Archives

# The scattering operators associated to are defined

The scattering operators associated to are defined as follows: for any and , , , there exists a unique solution to the following equation such that Then the scattering operator is defined by Here is an elliptic pseudodifferential operator of order , which is conformally covariant on the boundary. Moreover, can be extended meromorphically to , where K is the same as above. The poles at , or are of first order. For simplicity we define the renormalised scattering operators by If g is approximate Einstein, i.e. then for are GJMS operators. In particular, is the Yamabe operator. See [12] for more details. If the conformal infinity is of positive Yamabe type, the scatting operators are studied by Guillarmou and Qing [16]. They showed that

A new interpretation of the renormalised scattering operator is given by Case–Chang in [2] as a generalised Dirichlet-to-Neumann map on naturally associated smooth metric measure spaces. The authors exhibited some Vincristine sulfate identities for on the boundary in terms of energies in the compact space . This connects the positivity of renormalised scattering operators to the positivity of curvature terms for the compactified metric . In particular, while a Poincaré–Einstein metric has positive conformal infinity, Qing [31] showed that there exists a suitable compactification such that has positive scalar metric, which implies that has positive spectrum for from Case–Chang's energy identity.
In this paper, we consider a complex manifold X of complex dimension , with strictly pseudoconvex boundary . The complex structure on X naturally induces a CR-structure on M, where , and is defined by . Let ρ be a smooth boundary defining function such that on . Assume the function is plurisubharmonic. We consider the Kähler metric g induced by Kähler form The metric g is asymptotically complex hyperbolic in the sense that the holomorphic sectional curvature has limit −4 when approaching to the boundary. More explicitly, g takes the form near M, where and have Taylor series in ρ at . In particular, gives the contact form on M and induces a pseudo-Hermitian metric on H. Moreover, the conformal class of the boundary pseudo-Hermitian structure is independent of choice of boundary defining function. A standard example is the complex hyperbolic space : it is the ball equipped with Kähler metric induced from the boundary defining function .
The spectrum and resolvent of the Laplacian operator were studied by Epstein–Melrose–Mendoza [6] and Vash–Wunsch [33]. Actually in both papers, the authors studied more general ACH manifolds. Similar like the real case, the spectrum of consists of two disjoint parts: the absolute continuous spectrum and the pure point spectrum . More explicitly, The resolvent is a bounded operator on for , and has finite meromorphic extension to . The smoothness of ρ implies that the metric has even asymptotic expansion in the sense of [6]. If g is Kähler–Einstein, or equivalently if ρ is a solution to the complex Monge–Ampére equation, then generally speaking ρ is not smooth up to boundary and it has logarithmic terms in the Taylor expansion at boundary with respect to smooth coordinates. See [23] for more details. The analysis of spectrum and resolvent is still valid except that the mapping property of changes a bit.
The scattering operators associated to is defined in a similar way as the real case. For any and , , and , consider the equation There exists a unique solution u such that Then the scattering operator is defined by which is a pseudo-differential operator of Heisenberg class of order , conformally covariant and having meromorphic extension to . See [6], [17], [29] for more details on the scattering theory and [1], [5], [10], [30], [32] for the Heisenberg calculus. For simplicity, we also define the renormalised scattering operators as If g is approximate Einstein (see Definition 1), then for are CR-GJMS operators of order 2k. In particular, is the CR-Yamabe operator. See [18]. This gives a different approach to construct the CR-invariant powers of sub-Laplacian studied by Grover–Graham [13] as well as the Q-curvature by Fefferman–Hirachi [8].