Our main purpose is to introduce the concepts of upper and lower weakly (Λ, sp)- continuous multifunctions. In particular, some characterizations of upper and lower weakly (Λ, sp)- continuous multifunctions are investigated.
In this paper, we study block Jacobi operators on $\mathbb{Z}$ with quasi-periodic meromorphic potential. We prove the non-perturbative Anderson localization for such operators in the large coupling regime.
Spectra of functionals $$Φ(u)=\frac{\left\langle u^{(n)}u^{(n)}\right\rangle}{\left\langle u^{(n-p)}u^{(n-p)}\right\rangle}$$ in spaces ${\mathop{W}\limits^\circ}^2_n$ are considered for different $n$. One has shown that for even functions in $\mathop{W}\limits^\circ{}^2_n$ and $\mathop{W}\limits^\circ{}^2_m$ spectra of functionals do not intersect for $m=n+1, n+2$. The neccesary conditions for two spectra to intersect are written for $Δ=m-n>2$.
For any subgraph of a graph, the Laplacian with Neumann boundary condition was introduced by Chung and Yau [CY94]. In this paper, motivated by the Riemannian case, we introduce the Cheeger constants for Neumann problems and prove corresponding Cheeger estimates for first nontrivial eigenvalues.
We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations on some matrix-valued functions and few number of integrations, the discrete part is much more complicated.
We consider the differential system $y'-x^{-1}Ay-q(x)y=ρBy $ with $n\times n$ matrices $A,B, q(x)$, where $A,B$ are constant, $B$ is diagonal, $A$ and $q(x)$ are off-diagonal, $q(\cdot)\in W^1_1[0,\infty)$. Some distinguished fundamental system of solutions is constructed. Also, we discuss the inverse scattering problem and obtain the uniqueness result.
The self-adjoint and $m$-sectorial extensions of coercive Sturm-Liouville operators are characterised, under minimal smoothness conditions on the coefficients of the differential expression.
New estimates for eigenvalues of non-self-adjoint multi-dimensional Schrödinger operators are obtained in terms of $L_{p}$-norms of the potentials. The results extend and improve those obtained previously. In particular, diverse versions of an assertion conjectured by Laptev and Safronov are discussed. Schrödinger operators with slowly decaying potentials are also considered.
The objective in this paper is to demonstrate that four of the most used techniques in applied mathematics, viz., Fourier series, Fourier transform, Laplace transform and the Fourier-Laplace transform can be introduced using eigenvalue problems for first order differential operators with discrete/continuous spectra.
For $s\textgreater{}0$, let $H\_0=(-Δ)^s$ be the fractional Laplacian. In this paper, we obtainLieb-Thirring type inequalities for the fractional Schrödinger operator defined as $H=H\_0+V$,where $V \in L^p(\mathbb{R}^d), p\ge 1, d\ge 1,$ is a complex-valued potential.Our methods are based on results of articles by Borichev-Golinskii-Kupin \cite{BoGoKu} and Hansmann \cite{Ha1}.
We prove the Bari-Markus property for spectral projectors of non-self-adjoint Dirac operators on a finite interval with square-integrable matrix-valued potentials and some separated boundary conditions.
In the terms of triples $D^+\to H\to D^-$ of Hilbert spaces we construct an analogue of Friedrichs's extension for operator matrices. Also we establish some general approach to construction of variational principles for such matrices.
The paper presents a lower bound for the number of negative eigenvalues of an integral operator with continuous kernel K lying below a nonpositive number t. The estimate is given in terms of some integrals of K.
In this paper we describe the asymptotic behaviour of the spectral norm of the product of two finite Toeplitz matrices as the matrix dimension goes to the infinity. These Toeplitz matrices are generated by positive functions with Fisher-Hartwig singularities of negative order. Since we have positive operators it is known that the spectral norm is also the largest eigenvalue of this product.
We give a new sufficient condition for existence and completeness of wave operators in abstract scattering theory. This condition generalises both trace class and smooth approaches to scattering theory. Our construction is based on estimates for the Cauchy transforms of operator valued measures.
We establish in this paper an upper bound on the second eigenvalue of n-dimensional spheres in the conformal class of the round sphere. This upper bound holds in all dimensions and is asymptotically sharp as the dimension increases.
We show that a if a Riemannian manifold admits a universal cover with bounded geometry and if 0 does not belong to the spectrum or is an isolated point in the spectrum of the Laplacian on $\ell$-forms, then there exists $1<p<2$ such that for all $p<r<p^{\prime}$ the Hodge - de Rham decomposition for $L^{r}$-forms holds ($p^{\prime}$ denotes the conjugate of $p$).
Abstract We describe the supercuspidal representations of Sp 4 ( F ), for F a non-archimedean local field of residual characteristic different from two, and determine which are generic.
We define the concept of completely regular ordinary differential operators and give various criteria for operators to belong to this class. We give also criteria for Birkhof regularity of ordinary differential operators in terms of the growth of the Green function and basis property.