If we have nonnegative $V \in L^1\_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for example, E.B. Davies in *Heat Kernels and Spectral Theory*. He also proves that $H$ is the infinitesimal generator of an ultracontractive symmetric Markov semigroup $e^{-Ht}$ which is given by integration against a kernel $p(x,y,t)$:


$$e^{-Ht}[f] = \int\_{\mathbb{R}^{n}} p(x,y,t)f(y)\,dy \qquad f \in L^2(\mathbb{R}^{n})$$


Now I have seen several authors use, directly or indirectly, the fact that for $y$ fixed, $p(\cdot,y,t)$ is a weak solution of the equation $(\partial\_{t} + H)u = 0$. But how exactly do we know this is true? I can't find anything about this in Davies.



(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation
There is an old [paper by Ball](http://people.maths.ox.ac.uk/~ball/Papers/Ball_AMS_77.pdf) where he discusses the abstract connections between weak solutions and semigroup orbits. He proves that the mild solution is essentially always the weak solution for strongly continuous semigroups.


Theorem 3.1.7 from the [monograph by Courtain and Zwart](http://books.google.it/books?id=G_x-F-l2V2UC&lpg=PP1&hl=de&pg=PP1#v=onepage&q&f=false) is also proving this.


