Let
$(\tau ,{V}_{\tau})$ be a spinor
representation of
$\mathrm{Spin}(n)$
and let
$(\sigma ,{V}_{\sigma})$ be a spinor
representation of
$\mathrm{Spin}(n1)$ that
occurs in the restriction
${\tau}_{\mathrm{Spin}(n1)}$.
We consider the real hyperbolic space
${H}^{n}(\mathbb{R})$ as the rank one
symmetric space
${\mathrm{Spin}}_{0}(1,n)\u2215\mathrm{Spin}(n)$ and
the spinor bundle
$\mathrm{\Sigma}{H}^{n}(\mathbb{R})$
over
${H}^{n}(\mathbb{R})$ as the
homogeneous bundle
${\mathrm{Spin}}_{0}(1,n){\times}_{\mathrm{Spin}(n)}{V}_{\tau}$.
In this paper we characterize eigenspinors of the algebra of invariant differential operators
acting on
$\mathrm{\Sigma}{H}^{n}(\mathbb{R})$
which can be written as the Poisson transform of
${L}^{p}$sections of the
bundle
$\mathrm{Spin}(n){\times}_{\mathrm{Spin}(n1)}{V}_{\sigma}$ over
the boundary
${S}^{n1}\simeq \mathrm{Spin}(n)\u2215\mathrm{Spin}(n1)$
of
${H}^{n}(\mathbb{R})$, for
$1<p<\infty $.
