given by $$\lambda_n=-D\left(\frac{2\pi n}{L}\right)^2,\qquad n\in\mathbb{Z},$$ with equal right and left eigenmodes, due to the Hermiticity of ${\mathcal{L}}$, given by $$r_n(x)=\frac{1}{\sqrt{L}}e^{i\frac{2\pi n}{L}x},$$ and normalised according to $$\begin{aligned}
\left\langle r_n|r_m \right\rangle = \int_{-L/2}^{L/2}r^*_n(x)r_m(x)\, dx=\delta_{nm}.\end{aligned}$$

Since the spectrum exists and can be normalized appropriately, we may use our results of Sec. \[SecRes\]. Adding resets at a rate $\Gamma$ to the position 0, and defining the states $\ket{0}$ and $\bra{-}$ by $\left\langle g|0 \right\rangle=g(0)$ and $$\left\langle -|f \right\rangle=\left\langle l_0|f \right\rangle=\int_{-L/2}^{L/2}f(x)\, dx,$$ respectively, we find the new stationary state, which is now a probability density, to be given by the sum $$\begin{aligned}
p_{\text{ss}}^\Gamma(x)=\frac{\Gamma}{L}\sum_{n=-\infty}^{\infty}\frac{e^{i\frac{2\pi n}{L}x}}{D\left(\frac{2\pi n}{L}\right)^2+\Gamma}.\end{aligned}$$ Taking the infinite-size limit, we then find $$p_{\text{ss}}^\Gamma(x)=\Gamma\int_{-\infty}^{\infty}\frac{dk}{2\pi}\frac{e^{ikx}}{Dk^2+\Gamma},$$ which can be solved using residues to give $$p_{\text{ss}}^\Gamma(x)=\frac{1}{2}\sqrt{\frac{\Gamma}{D}}e^{-\sqrt{\frac{\Gamma}{D}}|x|}.$$ This agrees with the result of [@evans2011] and is similar to the distribution found for the discrete random walk.

Open quantum systems with resets {#OQS}
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We next consider adding resets to open quantum systems interacting with an environment. Under appropriate conditions on the timescales of the dynamics in the environment and the strength of the interactions, the environment can be suitably viewed as memoryless, allowing us to consider the system to be a quantum generalization of the Markovian systems considered earlier .

