Here are some figures and animations to accompany my paper with Ben Pineau, which can be found here.

In the numerics, we consider the one-dimensional wave operator $\partial_t^2 + P_V$ where $P_V = -\partial_x^2 + V(x)$, with $V(x)$ a real-valued exponentially decaying potential (analytic at infinity).

For each potential, we numerically estimate the cosine wave propagator $U_V(t) := \cos(t \sqrt{P_V})$.

We then compute the flat trace of $U_V(t)$ for $t > 0$, that is, $Tr(U_V(t) - U_0(t))$. If $K(t,x,y)$ is the Schwartz kernel of $U_V(t) - U_0(t)$, then $Tr(U_V(t) - U_0(t)) = \int_x K(t,x,x)\, dx$.

For each potential, we numerically compute $K(t,x,x)$ and $\int_x K(t,x,x)\, dx$.

In our paper, we prove that (for $t > 0$) \[ Tr(U_V(t) - U_0(t)) = \tfrac{1}{2} \sum_{\lambda_j} e^{-it\lambda_j} + \mathcal{A}_{+}(t) + \mathcal{A}_{-}(t) - \tfrac{1}{2}, \] where $\lambda_j$ are the scattering resonances of $P_V$ and $\mathcal{A}_{\pm}(t)$ are functions depending on the exponential decay of $V(x)$.

Pöschl–Teller Potential ($\ell = 1$)

Here we consider the Pöschl–Teller potential $V_{PT}(x) = (\ell^2 + 1/4) / \cosh(x)^2$ with $\ell = 1$.

In this case, the resonances are $\{\lambda_j^\pm = -i(j + 1/2) \pm \ell : j \in \mathbb{Z}_{\ge 0}\}$.

Our theorem (after some algebra) shows that \[ Tr(U_V(t) - U_0(t)) = \tfrac{1}{2}\left(\frac{\cos(\ell t) - e^{-t/2}}{\sinh(t/2)} - 1\right). \] This matches the numerically computed curve shown above.

Pöschl–Teller Potential ($\ell = 3$)

Here we consider the Pöschl–Teller potential $V_{PT}(x) = (\ell^2 + 1/4) / \cosh(x)^2$ with $\ell = 3$.

In this case, the resonances are further from the imaginary axis, resulting in more oscillation of the flat trace of the propagator, as seen above.

Pöschl–Teller Potential ($\ell = 0$)

Here we consider the Pöschl–Teller potential $V_{PT}(x) = (\ell^2 + 1/4) / \cosh(x)^2$ with $\ell = 0$.

In this case, the resonances lie along the negative imaginary axis, so there is no oscillation of the flat trace of the propagator, as seen above.

Nonsymmetric Double Pöschl–Teller Potential

Here we consider $V(x) = \tfrac{1}{4} / \cosh(x-4)^2 + 3 / \cosh(x+4)^2$.

In this case, resonances lie closer to the real axis due to the wave slowly escaping the double well, leading to slower decay in the flat trace of the propagator.