Abstract

We consider a one-parameter family of degenerately elliptic operators ${\mathcal{ℒ}}_{\gamma }$ on the closed disk $\mathbb{𝔻}$, of Keldysh (or Kimura) type, which appears in prior work by the authors and Mishra (2023), related to the geodesic X-ray transform. Depending on the value of a constant $\gamma \in ℝ$ in the subprincipal term, we prove that either the minimal operator is self-adjoint (case $\left|\gamma \right|\ge 1$), or that one may construct appropriate trace maps and Sobolev scales (on $\mathbb{𝔻}$ and ${\mathbb{𝕊}}^1=\partial \mathbb{𝔻}$) on which to formulate mapping properties, Dirichlet-to-Neumann maps, and extend Green’s identities (case $\left|\gamma \right|<1$). The latter can be reinterpreted in terms of a boundary triple for the maximal operator, or a generalized boundary triple for a distinguished restriction of it. The latter concepts, objects of interest in their own right, provide avenues to describe sufficient conditions for self-adjointness of extensions of ${\mathcal{ℒ}}_{\gamma ,\min <!--FUNCTION\; APPLICATION-->}$ that are parametrized in terms of boundary relations, and we formulate some corollaries to that effect.

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