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Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth sections with compact support. I want to define and study the exponential $e^{tD}$ of $D$.

To this end, I thought I would study the following Cauchy problem on $\mathbb{R} \times M$: let $\psi_0 \in \Gamma_c(M,E)$, we are looking for a solution $\psi \in \Gamma(\mathbb{R} \times M,E)$ to the equation: $$ \left( \frac{\partial}{\partial t} - D \right) \psi = 0 $$ with the Cauchy boundary condition: $$\psi(t=0, \cdot) = \psi_0.$$

Although global smooth Cauchy problems are quite complicated (especially on general manifolds), I cannot help but feel that this particular one (i.e. on the cylindrical manifold $\mathbb{R} \times M$) should have a general global solution. This to me is very similar to the existence of flows of vector fields, which are diffeomorphisms. I have sadly not been able to find any theorems pertaining to this very specific problem, with the exception of theorems for the propagation of waves on globally hyperbolic manifolds. But these theorems only apply to hyperbolic differential operators, and not general ones.

My questions are the following:

1) Has this problem ever been studied? (especially for $D$ first order)

2) If so, are there any conditions that must be satisfied so that $\psi$ exists and is unique?

3) Most importantly, when is the support of $\psi(t,\cdot)$ compact for all $t$?

Thank you for reading this far!

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  • $\begingroup$ If you allow 2nd order $D$, you can get the Schrodinger equation, and then there are many books written about the problem of scattering theory for those, i.e. your problem is too hard to have a general solution. $\endgroup$
    – Ben McKay
    Commented Oct 8, 2018 at 16:03
  • $\begingroup$ 3) Often not (en.wikipedia.org/wiki/Heat_kernel). To get an idea of what's involved in constructing $e^{tD}$ using the spectral theory of $D$ (at least in the case when it has self-adjoint closed extensions), have a look at this old blog post by Terry Tao: terrytao.wordpress.com/2011/12/20/… $\endgroup$
    – Igor Khavkine
    Commented Oct 8, 2018 at 19:09
  • $\begingroup$ 3) Though sometimes (Chernoff 1973). $\endgroup$
    – Igor Khavkine
    Commented Oct 8, 2018 at 19:16
  • $\begingroup$ I believe that such a problem is well posed only if $\partial_t - D$ is hyperbolic. In other words, if $a^i\partial_i$ is the first order term for $D$, then for each $x \in M$, the roots $\xi$ of the polynomial $p(x, \xi) = \det a^i(x)\xi_i$ are all real. However, this condition is in general not sufficient. It is known to be sufficient under certain conditions. The best known are symmetric hyperbolic, where $a^i$ are symmetric matrices, and strictly hyperbolic, where the roots of $p$ are real and distinct. $\endgroup$
    – Deane Yang
    Commented Oct 8, 2018 at 20:15
  • $\begingroup$ @IgorKhavkine , thank you!! The Chernoff paper is exactly what I need for what I'm working on! But I think I'm going to leave the question open, just in case somebody wants to contribute with a more general theorem (although I'm starting to doubt it's even possible). $\endgroup$
    – user209029
    Commented Oct 9, 2018 at 19:25

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