  At any energy E > 0, we construct a sequence of bounded potentials $V^E_{n},
n\in\N$, supported in an annular region $B_{out}\setminus B_{inn}$ in
three-space, which act as approximate cloaks for solutions of Schr\"odinger's
equation: For any potential $V_0\in L^\infty(B_{inn})$ such that E is not a
Neumann eigenvalue of $-\Delta+V_0$ in $B_{inn}$, the scattering amplitudes
$a_{V_0+V_n^E}(E,\theta,\omega)\to 0$ as $n\to\infty$. The $V^E_{n}$ thus not
only form a family of approximately transparent potentials, but also function
as approximate invisibility cloaks in quantum mechanics. On the other hand, for
$E$ close to interior eigenvalues, resonances develop and there exist {\it
almost trapped states} concentrated in $B_{inn}$. We derive the $V_n^E$ from
singular, anisotropic transformation optics-based cloaks by a
de-anisotropization procedure, which we call \emph{isotropic transformation
optics}. This technique uses truncation, inverse homogenization and spectral
theory to produce nonsingular, isotropic approximate cloaks. As an intermediate
step, we also obtain approximate cloaking for a general class of equations
including the acoustic equation.
