M. Langer and H. Woracek,
The exponential type of the fundamental solution of an indefinite Hamiltonian system,
Complex Anal. Oper. Theory 7 (2013), 285–312

Abstract:
The fundamental solution of a Hamiltonian system whose Hamiltonian H is positive definite and locally integrable is an entire function of exponential type. Its exponential type can be computed as the integral over √(det H). We show that this formula remains true in the indefinite (Pontryagin space) situation, where the Hamiltonian is permitted to have finitely many inner singularities. As a consequence, we obtain a statement on non-cancellation of exponential growth for a class of entire matrix functions.

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