Maslov Crossing Forms in Spectral Theory

• Maslov Crossing Forms are key constructs in symplectic geometry that measure intersections of Lagrangian subspaces and underpin the calculation of the Maslov index.
• They enable rigorous computation of spectral flow and eigenvalue derivatives in Hamiltonian PDEs, Schrödinger operators, and boundary perturbation scenarios.
• Applications include stability analysis of nonlinear waves and domain deformations, linking local crossing signatures to global spectral invariants.

The Maslov crossing form is a fundamental mathematical construct in symplectic geometry, particularly in spectral theory, the analysis of Hamiltonian systems, and index theory. It provides a precise mechanism for quantifying the local behavior of intersections between Lagrangian subspaces along a path in the Lagrangian Grassmannian. The Maslov crossing form underlies the calculation of the Maslov index, a topological invariant that encodes spectral and dynamical information in Hamiltonian partial differential equations (PDEs), Schrödinger operators, and stability problems for nonlinear waves. Its algebraic structure and analytic properties enable both local and global spectral information to be systematically computed, including in the presence of degeneracies.

1. Definition and Algebraic Structure

Let $(\R^{2n},\omega)$ denote a finite-dimensional real symplectic vector space with standard symplectic form

$$\omega(u,v) = \langle J\,u, v \rangle, \qquad J = \begin{pmatrix}0 & I_n \ -I_n & 0 \end{pmatrix}.$$

A $n$-dimensional subspace $\Lambda \subset \R^{2n}$ is called Lagrangian if $\omega|_{\Lambda \times \Lambda} = 0$.

Given a smooth path $\Lambda:[a,b]\to L(n)$ of Lagrangian subspaces and a fixed reference Lagrangian $V$, a crossing at $t_0\in [a,b]$ occurs if $\dim(\Lambda(t_0)\cap V)>0$.

Near a regular (transverse) crossing, $\Lambda(t)$ can locally be written as $\{w + R_t w : w \in \Lambda(t_0)\}$, with $R_{t_0}=0$ and $R_t$ mapping into a complementary Lagrangian. The first-order Maslov crossing form at $t_0$ is the symmetric bilinear form defined (on $w_0\in \Lambda(t_0)\cap V$) as

$$\mathfrak m_{t_0}(w_0) = \omega(Z'(t_0)h_0, Z(t_0)h_0) = \langle Z(t_0)^T J Z'(t_0) h_0, h_0 \rangle,$$

where $Z(t)$ is a frame for $\Lambda(t)$ and $w_0=Z(t_0)h_0$. Regularity is characterized by nondegeneracy of this quadratic form (Curran et al., 2024, Jones et al., 2015, Waterstraat, 2014, Cox et al., 2022).

2. Higher-Order Crossing Forms and Degenerate Crossings

When degeneracies occur—that is, when the first-order crossing form vanishes identically—the structure of the crossing is captured by higher-order forms following the Piccione–Tausk and Giambò–Portaluri framework. For $k\geq 2$, the $k$-th order crossing form is

$$\mathfrak m^{(k)}_{t_0}(w_0) = \omega(w^{(k)}(t_0), w_0) = \langle J\,w^{(k)}(t_0), w_0\rangle,$$

where $w(t)$ is a $V$-root function with $w(t_0) = w_0$ and $w^{(i)}(t_0)\in V$ for $i<k$. The residual subspaces $W_k$ describe the hierarchy of kernel spaces as

$$W_1 = V\cap\Lambda(t_0), \quad W_{k+1}\subset W_k = \ker\mathfrak m^{(k-1)}_{t_0}.$$

The contribution of a degenerate (non-regular) crossing to the Maslov index is governed by the lowest-order non-vanishing odd crossing form and its signature (Curran et al., 2024, Cox et al., 2022).

3. Computational Frameworks and Applications

The Maslov crossing form facilitates explicit computations in a variety of spectral and dynamical settings, including:

• Spectral flows and eigenvalue counts: For Hamiltonian PDEs, Schrödinger operators, and boundary-value problems, regular and non-regular crossings correspond to changes in the spectrum as parameters vary (e.g., spatial location, boundary phase, domain deformation).
• Spectral instability criteria: Using a homotopy (“Maslov box”) argument, the signature data from crossing forms directly yield lower bounds for the number of unstable eigenvalues and furnish Vakhitov–Kolokolov-type stability criteria for nonlinear wave solutions. Specifically, the signature of the higher-order crossing form at a degenerate crossing determines corrections to the Maslov index and, thereby, spectral flow (Curran et al., 2024, Cox et al., 2022).
• Derivative formulas for eigenvalues: In the setting of $\theta$-periodic Schrödinger operators, the Maslov crossing form evaluates to the derivative of eigenvalues with respect to the boundary phase parameter, producing explicit analytic perturbation formulas based on boundary traces and the symplectic form (Jones et al., 2015, Latushkin et al., 2016).

A summary table of key settings and crossing form roles:

Problem Type	Crossing Form Role	Relevant Reference
Fourth-order NLS spectral stability	High-order forms compute spectral instability and index corrections	(Curran et al., 2024)
$\theta$-periodic Schrödinger eigenvalue flow	First-order form yields $\frac{d\lambda_j}{d\theta}$	(Jones et al., 2015)
Shape perturbation of domains	Maslov crossing form gives Hadamard-type boundary variation formula	(Latushkin et al., 2016)
Homoclinic Hamiltonian systems	Crossing form relates spectral flow to Maslov index	(Waterstraat, 2014)

4. Signature, Spectral Flow, and the Maslov Index

The signature $\operatorname{sign}\mathfrak m_{t_0}$ (number of positive minus negative squares) associated with the crossing form at regular crossings accumulates along a path as an integer-valued index, the Maslov index. This index measures the net number and orientation (directional flow) of eigenvalue crossings through a spectral level or parameter threshold. When non-regular (degenerate) crossings are present, the signature of the lowest non-vanishing odd order form is used, as specified by the generalization due to Piccione–Tausk (Curran et al., 2024, Cox et al., 2022).

In operator-theoretic settings, the crossing form

$$\Gamma(A, x_0)[u] = (\dot A_{x_0} u, u)_H, \quad u \in \ker A_{x_0}$$

provides a direct relation between the spectral flow and the Maslov index via

$$\mathrm{sf}(A) = \sum_{x_0} \operatorname{sign} \Gamma(A, x_0)$$

where the sum is over all regular crossings (Waterstraat, 2014).

5. Boundary Value and PDE Applications

The Maslov crossing form plays a central role in problems with boundary parameters or domain deformations:

• In shrinking domain problems for multidimensional Schrödinger operators under Dirichlet or Robin-type boundary conditions, the first derivative of eigenvalues with respect to the deformation parameter is given by evaluation of the Maslov crossing form on normalized eigenfunction traces. This generalizes the Rayleigh–Hadamard formula and establishes a direct link between spectral flow and the Arnold–Maslov–Keller index (Latushkin et al., 2016).
• The approach allows for spectral flow calculations in infinite dimensions by reducing to finite-dimensional intersection theory in the boundary (trace) space, where the Maslov index captures eigenvalue flow, and the crossing form encodes the directionality and regularity of crossings.

6. Example: Nonlinear Schrödinger and Stability Indices

In the fourth-order nonlinear Schrödinger (NLS) context, spectral stability of solitons is analyzed by tracking crossings between stable and unstable bundles in an augmented symplectic phase space over a $(x, \lambda)$ Maslov box. The presence of a non-regular corner crossing corresponds to vanishing first-order forms, necessitating computation of higher-order forms (up to fifth order in highly degenerate cases). These form signatures determine the correction term $\mathfrak c\in \{-1, 0, 1\}$ in the index-count formula, which, together with conjugate point data for reduced fourth-order operators, yields a computable lower bound for real unstable eigenvalues: $n_+(N)\;\geq\;|P - Q - \mathfrak c|$ where $P,Q$ are positive eigenvalue counts for the constituent operators and $\mathfrak c$ is determined by explicit integrals involving solutions of associated inhomogeneous equations (Curran et al., 2024). This framework underpins the verification of the Vakhitov–Kolokolov criterion in the spatially inhomogeneous, higher-order setting.

7. Connections, Generalizations, and Further Developments

The Maslov crossing form is deeply connected to the theory of symplectic invariants, spectral flow, and the analysis of stability in Hamiltonian systems. Its formalism unifies local differential-geometric data (quadratic forms at crossings) with global topological invariants (the Maslov index), permitting comprehensive treatment of both regular and degenerate intersections. Modern generalizations encompass infinite-dimensional Hilbert settings, operator-valued crossings, and nontrivial boundary conditions, with current research focusing on applications to high-order PDEs, bifurcation theory, and constrained variational problems (Curran et al., 2024, Cox et al., 2022, Waterstraat, 2014, Latushkin et al., 2016).

These methodologies are foundational in contemporary analysis of spectral and dynamical stability, with robust computational algorithms available for explicit high-order crossing forms, even in degenerate cases arising in PDE spectral problems. As such, the Maslov crossing form constitutes a principal tool for researchers engaging in symplectic techniques in mathematical physics, analysis, and geometric spectral theory.