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DCDS

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem,
namely, the wave equation with an additional diffusion term that blows up at time zero. A solution to this wave extension
problem is obtained from the Schrödinger group by means of an oscillatory subordination formula, which also allows us
to find kernel representations for such solutions. Asymptotics of related oscillatory
integrals are analysed in order to determine the correct domains for initial data in the
general extension problem involving non-negative self-adjoint operators. An alternative approach using Bessel functions
is also described.

keywords:
Schrödinger group
,
oscillatory integrals
,
Bessel functions.
,
wave equation
,
Fractional Laplacian

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