  It is well known that (possibly non-unique) suitable field dynamics can be
prescribed in spacetimes with timelike boundaries by means of appropriate
boundary conditions. In Ref. [J. Math. Phys. {\bf 21}, 2802 (1980)], Wald
derived a conserved energy functional for each prescribed dynamics. This
conserved energy is related to the positive self-adjoint extensions of the
spatial part $A$ of the wave equation $\partial^2\Phi/\partial t^2=-A\Phi$ ($A$
may not be, in principle, essentially self-adjoint). This is quite surprising
since the canonical energy is not conserved in these cases. In this paper, we
rederive this energy functional from an action principle (with appropriate
boundary terms) following Ref. [Phys. Rev. D, {\bf 69}, 085005, (2004)] and
consider field dynamics arising from non-positive self-adjoint extensions of
$A$. The spectrum of the resulting theory fails to be positive and unstable
mode solutions for classical fields come to light. By studying fields in
half-Minkowski spacetime, we illustrate that these unstable classical solutions
come as a consequence of an inverted parabolic potential governing their
dynamics. From the quantum mechanical point of view, this leads to an effective
inverted harmonic oscillator at the boundary. We then explore these unstable
modes behavior, as well as their instabilities, at the quantum level.
