= Solution
The continuum Hamiltonian is that of a spinless one-dimensional p-wave <topological superconductor>. Its topological phase has $\mu>0$, whereas the empty vacuum may be modeled as the trivial phase with $\mu<0$. With superconductor at $x<0$ and vacuum at $x>0$, this is
$$
\boxed{\operatorname{sgn}\mu(x)=-\operatorname{sgn}x.}
$$
Near the transition $\mu=0$, momenta are small and the quadratic term may be neglected. In Pauli-matrix notation, the long-wavelength <Bogoliubov--de Gennes Hamiltonian> is
$$
H_{\rm lw}=-\mu(x)\tau_z-i\hbar\Delta\tau_x\partial_x.
$$
The zero-energy equation becomes
$$
\partial_x\psi=\frac{\mu(x)}{\hbar\Delta}\tau_y\psi.
$$
Choose a constant spinor $\chi_+$ with $\tau_y\chi_+=\chi_+$. Then
$$
\boxed{
\psi(x)=\mathcal N
\exp\left(\int_0^x\frac{\mu(s)}{\hbar\Delta}\,ds\right)\chi_+}
$$
decays on both sides because $\mu$ changes from positive to negative. For asymptotically constant $|\mu|$, the <Continuum p-wave Majorana interface mode> has width
$$
\boxed{\xi=\frac{\hbar\Delta}{|\mu|}.}
$$
Its characteristic momentum is $|p|\sim|\mu|/\Delta$. The neglected kinetic energy is small compared with $|\mu|$ when $|\mu|\ll m\Delta^2$ up to a factor of two, equivalently when $\xi\gg\hbar/(m\Delta)$.
In the Nambu basis, <Particle-hole symmetry of a Bogoliubov--de Gennes Hamiltonian> lets a zero-energy eigenvector be chosen self-conjugate, $v(x)=u(x)^*$. The corresponding quasiparticle operator is
$$
\boxed{\gamma=\int dx\,[u(x)c(x)+u(x)^*c^\dagger(x)].}
$$
Taking the adjoint and interchanging the two terms gives $\gamma^\dagger=\gamma$. With the usual normalization it obeys $\{\gamma,\gamma\}=2$, so it is a <Majorana fermion operator> localized at the interface.
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