On a Riemannian manifold $M$, the Laplace operator $L$ is uniquely defined.
If $M$ is not compact, then $L$ admits a continuous spectrum.
Is there a way of "changing" $M$ and/or $L$ in say $M'$ and $L'$, such that the spectrum of $L'$ is now purely discrete and contains the spectrum of $L$? I am interested in examples of this type.
Feel free to restrict to homogenous varieties or any other geometric gadget, if have an example of similar nature in mind.
Motivation: More general it is true that, given an self adjoint operator $T$ on a Hilbert space, does there exists a decomposition such that $T_1 + T_2 = T$, where $T_1$ has the same point spectrum as $T$. I am not looking for this. Is there a better geometric construction?