system is structurally stable if all eigenfrequencies are nondegenerate, while degeneracies are structurally stable only when the colliding eigenmodes have the same energy sign; if opposite signs meet, the system is structurally unstable (Hagstrom et al., 2010).

In this sense, a Krein collision is the collision of neutral modes of opposite signature. The standard spectral consequence is that two purely imaginary eigenvalues meet, lose definiteness, and then leave the imaginary axis. In Hamiltonian bifurcation language this is the Hamiltonian-Hopf mechanism: purely imaginary eigenvalues of opposite Krein signature collide and generically move off the imaginary axis, producing eigenvalues with nonzero real part (Kapitula et al., 2019). The basic dichotomy is therefore sharp: same-sign collisions generically remain on the imaginary axis, whereas opposite-sign collisions are the destabilizing configuration (Kapitula et al., 2019).

The signature itself depends on the operator class. For first-order Hamiltonian problems, the negative Krein index on the imaginary axis reduces to the sign of the self-adjoint operator $A_0$ restricted to the eigenspace; for quadratic pencils it becomes the sign of $A_0+\lambda_0^2A_2$ on the eigenspace (Kapitula et al., 2019). In finite-dimensional complex $G$-Hamiltonian systems, the signature is read from an indefinite Hermitian form rather than a positive-definite inner product, but the role of opposite-sign collisions is unchanged (Zhang et al., 2016).

2. Signature as energy, action, or adjoint pairing

Across the literature, the signature attached to a neutral spectral component is not represented by a single universal formula. What is preserved is the role of the sign, not the specific bilinear form from which it is computed.

In a complex $G$-Hamiltonian system

$\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$

with $G$ nonsingular Hermitian and $S$ Hermitian, the defining algebraic relation is

$A^*G+GA=0,$

and the indefinite $\mathcal{PT}$0-product is

$\mathcal{PT}$1

For a simple eigenvalue $\mathcal{PT}$2 on the imaginary axis with eigenvector $\mathcal{PT}$3, the sign of $\mathcal{PT}$4 is the Krein signature. If $\mathcal{PT}$5, then

$\mathcal{PT}$6

where

$\mathcal{PT}$7

This identifies the signature with the sign of the action $\mathcal{PT}$8, and the two-stream analysis therefore states the instability criterion as resonance between a positive-action mode and a negative-action mode (Zhang et al., 2016).

In the $\mathcal{PT}$9-symmetric nonlinear Schrödinger setting, the non-Hermitian structure changes the definition fundamentally. For a simple isolated eigenvalue $A_0$0 of the spectral and adjoint spectral problems, with eigenvector $A_0$1 and adjoint eigenvector $A_0$2, the Krein quantity is

$A_0$3

and the Krein signature is the sign of this quantity (Chernyavsky et al., 2017). This is a structural departure from the Hamiltonian case: unless $A_0$4 or $A_0$5, the adjoint eigenvector is not directly related to the eigenvector, so the signature cannot be read off from the eigenvector alone (Chernyavsky et al., 2017).

For continuous-spectrum Vlasov–Poisson, the signature is assigned pointwise along the continuum. After action-angle diagonalization, the analog of Krein signature is determined by

$A_0$6

so the sign of $A_0$7 gives the sign of the continuum energy density at label $A_0$8 (Hagstrom et al., 2010). For embedded discrete modes, the signature is instead given by

$A_0$9

where $A_0+\lambda_0^2A_2$0 is the real part of the dielectric function on the real axis (Hagstrom et al., 2010).

3. $A_0+\lambda_0^2A_2$1-symmetric Krein collisions

The paper on $A_0+\lambda_0^2A_2$2-symmetric states develops a Krein-signature theory for stationary states of the one-dimensional $A_0+\lambda_0^2A_2$3-symmetric nonlinear Schrödinger equation

$A_0+\lambda_0^2A_2$4

with

$A_0+\lambda_0^2A_2$5

and stationary state

$A_0+\lambda_0^2A_2$6

Linearization produces the spectral problem

$A_0+\lambda_0^2A_2$7

with

$A_0+\lambda_0^2A_2$8

Its central concern is the mechanism by which two isolated imaginary eigenvalues of the linearization collide and either remain on the imaginary axis or bifurcate into a complex quartet (Chernyavsky et al., 2017).

For simple isolated nonzero imaginary eigenvalues, both eigenvector and adjoint eigenvector can be normalized to be $A_0+\lambda_0^2A_2$9-symmetric, and this symmetry implies that the Krein quantity is real. Lemma 4 states three basic facts: $G$0 is real if $G$1; $G$2 if $G$3; and $G$4 if $G$5 (Chernyavsky et al., 2017). Thus a simple isolated neutral eigenvalue has a nonvanishing real Krein quantity, while non-imaginary eigenvalues have vanishing Krein quantity.

At a collision point $G$6, two simple isolated eigenvalues $G$7 may coalesce into a defective eigenvalue

$G$8

of geometric multiplicity one and algebraic multiplicity two. With the Jordan-chain assumptions

$G$9

and the corresponding adjoint relations, the solvability condition implies

$G$0

The defective collision is therefore marked by vanishing Krein quantity (Chernyavsky et al., 2017).

The main theorem states that, under assumptions (A1)–(A3), the defective-eigenvalue assumption, and the nondegeneracy condition

$G$1

there exists $G$2 such that for $G$3, $G$4 small, there are two simple eigenvalues $G$5. On one side of $G$6, these eigenvalues remain on the imaginary axis and satisfy

$G$7

while on the other side they leave $G$8 (Chernyavsky et al., 2017). Opposite Krein signatures are therefore a necessary condition for instability-producing collisions in this non-Hermitian setting.

The paper is explicit that this condition is necessary, not sufficient. If the nondegeneracy coefficient

$G$9

vanishes, then “the perturbation theory must be extended to the next order,” and the defective eigenvalue may split along $\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$0 on both sides of the bifurcation point (Chernyavsky et al., 2017). This is the paper’s main caveat against interpreting opposite signatures as a sufficient instability criterion.

4. Continuous-spectrum analogues in Vlasov–Poisson

For the linearized Vlasov–Poisson equation about a homogeneous equilibrium $\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$1, the relevant time-evolution operator is

$\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$2

and in the simplified notation often used in the paper,

$\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$3

The operator is noncanonical Hamiltonian and has a continuous spectrum filling the imaginary axis, except for possible missing points corresponding to embedded point modes (Hagstrom et al., 2010). Consequently, the phrase “two eigenvalues collide” is no longer sufficient; one must formulate a Krein theory for the continuum.

The dispersion relation is

$\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$4

and spectral instability occurs if and only if there exists $\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$5 in the upper half-plane with $\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$6 (Hagstrom et al., 2010). On the real axis, the Penrose boundary values are expressed through the Hilbert transform: $\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$7 This yields the Penrose contour

$\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$8

or equivalently

$\dot{\mathbf{x}}=A\mathbf{x}, \qquad A=iG^{-1}S,$9

whose winding number around the origin counts unstable roots (Hagstrom et al., 2010).

The continuous-spectrum analogue of Krein signature is defined pointwise. Suppose $G$0. Then the signature of the point $G$1 is

$G$2

If $G$3, the continuum at that $G$4 is positive signature; if $G$5, it is negative signature (Hagstrom et al., 2010). This replaces the discrete assignment of a sign to each isolated eigenvector by a sign attached to each continuum label.

The paper identifies two continuous-spectrum analogues of the classical Krein collision. The first is a change of signature along the continuous spectrum. The second is the interaction of a discrete mode embedded in the continuous spectrum with continuum of opposite signature (Hagstrom et al., 2010). These are the distributed and embedded-mode versions of the classical opposite-sign collision.

The structural conclusions depend sharply on the perturbation class. Under unrestricted perturbations of the equilibrium in $G$6 and $G$7, every stable equilibrium is structurally unstable, because one can construct arbitrarily small perturbations whose Hilbert transform is order one at a targeted zero of $G$8, thereby changing the Penrose winding number (Hagstrom et al., 2010). Under dynamically accessible perturbations, however, the paper proves a Krein-like theorem: a stable equilibrium is structurally stable if there is only one solution of $G$9, whereas if there are multiple solutions, the equilibrium is structurally unstable and the unstable modes come from the zeros of $S$0 that satisfy $S$1 (Hagstrom et al., 2010). In signature language, single-sign continuum signature implies structural stability under dynamically accessible perturbations, while signature change implies arbitrarily nearby unstable rearrangements.

For embedded discrete modes, the paper states that if a discrete mode embedded in the continuous spectrum is surrounded by the opposite signature there is an infinitesimal perturbation in $S$2 norm that makes $S$3 unstable (Hagstrom et al., 2010). It also proves the converse-type statement that if $S$4 is stable there are no discrete modes with signature the same as the signature of the continuum (Hagstrom et al., 2010). Relative signature therefore remains decisive even when one spectral component is continuous.

5. Positive- and negative-action resonance in the warm two-stream problem

The warm two-stream instability is analyzed through a one-dimensional warm two-fluid model which, after taking a single Fourier mode $S$5, reduces to a finite-dimensional complex ODE

$S$6

with an explicit $S$7 complex matrix $S$8 (Zhang et al., 2016). The exact dispersion relation is

$S$9

The striking feature emphasized in the paper is the band structure of the instability diagram in the warm case: the unstable region lies between two boundaries, rather than below a single threshold as in the cold model (Zhang et al., 2016).

The structural explanation is the complex $A^*G+GA=0,$0-Hamiltonian form

$A^*G+GA=0,$1

with explicit Hermitian matrices $A^*G+GA=0,$2 and $A^*G+GA=0,$3, so that

$A^*G+GA=0,$4

A quoted theorem states that the eigenvalues of a $A^*G+GA=0,$5-Hamiltonian matrix are symmetric with respect to the imaginary axis, and because $A^*G+GA=0,$6, the eigenfrequencies $A^*G+GA=0,$7 are symmetric with respect to the real axis (Zhang et al., 2016). Another theorem quoted in the paper is the Krein–Gel'fand–Lidskii theorem: the $A^*G+GA=0,$8-Hamiltonian system is strongly stable if and only if all eigenvalues of $A^*G+GA=0,$9 lie on the imaginary axis and are definite (Zhang et al., 2016).

In this framework, a Krein collision is the collision, on the imaginary axis in $\mathcal{PT}$00-space, of eigenvalues with opposite Krein signatures; in frequency language it is a collision of real eigenfrequencies whose corresponding modes have opposite action (Zhang et al., 2016). The paper states the resulting physical criterion in explicit form: the system becomes unstable when and only when a positive-action mode resonates with a negative-action mode (Zhang et al., 2016). It also stresses that it is not accurate to say simply that instability occurs when a negative-energy mode resonates with a positive-energy mode, because the rigorous quantity is the sign of $\mathcal{PT}$01, not the sign of $\mathcal{PT}$02 alone (Zhang et al., 2016).

The band structure is then interpreted as a region bounded by two distinct Krein collisions. For the scan with

$\mathcal{PT}$03

the system is stable at $\mathcal{PT}$04; at

$\mathcal{PT}$05

a positive-action mode collides with a negative-action mode, marking the upper instability boundary; at

$\mathcal{PT}$06

another collision marks the lower instability boundary, after which the spectrum returns to the imaginary axis and the system is stable again (Zhang et al., 2016). The same interpretation applies to the scan in $\mathcal{PT}$07, with collisions at

$\mathcal{PT}$08

(Zhang et al., 2016). The instability band is therefore not an incidental root-geometry feature but the interval between two opposite-sign collisions.

6. Krein matrix, Hamiltonian-Krein index, and collision diagnostics

The reformulated Krein matrix provides a finite-dimensional meromorphic diagnostic for Krein collisions in star-even polynomial operator pencils

$\mathcal{PT}$09

with $\mathcal{PT}$10, where even coefficients are Hermitian and odd coefficients are skew-Hermitian, so that

$\mathcal{PT}$11

and the spectrum is symmetric with respect to the imaginary axis (Kapitula et al., 2019). The reformulation is designed to allow operator coefficients with nontrivial kernel and to handle quadratic star-even operators directly (Kapitula et al., 2019).

Given a finite-dimensional subspace $\mathcal{PT}$12 with orthonormal basis $\mathcal{PT}$13, the projection-based Krein matrix is

$\mathcal{PT}$14

For purely imaginary spectral parameter $\mathcal{PT}$15, the paper redefines it by multiplying by $\mathcal{PT}$16: $\mathcal{PT}$17 Its scalar eigenvalues $\mathcal{PT}$18 are the Krein eigenvalues, and

$\mathcal{PT}$19

Zeros of $\mathcal{PT}$20 detect eigenvalues represented in $\mathcal{PT}$21, while poles correspond to eigenvalues lying in $\mathcal{PT}$22 (Kapitula et al., 2019).

The paper’s central graphical rule is that for a simple zero $\mathcal{PT}$23, the slope determines Krein signature: after the $\mathcal{PT}$24 normalization, positive slope means positive Krein signature and negative slope means negative Krein signature (Kapitula et al., 2019). This turns Krein signature into a property that can be read directly from a finite-dimensional Hermitian meromorphic matrix.

The Hamiltonian-Krein index is defined by

$\mathcal{PT}$25

where $\mathcal{PT}$26 is the total number of positive real polynomial eigenvalues, $\mathcal{PT}$27 the total number of polynomial eigenvalues with positive real part and nonzero imaginary part, and $\mathcal{PT}$28 the total number of purely imaginary eigenvalues with negative Krein signature (Kapitula et al., 2019). For first-order pencils,

$\mathcal{PT}$29

and for quadratic pencils,

$\mathcal{PT}$30

(Kapitula et al., 2019). In this framework, the Hamiltonian-Krein index gives the total count of potentially dangerous spectral objects, while the Krein matrix locates them and identifies their signatures.

A particularly concrete insight is the zero/pole interpretation of collisions. With the choice

$\mathcal{PT}$31

dangerous imaginary eigenvalues appear as zeros, while benign positive-signature ones may remain as poles (Kapitula et al., 2019). A Krein collision can therefore often be seen as a zero meeting a pole. In the periodic-wave application, when a simple zero coincides with a simple removable singularity, perturbation can convert this geometry into either two purely imaginary eigenvalues of opposite Krein signatures or a pair with nonzero real part, namely a Hamiltonian-Hopf bifurcation (Kapitula et al., 2019). The paper also distinguishes semisimple and nonsemisimple collisions: higher-order vanishing of Krein eigenvalues signals Jordan chains and higher-order degeneracy (Kapitula et al., 2019).

7. Scope, caveats, and recurrent misconceptions

A recurring theme across these works is that opposite signature is typically a necessary ingredient for destabilizing collisions, but not an unrestricted sufficiency statement. In the $\mathcal{PT}$32-symmetric nonlinear Schrödinger problem, opposite Krein signatures are necessary for instability bifurcation from a defective double eigenvalue, but without the nondegeneracy condition the eigenvalues may still split along $\mathcal{PT}$33 rather than form a complex quartet (Chernyavsky et al., 2017). The numerical examples explicitly include a true defective opposite-sign collision that does not produce instability, interpreted as failure of the nondegeneracy condition (Chernyavsky et al., 2017).

Another caveat concerns the nature of the collision. The $\mathcal{PT}$34-symmetric theory addresses nonzero isolated imaginary eigenvalues colliding into a defective double eigenvalue, and the paper explicitly states that collisions at the origin are outside the predictive scope of the Krein quantity developed there (Chernyavsky et al., 2017). It also notes that if the collision point is semisimple rather than defective, the reduced matrices are non-Hermitian and the familiar Hamiltonian conclusions do not transfer cleanly (Chernyavsky et al., 2017).

For Vlasov–Poisson, the perturbation class is decisive. Under unrestricted $\mathcal{PT}$35 perturbations, every equilibrium is structurally unstable, so no informative Krein theorem survives; the meaningful signature-based statement is recovered only after restricting to dynamically accessible perturbations, i.e. area-preserving rearrangements (Hagstrom et al., 2010). This shows that in continuous-spectrum problems, “opposite-sign interaction” is inseparable from the admissible perturbation geometry.

The plasma two-stream analysis adds a different correction to common language. The destabilizing resonance is rigorously between positive-action and negative-action modes, not merely between positive- and negative-energy modes (Zhang et al., 2016). This distinction matters because the signature is the sign of $\mathcal{PT}$36, not of $\mathcal{PT}$37 alone.

Taken together, these results present Krein collisions not as a single formula but as a structural principle. In discrete Hamiltonian systems they are opposite-sign collisions of neutral modes; in $\mathcal{PT}$38-symmetric problems they are detected through an adjoint-based Krein quantity; in Vlasov–Poisson they appear as signature change in the continuum or interaction between an embedded mode and surrounding continuum; and in operator-pencil formulations they can be rendered as zero/pole interactions of a meromorphic Krein matrix (Chernyavsky et al., 2017, Hagstrom et al., 2010, Zhang et al., 2016, Kapitula et al., 2019).