Hamiltonian-Krein Index Analysis

• Hamiltonian-Krein index analysis is a rigorous framework that predicts and classifies spectral instabilities in Hamiltonian systems via the study of Krein signatures and eigencurve topology.
• It quantifies instability by counting eigenvalues with positive real parts using index theorems and practical computation tools like the Evans function and Krein matrix.
• The methodology integrates analytical and numerical approaches to assess stability in diverse applications such as plasma physics, fluid dynamics, and Bose-Einstein condensates.

Hamiltonian-Krein index analysis provides a comprehensive framework for predicting, classifying, and counting spectral instabilities in linearized Hamiltonian systems, operator pencils, and PDE eigenvalue problems. It establishes a rigorous correspondence between the spectrum of linearized systems—particularly the emergence of growth/decay due to unstable modes—and algebraic signatures associated with quadratic forms or eigencurve topology. The methodology has become central in stability investigations in plasma physics, fluid dynamics, atomic Bose-Einstein condensates, nonlinear wave PDEs, and related fields.

1. Theoretical Framework and Krein Signature

The central object of study is the linear Hamiltonian eigenvalue problem,

$J L u = \lambda u,$

where $J$ is a bounded, invertible, skew-adjoint (often symplectic) operator and $L$ is self-adjoint with compact resolvent. The spectrum of $JL$ is symmetric with quartets $\{\lambda, -\lambda, \bar\lambda, -\bar\lambda\}$. Spectral stability requires all eigenvalues to lie on the imaginary axis.

The Krein signature encodes the potential for structural instability of imaginary eigenvalues. For a simple eigenvalue $\lambda = i\omega$ ($\omega > 0$),

$$\text{Krein signature} = \mathrm{sign}\bigl( \langle L u, u \rangle \bigr),$$

or, equivalently,

$K(\lambda) = -i\lambda \langle J u, u \rangle,$

for (1) $J L u = \lambda u$, with self-adjoint $L$. This quantity is real for purely imaginary $\lambda$ and distinguishes positive- and negative-action modes, which is essential for understanding resonance-driven instabilities (1711.02191, Zhang et al., 2016).

2. Krein Collisions and Quartet Instability Mechanism

A fundamental theorem (Krein–Gelʹfand–Lidskii) asserts that for a G-Hamiltonian matrix, spectral and structural stability is maintained as long as all eigenvalues are definite on the imaginary axis. Instability arises only via "Krein collision"—the coalescence of two such eigenvalues of opposite signature. The generic result is the emergence of a quartet of complex eigenvalues, corresponding to growing and decaying modes. Denoting the participation parameters by $\mu$,

$$\lambda_1(\mu),\ \lambda_2(\mu) \in i\mathbb{R},\quad \sigma(\lambda_1)\sigma(\lambda_2) = -1$$

yield, post-collision ($\mu > \mu_c$),

$$\{\lambda, \overline\lambda, -\lambda, -\overline\lambda\},\quad \Re\lambda \neq 0$$

which is a generic route to Hamiltonian-Hopf bifurcation (Zhang et al., 2016, 1711.02191).

3. Index Theorems and Counting Instabilities

The Hamiltonian-Krein index (also termed instability index) quantifies how many eigenvalues with $\Re \lambda > 0$ exist: $K_{\rm Ham} = k_r + 2 k_c + 2 k_i^-$ where $k_r$ is the count of real eigenvalues with positive real part, $k_c$ the count of complex quartets, and $k_i^-$ the total negative Krein index of neutral modes. Alternatively, for operator pencils and systems with nontrivial kernel, index formulae relate this count to algebraic properties:

• For problems in canonical form,

$K_{\rm Ham} = n(L) - n(D)$

where $n(L)$ is the number of negative eigenvalues of $L$ and $n(D)$ the count for the finite-dimensional constraint matrix arising from the kernel of $L$ (Kollár et al., 2012, Kapitula et al., 2012, Kostenko et al., 2017).

• For quadratic pencils, e.g.,

$$K_H = n(A_0) + n(A_2) - n( [A_2 - A_1 A_0^{-1} A_1] |_{\ker{A_0}} )$$

(Kapitula et al., 2019).

4. Graphical and Algebraic Krein Signature Computation

The graphical Krein signature assigns $\pm 1$ to crossings of eigencurves $\mu(\lambda)$ at characteristic values: $$\kappa(\lambda_0) = \mathrm{sign}( \mu'(\lambda_0) )$$ Higher-order tangencies and multi-dimensional Jordan blocks can be treated analytically through sequences of derivatives and Gram forms on root-vector chains, using

$W_{ij} = \langle u^{[i]}, K u^{[j]} \rangle$

for canonical Jordan chains $\{u^{[0]}, \ldots, u^{[m-1]}\}$ (Kollár et al., 2012, Kollár et al., 2012). This graphical approach is functionally equivalent to algebraic methods in Krein space and generalizes naturally to polynomial pencils and systems with complicated kernel structure.

The Evans-Krein function, an analytic determinant tracking both eigenvalues and signatures, provides a computationally efficient and robust route for numerical index assessment (Kollár et al., 2012).

5. Krein Matrix Methodology and Practical Applications

The Krein matrix $K(z)$ is a meromorphic, finite-dimensional matrix pencil constructed by projection onto the negative subspace (including kernel) of the underlying self-adjoint operator. Its determinant vanishes at the nonzero eigenvalues of the full problem. Signatures are classified by the slope of Krein eigenvalues $r_j(z)$, with negative slope indicating negative Krein signature modes: $$K(z) c = 0,\quad \text{with}\quad r_j'(z_0) > 0\ \Rightarrow\ \text{negative signature}$$ This reduction enables localization of both unstable and negative-signature neutral modes without full diagonalization of infinite-dimensional linearizations (Kapitula et al., 2012, Kapitula et al., 2019).

Applications include:

• Stability analysis of dark-soliton and multi-vortex states in Bose-Einstein condensates, where the Krein matrix directly isolates instability boundaries and the emergence of quartet eigenvalues (Kapitula et al., 2012).
• Small-amplitude periodic waves and n-pulse problems in nonlinear Hamiltonian PDEs (e.g., fifth-order KdV equations, suspension-bridge models), where the Krein matrix provides explicit asymptotic formulae for the location and signature of small or near-zero eigenvalues (Kapitula et al., 2019).

6. Implementation and Extensions

Algorithmic implementation proceeds via:

• Construction of the variational structure (identifying $J$ and $L$),
• Computation of simple imaginary eigenvalues and evaluation of the Krein signatures via quadratic forms,
• Use of the Krein matrix or graphical signature theory for numerics.

Special attention is required when dealing with operator pencils with singular or unbounded coefficients; e.g., KdV-type problems where the skew-adjoint operator $J = \partial_x$ lacks a bounded inverse are treated by regularization and sandwiching techniques (Kapitula et al., 2012, Kapitula et al., 2019).

Extensions to PT-symmetric systems require adjoint problems and modified signature criteria, but the essential methodology—classification by signature, index tracking, quartet emergence—remains unchanged, with natural generalization (1711.02191).

7. Interplay with Spectral Flow, Maslov/Evans Indices, and Global Criteria

Spectral flow, via the Evans function or Maslov index, can be interpreted directly in terms of the count of Krein signature changes: crossings of eigencurve branches through zero or coalescences of opposite-signature modes map to topological index jumps (Kollár et al., 2012, Kollár et al., 2012). This provides unified proofs of classical criteria such as Vakhitov–Kolokolov, Grillakis–Shatah–Strauss, and links the algebraic index to global geometric invariants in variational PDE settings.

The conclusion is that Hamiltonian-Krein index analysis offers a unified and predictive framework for modal instability: instability can occur only via the resonance of modes with opposite Krein signatures, and the number and structure of unstable modes are fixed entirely by an algebraic count associated with constrained quadratic forms and eigencurve topology. The machinery generalizes to complex, infinite-dimensional, and non-self-adjoint systems with appropriately modified index theorems and computational tools (Zhang et al., 2016, Kostenko et al., 2017, Kollár et al., 2012, Kapitula et al., 2012, Kollár et al., 2012, Kapitula et al., 2012, Kapitula et al., 2019).