Lieb-Thirring inequalities for the damped wave equation

The operator approach for the damped wave equation provides us with uniqueness and regularity of its solutions since the operator generates a C0-semigroup. Moreover the behavior of its spectrum gives us information about the stability of the solutions. Using the correspondence between the spectra of this operator and the Schrödinger operator we obtain numerous bounds on the eigenvalues and criteria for their existence or absence even in the case of complex damping. These results are demonstrated on the analytically computable case when the damping is a finite rectangular well.