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Register a new userGraphical Krein Signature Theory and Evans-Krein Functions
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Two concepts, very different in nature, have proved to be useful in analytical and numerical studies of spectral stability: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half-plane leading to dynamical instability under perturbation of the system, and (ii) the Evans function, an analytic function detecting the location of eigenvalues. One might expect these two concepts to be related, but unfortunately examples demonstrate that there is no way in general to deduce the Krein signature of an eigenvalue from the Evans function. The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well-known in the spectral theory of polynomial operator pencils. This interpretation avoids altogether the need to view the Krein signature in terms of root subspaces and their relation to indefinite quadratic forms. To demonstrate the utility of this graphical interpretation of the Krein signature, we use it to define a simple generalization of the Evans function -- the Evans-Krein function -- that allows the calculation of Krein signatures in a way that is easy to incorporate into existing Evans function evaluation codes at virtually no additional computational cost. The graphical Krein signature also enables us to give elegant proofs of index theorems for linearized Hamiltonians in the finite dimensional setting: a general result implying as a corollary the generalized Vakhitov-Kolokolov criterion (or Grillakis-Shatah-Strauss criterion) and a count of real eigenvalues for linearized Hamiltonian systems in canonical form. These applications demonstrate how the simple graphical nature of the Krein signature may be easily exploited.
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We consider harmonic Toeplitz operators $T_V = PV:{mathcal H}(Omega) to {mathcal H}(Omega)$ where $P: L^2(Omega) to {mathcal H}(Omega)$ is the orthogonal projection onto ${mathcal H}(Omega) = left{u in L^2(Omega),|,Delta u = 0 ; mbox{in};Omegaright}$, $Omega subset {mathbb R}^d$, $d geq 2$, is a bounded domain with $partial Omega in C^infty$, and $V: Omega to {mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V in S_{p, {rm w}}$, the weak counterpart of $S_p$. Next, we assume that $Omega$ is the unit ball in ${mathbb R}^d$, and $V = overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $partial Omega$ or $V$ is compactly supported in $Omega$. Further, we consider general $Omega$ and $V geq 0$ which is regular in $Omega$, and admits a power-like decay of rate $gamma > 0$ at $partial Omega$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-gamma$, self-adjoint in $L^2(partial Omega)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K geq 0$, self-adjoint in $L^2(Omega)$; it is known that ${rm Ker},K = {mathcal H}(Omega)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V in C(overline{Omega};{mathbb R})$, and show that $sigma_{rm ess}(K+V) = V(partial Omega)$. Assuming that $V geq 0$ and $V{|partial Omega} = 0$, we study the asymptotic distribution of the eigenvalues of $K pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.
We study spectral properties for $H_{K,Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-Delta+V$ defined on $C^infty_0(Omega)$, where $V$ is measurable, bounded and nonnegative, in a bounded open set $Omegasubsetmathbb{R}^n$ belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class $C^{1,r}$, $r>1/2$. In particular, in the aforementioned context we establish the Weyl asymptotic formula [ #{jinmathbb{N} | lambda_{K,Omega,j}leqlambda} = (2pi)^{-n} v_n |Omega| lambda^{n/2}+Obig(lambda^{(n-(1/2))/2}big) {as} lambdatoinfty, ] where $v_n=pi^{n/2}/ Gamma((n/2)+1)$ denotes the volume of the unit ball in $mathbb{R}^n$, and $lambda_{K,Omega,j}$, $jinmathbb{N}$, are the non-zero eigenvalues of $H_{K,Omega}$, listed in increasing order according to their multiplicities. We prove this formula by showing that the perturbed Krein Laplacian (i.e., the Krein--von Neumann extension of $-Delta+V$ defined on $C^infty_0(Omega)$) is spectrally equivalent to the buckling of a clamped plate problem, and using an abstract result of Kozlov from the mid 1980s. Our work builds on that of Grubb in the early 1980s, who has considered similar issues for elliptic operators in smooth domains, and shows that the question posed by Alonso and Simon in 1980 pertaining to the validity of the above Weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting.
A Banach involutive algebra is called a Krein C*-algebra if there is a fundamental symmetry (an involutive automorphism of period 2) such that the C*-property is satisfied when the original involution is replaced with the new one obtained by composing the automorphism with the old involution. For a given fundamental symmetry, a Krein C*-algebra decomposes as a direct sum of an even part (a C*-algebra) and an odd part (a Hilbert C*-bimodule on the even part). Our goal here is to develop a spectral theory for commutative unital Krein C*-algebras when the odd part is a symmetric imprimitivity C*-bimodule over the even part and there exists an additional suitable exchange symmetry between the odd and even parts.
We introduce a notion of Krein C*-module over a C*-algebra and more generally over a Krein C*-algebra. Some properties of Krein C*-modules and their categories are investigated.
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 $tau inmathbb Rcup{infty}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Kreins 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 $Rto infty$, 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 $n+2$.
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