Consider a Riemannian manifold and let

- $\mathrm{id}$ be the identity operator, let
- $\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let
- $t > 0$ be a parameter.

Does the operator $$A := \mathrm{id}-t\Delta$$ have an established name? Or alternatively, is there an established name for the corresponding Green's functions? Where do these objects show up? I've been working with this beast a lot lately and it would be very useful to at least have a Googleable name.

The operator shows up in a couple places, for instance if you take the heat equation
$$\dot{u} = \Delta u$$
and discretize it using backward Euler then you get
$$\frac{u-u_0}{t}=\Delta u$$
or $(\mathrm{id}-t\Delta)u=u_0$. So the Green's function can be viewed as a 1-step approximation to the heat kernel. (The operator *cannot*, however, be viewed as the first part of the Taylor series for the solution operator $e^{t\Delta}$ -- that would be $\mathrm{id} + t\Delta$.)

If you replace $\mathrm{id}$ with the identity matrix $I$ and $\Delta$ with the graph Laplacian $L$, then on a finite graph you can pick $t$ small enough that $I-tL$ can be viewed as the limit of the Neumann series $$\sum_{k=0}^\infty t^k L^k.$$

However, I suspect there are **much** better interpretations of the original operator $A$, which is why I'm asking the question!

Thanks.

pseudo-Riemanniancase the physicists would probably refer to it as the(massive) Klein--Gordon operator. $\endgroup$ – mathphysicist Aug 17 '12 at 19:13