Spectral theory of dissipative q-Sturm-Liouville problems

This paper is devoted to studying a q-analogue of Sturm-Liouville operators. We formulate a dissipative q-difference operator in a Hilbert space. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which is based on the method of Pavlov and define its characteristic function. Finally, we prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative q-Sturm-Liouville difference operator.

Functional Model of Dissipative Fourth Order Differential Operators

In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

Dissipative operators with finite dimensional damping

SynopsisLetG: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ andT(t) be theCosemigroup generated byG. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {xε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions forGare satisfied. Proofs are given in terms of the Cayley transformationT= (G+I)(G−I)−1ofG. The results are applied to the damped wave equationutt+ γutx+uxxxx+ ßuxx= 0, 0 ≦t< ∞ 0 <x< 1, β, γ ≧ 0, with boundary conditionsu(0,t) =ux(0,t) =uxx(1,t) =uxxx(1,t) = 0.