Analyticity and Riesz basis property of semigroups associated to damped vibrations

Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of the operator matrix $A$ associated with the second order problem above. We develop sufficient conditions for analyticity of the associated semigroup and for the existence of a Riesz basis consisting of eigenvectors and associated vectors of $A$ in the phase space

Dirac-Krein systems on star graphs

We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $R\to \infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $\kappa_0$, which we call dislocation index, at most by $n+2$.Comment: Accepted for publication in IEO

Spectral inclusions of perturbed normal operators and applications

We consider a normal operator $T$ on a Hilbert space $H$. Under various assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$ where $A$ is $T$-bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between certain hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbola. The case of infinite gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$, then we obtain stronger results for the localisation of the spectrum of $T+A$

On the Spectral Decomposition of Dichotomous and Bisectorial Operators

For an unbounded operator $S$ on a Banach space the existence of invariant subspaces corresponding to its spectrum in the left and right half-plane is proved. The general assumption on $S$ is the uniform boundedness of the resolvent along the imaginary axis. The projections associated with the invariant subspaces are bounded if $S$ is strictly dichotomous, but may be unbounded in general. Explicit formulas for these projections in terms of resolvent integrals are derived and used to obtain perturbation theorems for dichotomy. All results apply, with certain simplifications, to bisectorial operators

Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials

The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when the coefficients are complex. Here the main result is a collection of various criteria which allow us to decide to which class of Sims' scheme a given Sturm-Liouville problem with complex coefficients belongs. This is subsequently applied to a second order differential equation defined on a ray in $\mathbb C$ which is motivated by the recent intensive research connected with $\mathcal P \mathcal T$-symmetric Hamiltonians

Simplicity of extremal eigenvalues of the Klein-Gordon equation

We consider the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials. If the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given