Recently I've been trying to understand better the one-dimensional heat equation with a potential \begin{equation} \partial_tu=\frac{1}{2}\Delta_xu-Vu\tag{1} \end{equation} defined on $\mathbb{R}\times[0,T]$ with initial condition $u(x,0)=f(x)$, where $V\colon\mathbb{R}\times[0,T]\to\mathbb{R}$ is a real function, called the potential.
Regarding physical intuition, I have read that physically the potential term $-Vu$ can be thought of as a term approximately accounting for a temperature-dependent radiative loss of heat, with $V(x,t)$ describing how intense that radiation is with respect to position and time.
Now, since the heat equation with a potential is the Wick rotated version of the Schrödinger equation (i.e. the equation obtained via the change of variables $t\mapsto i\tau$), I thought it might be a good idea to look for solutions of it for potentials $V$ that are physically relevant in the context of the Schrödinger equation.
Question. What is known about explicit analytic solutions (besides Feynman–Kac) of Equation (1) for particular potentials $V$?
Here I have in mind something like assembling a version of Wikipedia's List of quantum-mechanical systems with analytical solutions for the heat equation with a potential. For example, fixing $k\in\mathbb{R}$, are there known analytical solutions when:
- $V(x,t)$ is constant with value $k$?
- $V(x,t)$ takes the value $0$ on an interval $[-L,L]$ and a constant value $k$ outside $[-L,L]$ (in particular, when $k\to\infty$, does the solution in this case resemble the solution of the heat equation on $[-L,L]$ with zero Dirichlet boundary conditions?)
- $V(x,t)$ is a step potential with $V(x,t)=0$ for $x<0$ and $V(x,t)=k$ for $x\geq0$.
- $V(x,t)$ is a harmonic potential of the form $kx^2$.
- $V(x,t)$ is a Coulomb potential of the form $k/x$.
- $V(x,t)$ is a Yukawa potential of the form $ke^{-\alpha x}/x$ with $k,\alpha\in\mathbb{R}$.