A Linearization for a Class of λ-Nonlinear Boundary Eigenvalue Problems

AbstractIn this paper a new linearization of boundary eigenvalue problems for systems ỹ′+Ã0ỹ=λÃ1ỹ of n first order differential equations with λ-polynomial boundary conditions is proposed. The linearized problem is again a boundary eigenvalue problem for a system y′+A0y=λA1y of first order differential equations of dimension n+n̂ where n̂ is the total polynomial degree of the boundary conditions. As a particular case, we consider systems of first order differential equations induced by nth order differential equations Nη=λPη, and we give an application to the Orr–Sommerfeld equation with λ-quadratic boundary conditions

The damped wave equation with unbounded damping

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.Comment: 26 pages, 2 figure

Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.Comment: 32 pages, 3 figure

Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator

Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint spectral sets associated with the operator L and/or the operator A. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed. The sharp norm bounds obtained for the operator angles generalize the celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjoint perturbations.Comment: (http://www.iumj.indiana.edu/IUMJ/FULLTEXT/2010/59/4225

Numerical Range and Quadratic Numerical Range for Damped Systems

We prove new enclosures for the spectrum of non-selfadjoint operator matrices associated with second order linear differential equations $\ddot{z}(t) + D \dot{z} (t) + A_0 z(t) = 0$ in a Hilbert space. Our main tool is the quadratic numerical range for which we establish the spectral inclusion property under weak assumptions on the operators involved; in particular, the damping operator only needs to be accretive and may have the same strength as $A_0$. By means of the quadratic numerical range, we establish tight spectral estimates in terms of the unbounded operator coefficients $A_0$ and $D$ which improve earlier results for sectorial and selfadjoint $D$; in contrast to numerical range bounds, our enclosures may even provide bounded imaginary part of the spectrum or a spectral free vertical strip. An application to small transverse oscillations of a horizontal pipe carrying a steady-state flow of an ideal incompressible fluid illustrates that our new bounds are explicit.Comment: 27 page

Everything is possible for the domain intersection dom T \cap dom T*

This paper shows that for the domain intersection \dom T\cap\dom T^* of a closed linear operator and its Hilbert space adjoint everything is possible for very common classes of operators with non-empty resolvent set. Apart from the most striking case of a maximal sectorial operator with \dom T\cap\dom T^*=\{0\}, we construct classes of operators for which \dim(\dom T\cap\dom T^*)= n \in \dN_0; \dim(\dom T\cap\dom T^*)= \infty and at the same time \codim(\dom T\cap\dom T^*)=\infty; and \codim(\dom T\cap\dom T^*)= n \in \dN_0; the latter includes~the case that \dom T\cap\dom T^* is dense but no core of $T$ and $T^*$ and the case \dom T=\dom T^* for non-normal $T$. We also show that all these possibilities may occur for operators $T$ with non-empty resolvent set such that either W(T)=\dC, $T$ is maximal accretive but not sectorial, or $T$ is even maximal sectorial. Moreover, in all but one subcase $T$ can be chosen with compact resolvent.Comment: 34 page

Eigenvalues of Magnetohydrodynamic Mean-Field Dynamo Models: Bounds and Reliable Computation

This paper provides the first comprehensive study of the linear stability of three important magnetohydrodynamic (MHD) mean-field dynamo models in astrophysics, the spherically symmetric $\alpha^2$-model, the $\alpha^2\omega$-model, and the $\alpha\omega$-model. For each of these highly nonnormal problems, we establish upper bounds for the real part of the spectrum, prove resolvent estimates, and derive thresholds for the helical turbulence function $\alpha$ and the rotational shear function $\omega$ below which no MHD dynamo action can occur for the linear models (antidynamo or bounding theorems). In addition, we prove that interval truncation and finite section method, which are employed to regularize the singular differential expressions and the infinite number of coupled equations, are spectrally exact. This means that all spectral points are approximated and no spectral pollution occurs, thus confirming, for the first time, that numerical eigenvalue approximations for the highly nonnormal MHD dynamo problems are reliable