The conformal range (or the real Davis--Wielandt shell), which is a particular planar projection of the Davis--Wielandt shell, can be considered as the hyperbolic version of the numerical range; i. e. it is a ``field of values'' which can be interpreted as a subset of the asymptotically closed hyperbolic plane. Here we explain the analogue of the elliptical range theorem of $2\times2$ complex matrices for the conformal range.
Corry Corazon Marzuki, Aminah Utami, Mona Elviyenti
et al.
his study aims to determine the total vertex irregularity strength on a series parallel graph for and . Total labeling is said to be vertex irregular, if the weights for each vertices are different. Determination of the total vertex irregularity of series parallel graph is done by obtaining the largest lower bound and the smallest upper bound. The lower bound is obtained by analyzing the structure of the graph to obtain the largest minimum label of k and the upper bound is analyzed by labeling the vertices and edges of the graph, where the largest label is k and the values for each vertices weight is different. The result obtained for the total vertex irregularity strength of a series parallel graph is .
Non-self-adjoint second-order ordinary differential operators on a finite interval with complex weights are studied. Properties of spectral characteristics are established and the inverse problem of recovering operators from their spectral characteristics are investigated. For this class of nonlinear inverse problems an algorithm for constructing the global solution is obtained. To study this class of inverse problems, we develop ideas of the method of spectral mappings.
This research was devoted to investigate the inverse spectral problem of Sturm-Liouville operator with many frozen arguments. Under some assumptions, the authors obtained uniqueness theorems. At the end, a numerical simulation for the inverse problem was presented.
This paper is concerned with the characterizations of the Friedrichs extension for a class of singular discrete linear Hamiltonian systems. The existence of recessive solutions and the existence of the Friedrichs extension are proved under some conditions. The self-adjoint boundary conditions are obtained by applying the recessive solutions and then the characterization of the Friedrichs extension is obtained in terms of boundary conditions via linear independently recessive solutions.
We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.
For a tuple of square matrices $A_1,...,A_n$ the determinantal hypersurface is defined as \begin{eqnarray*} &σ(A_1,...,A_n)= \\ &\Big\{[x_1:\cdots :x_n]\in \C{\mathbb P}^{n-1}: det(x_1A_1+\cdots +x_nA_n)=0\Big \}. \end{eqnarray*} In this paper we develop a local spectral analysis near a regular point of reducibility of a determinantal hypersurface. We prove a rigidity type theorem for representations of Coxeter groups as an application
This article deals with the spectral analysis of the semiclassical Neumann magnetic Laplacian on a smooth bounded domain in dimension three. When the magnetic field is constant and in the semiclassical limit, we establish a four-term asymptotic expansion of the low-lying eigenvalues, involving a geometric quantity along the apparent contour of the domain in the direction of the field. In particular, we prove that they are simple.
We describe the spectrum structure for the restricted Dirichlet fractional Laplacian in multi-tubes, i.e. domains with cylindrical outlets to infinity. Some new effects in comparison with the local case are discovered. In this version, Theorem 4 is essentially improved.
This paper treatises the preservation of some spectra under perturbations not necessarily commutative and generalizes several results which have been proved in the case of commuting operators.
We prove that among all doubly connected domains of R^n (n>=2) bounded by two spheres of given radii, the Dirichlet heat content at any fixed time achieves its minimum when the spheres are concentric. This is shown to be a special case of a more general theorem concerning the optimal placement of a convex obstacle inside some larger domain so as to maximize or minimize the Dirichlet heat content.
An inverse spectral problem is studied for the matrix Sturm-Liouville operator on a finite interval with the general self-adjoint boundary condition. We obtain a constructive solution based on the method of spectral mappings for the considered inverse problem. The nonlinear inverse problem is reduced to a linear equation in a special Banach space of infinite matrix sequences. In addition, we apply our results to the Sturm-Liouville operator on a star-shaped graph.
In this paper, we define an analytical index for continuous families of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Banach space $X.$ We also consider the Weyl spectrum for continuous families of bounded linear operators and we study its continuity.
We consider the Dirac system on the interval $[0,1]$ with a spectral parameter $μ\in\mathbb{C}$ and a complex-valued potential with entries from $L_p[0,1]$, where $1\leq p <2$. We study the asymptotic behavior of its solutions in a stripe $|{\rm Im}\,μ|\le d$ for $μ\to \infty$. These results allows us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm--Liouville operators associated with the aforementioned Dirac system.
We show that a class of de Branges spaces, generated by means of generalized Fourier transforms associated with perturbed Bessel differential equations, has the properties of oversampling and aliasing.
We recall a Moure theory adapted to non self-adjoint operators and we apply this theory to Schr{ö}dinger operators with non real potentials, using different type of conjugate operators. We show that some conjugate operators permits to relax conditions on the derivatives of the potential that were required up to now.
The classical hierarchy of Toda flows can be thought of as an action of the (abelian) group of polynomials on Jacobi matrices. We present a generalization of this to the larger groups of $C^2$ and entire functions, and in this second case, we also introduce associated cocycles and in fact give center stage to this object.
Following the proof given by Froese and Herbst in [FH82] with another conjugate operator, we show for a class of real potential that possible eigenfunction of the Schrödinger operator has to decay sub-exponentially. We also show that, for a certain class of potential, this bound can not be satisfied which implies the absence of strictly positive eigenvalues for the Schrödinger operator.