I want to motivate the theory of $C_0$-semigroups to someone, and the following question was asked:
What is an example of a non-separable linear PDE?
Preferably a simple homogeneous one.
I want to motivate the theory of $C_0$-semigroups to someone, and the following question was asked:
What is an example of a non-separable linear PDE?
Preferably a simple homogeneous one.
See my comments above on some general ideas. For specific examples, perhaps a good starting point would be L. P. Eisenhart, "Enumeration of potentials for which one-particle Schrodinger equations are separable," Phys. Rev. 74, 87-89 (1948). So just take an arbitrary potential function $V(x)$ that is not in Eisenhart's list, and consider the Schroedinger equation
$$ [-i \partial_t + \triangle + V(x)]\psi(t,x) = 0$$
This is Linear, homogeneous, and aside from the trivial separability of the $t$ variable, satisfies your requirements.