Hierarchies of Huygens' operators and Hadamard's conjecture
The paper provides a systematic review and a unified approach to the problem of constructing linear hyperbolic partial differential operators that satisfy Huygens principle in the sense of J. Hadamard. A link to integrable systems, the soliton theory, and the bispectral problem appears as a natural aspect of this approach. In more detail, the paper concerns the equation \[ {\mathcal L}(\varphi)= \square\varphi+ \langle a(x),\nabla\varphi\rangle+ u(x)\varphi= 0, \] where \(\square= \langle \nabla, \nabla\rangle\) is the Laplace-Beltrami operator in a pseudo-Riemannian manifold and \(\nabla\) the relevant connection. The local integration theory is developed on a causal domain \(\Omega\): if \(\gamma(x,\xi)\) is the geodesic distance between \(x\), \(\xi\in\Omega\), then the forward (resp. backward) fundamental solution of \({\mathcal L}\) is defined as a distribution \(\Phi_{\pm}(x,\xi)\) satisfying \({\mathcal L}(\Phi_{\pm}(\cdot,\xi))(x)= \delta_\xi(x)\), \(\text{supp }\Phi_{\pm}(x, \xi)\subset J_{\pm}(\xi)\) and we speak of a Huygens operator if even \(\text{supp }\Phi_{\pm}(x,\xi)\subset C_{\pm}(\xi)\). (Here \(C_{\pm}(\xi)\) are half-conoids of the characteristic conoid \(C(\xi)= \{x\in \Omega:\gamma(x, \xi)= 0\}\), and \(J_{\pm}(\xi)\) are the corresponding open subsets of \(\Omega\) respectively bounded by \(C_{\pm}(\xi)\).) Concerning the new results, the operator \({\mathcal L}\) is called \(N\)-gauge related to \({\mathcal L}_0= \square+\langle a_0(x),\nabla\rangle+ u_0(x)\) if there exists a nonzero function \(\vartheta(x)\) such that \((\text{ad}_{{\mathcal L},{\mathcal L}_0})^N(\vartheta(x))= 0\) (\(\text{ad}_{{\mathcal L},{\mathcal L}_0}= {\mathcal L}\circ\vartheta- \vartheta\circ{\mathcal L}_0\)). By using the classical Hadamard criterion (finiteness of the Riesz-Hadamard expansion of \(\Phi_{\pm}\)), it follows that then \({\mathcal L}\) and \({\mathcal L}_0\) are either both Huygens operators, or not. This crucial result is thoroughly discussed for the particular case \({\mathcal L}=\square+ u(x)\) in the Minkowski space which yields a large hierarchy of Huygens operators associated with Burchnall-Chaundy polynomials, Coxeter root systems, spectral waves operators, and plane gravitational wave operators.
scientific article; zbMATH DE number 1230526