The Hamiltonian formulation of continuum Calogero-Moser models

Abstract: Recent well-posedness results have identified the Hardy space $L^2_+$ as the natural phase space for continuum Calogero-Moser models, both focusing and defocusing, on the line and on the torus. In this paper, we introduce a symplectic form on this phase space and so are able to realize these models as Hamiltonian systems. Moreover, we demonstrate that previously identified conserved quantities are mutually commuting, reinforcing the notion that these models are completely integrable. We further illustrate the utility of these structures by using them to give a new proof of global well-posedness in the critical space $L^2_+$, under the necessary mass restriction in the focusing case. Our work also brings to light several unforeseen connections: (i) the threshold for well-posedness coincides with that for the nondegeneracy of the symplectic form; (ii) this threshold is connected through Carleman's inequality to the isoperimetric problem in the plane; (iii) the transition from the line to the torus gives rise to a modified dynamical equation.

• The paper establishes an intrinsic Hamiltonian framework for continuum Calogero-Moser models by defining a strong, nondegenerate symplectic form on the Hardy space L²₊.
• It demonstrates that the derived Hamiltonians generate commuting flows with infinitely many conserved quantities, reinforcing the model's integrability.
• The work uncovers a deep link between analytic thresholds for well-posedness in critical spaces and classical isoperimetric inequalities through geometric analysis.

The Hamiltonian Formulation of Continuum Calogero-Moser Models

Introduction and Background

The continuum Calogero-Moser (CCM) models represent a class of integrable Hamiltonian PDEs governing the evolution of complex-valued fields $q(t,x)$, with spatial variable $x$ in either $\mathbb R$ (the real line) or $\mathbb T$ (the torus). These models extend the original discrete Calogero-Moser particle systems and provide a variant with a continuum of degrees of freedom, relevant both in mathematical physics and the analysis of integrable systems. The CCM equations couple second-order dispersion with a nonlocal cubic nonlinearity involving projections onto Hardy spaces, reflecting the intrinsic holomorphicity (chiral property) of the dynamics. There are both focusing and defocusing cases, distinguished by the choice of sign in the nonlinear term.

Historically, the well-posedness and invariance properties for these flows were not fully compatible with the standard Hilbert phase spaces or the classical Hamiltonian formulation. Recent works identified the Hardy space $L_+^2$ as the natural phase space, but a proper intrinsic symplectic/Hamiltonian structure—linking the equation's integrability to Hamiltonian flows—had not been fully elucidated.

This paper fills that gap, developing the intrinsic Hamiltonian and symplectic structures for continuum Calogero-Moser models and establishing analytical results fundamental for the rigorous theory of integrable PDEs in this class.

Hamiltonian and Symplectic Structure

The continuum Calogero-Moser equation on the line is given by

$i q_t = -q_{xx} \pm 2i q_x C_+(|q|^2),$

where $C_+$ is the Cauchy–Szegő projection onto the Hardy space $L^2_+(\mathbb{R})$. The "+"/"–" corresponds to the focusing and defocusing variants, respectively. The fundamental observation is that the natural phase space is $L_+^2(\mathbb{R})$, whose elements admit holomorphic extensions to the upper half-plane.

The authors construct a symplectic form on $L_+^2$, defined by

$x$0

where $x$1 incorporates both the standard complex structure and a nonlocal quadratic perturbation that enforces compatibility with the gauge-transformed variables and the nontrivial Poisson structure dictated by the CCM nonlinearity.

A key technical result is that this symplectic form is strong (in the infinite-dimensional sense) and nondegenerate precisely when the mass $x$2 is less than a critical threshold $x$3; in the focusing case, $x$4, while for the defocusing case $x$5. The same threshold appears in the well-posedness theory, signifying a deep connection between analytic stability and geometric nondegeneracy of the phase space structure.

A similar (but more subtle) construction is accomplished for the torus, where the absence of a natural ordering complicates the definition of the symplectic form due to the lack of a sign function. The authors resolve this by defining a new nonlocal operator $x$6 that serves as the appropriate primitive on the torus, ensuring the necessary properties for the Hamiltonian theory.

Hamiltonian Flows and Complete Integrability

With the above symplectic structures, the CCM equations are shown to be Hamiltonian flows for the functionals

$x$7

(on the line), and a carefully renormalized analogous Hamiltonian on the torus, which includes correction terms involving the mass and momentum to maintain compatibility with translation and phase symmetries and to guarantee well-posedness at critical regularity.

Crucially, the authors demonstrate that the CCM hierarchies feature infinitely many conserved quantities, $x$8, realized by the Lax pair construction. The Hamiltonian flows generated by these invariants are shown to be mutually commuting—a hallmark of Liouville integrability. This is established via a careful analysis of the Poisson bracket under the intrinsic symplectic form, including the computation of the symplectic gradient using Wirtinger derivatives and an explicit (though nontrivial) mapping $x$9 connecting gradients to vector fields.

Critical Mass Threshold and Geometric Connections

A striking aspect of the analysis is that the threshold for well-posedness in $\mathbb R$0 and for the nondegeneracy of the symplectic form coincides and is intimately connected to Carleman's inequality—a classical result in geometric analysis that forms the basis of the sharp isoperimetric inequality in the plane. On the torus, this yields a unifying perspective on the relationship between analytic and geometric properties of the model.

The connection is rendered precise by functional inequalities governing the size and distribution of $\mathbb R$1 within $\mathbb R$2, and their relation to the boundedness and invertibility of the nonlocal perturbative operators in the symplectic form.

Commuting Flows Method and Well-posedness in Critical Spaces

Applying the method of commuting flows—an advanced technique leveraging the complete integrability structure and Poisson commutativity—the paper provides an alternative proof of global well-posedness in the scaling-critical space $\mathbb R$3 (up to the mass threshold in the focusing case). The regularized Hamiltonians $\mathbb R$4 are constructed as smoothing approximations, converging (in the appropriate operator topologies) to the physical Hamiltonian while maintaining the commutativity and integrability properties.

The proof route demonstrates uniform control on orbits in $\mathbb R$5 via spectral tightness and equicontinuity; this is achieved by leveraging a new equivalence of Sobolev-type spaces adapted to the Lax operator (the dynamically changing "Fourier" variables) and the standard Sobolev scale.

Structural and Analytical Results

Key results include:

• The identification of a strong, nondegenerate symplectic form on phase space for $\mathbb R$6, with explicit degeneracy above this threshold in the focusing case.
• Proof that the Hamiltonians for the CCM model generate the flows via the new symplectic structure, on both the line and torus.
• Demonstration that the infinite sequence of conservation laws are in involution with respect to the Poisson bracket, reinforcing complete integrability.
• Reduction of the global well-posedness question to geometric properties of the symplectic form, with resolution via commuting flows and functional calculus arguments.
• The revelation of a tight correspondence between analytic thresholds in dispersive PDE well-posedness and classic isoperimetric inequalities.

Implications and Future Directions

The results lay a rigorous foundation for the geometric and analytic theory of continuum Calogero-Moser models. The correspondence between integrability, symplectic nondegeneracy, and analytic thresholds suggests new paths for examining other nonlocal integrable evolution equations, in particular for hierarchies where standard symplectic structures may be inaccessible.

The techniques developed regarding commutative Hamiltonians and the operator-theoretic framework on Hardy-Sobolev spaces are likely to impact ongoing research on invariant measures, statistical mechanics (Gibbs states for nonlinear dispersive equations), and the long-time dynamics of integrable PDEs.

Additionally, the explicit identification of the connection to Carleman's inequality opens the possibility of exploiting geometric analysis methods to study fine properties of solutions and thresholds for blow-up or global existence.

Future work is expected to address the dynamical invariance of Gibbs states for these models and to further elucidate the geometric underpinnings of integrability thresholds in connection to minimal surface theory.

Conclusion

This paper establishes a rigorous, fully-intrinsic Hamiltonian formulation for the continuum Calogero-Moser models on both the real line and the torus, with explicit symplectic forms, Hamiltonians, and integrable hierarchies grounded in the natural Hardy phase space. The analytic and geometric results are tightly intertwined, revealing new structural phenomena and offering robust tools for the analysis of integrable evolutionary PDEs at scaling-critical regularity.

Reference:

"The Hamiltonian formulation of continuum Calogero-Moser models" (2604.09479).