If both $A$ and $B$ are diagonalizable with eigen-values $a_1,\ldots,a_m$ and $b_1,\ldots,b_n$, respectively, then the characteristic polynomial of $A\otimes B$ must be $$ c(\l)=\det(\l-A\otimes B)=(-1)^{nm}\prod_{j=1}^m\prod_{k=1}^n(a_jb_k-\l)~. $$ Since diagonalizable operators are dense and the mapping that sends a pair $(A,B)$ to the coefficients of the characteristic polynomial of $A\otimes B$ is continuous, the above formula must hold for all pairs $(A,B)$.