“Hearing” the shape of a drum

Question

It is known that one cannot hear the shape of a drum, i.e. the map from domains in the plane to dirichlet eigenvalues is not one-to-one. But if we allow our differential operator to be any differential operator built up of finitely-many derivatives, can the map from domains in the plane to the eigenvalues of this new operator with dirichlet boundary conditions be one-to-one? If yes, how many dimensions is sufficient? Thanks!