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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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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.
We investigate the instability index of the spectral problem $$ -c^2y + b^2y + V(x)y = -mathrm{i} z y $$ on the line $mathbb{R}$, where $Vin L^1_{rm loc}(mathbb{R})$ is real valued and $b,c>0$ are constants. This problem arises in the study of stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough-Dodd equation). We show how to apply the standard approach in the situation under consideration and as a result we provide a formula for the instability index in terms of certain spectral characteristics of the 1-D Schrodinger operator $H_V=-c^2frac{d^2}{dx^2}+b^2 +V(x)$.
In the first (and abstract) part of this survey we prove the unitary equivalence of the inverse of the Krein--von Neumann extension (on the orthogonal complement of its kernel) of a densely defined, closed, strictly positive operator, $Sgeq varepsilon I_{mathcal{H}}$ for some $varepsilon >0$ in a Hilbert space $mathcal{H}$ to an abstract buckling problem operator. This establishes the Krein extension as a natural object in elasticity theory (in analogy to the Friedrichs extension, which found natural applications in quantum mechanics, elasticity, etc.). In the second, and principal part of this survey, we study spectral properties for $H_{K,Omega}$, the Krein--von Neumann extension of the perturbed Laplacian $-Delta+V$ (in short, the perturbed Krein Laplacian) 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 $C^{1,r}$, $r>1/2$.
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.
We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[0,infty)$.
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