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We consider a class of one-dimensional nonselfadjoint semiclassical pseudodifferential operators, subject to small random perturbations, and study the statistical properties of their (discrete) spectra, in the semiclassical limit h 0. We compare two types of random perturbation: a random potential vs. a random matrix. Hager and Sjöstrand showed that, with high probability, the local spectral density of the perturbed operator follows a semiclassical form of Weyl's law, depending on the value distribution of the principal symbol of our pseudodifferential operator. Beyond the spectral density, we investigate the full local statistics of the perturbed spectrum, and show that it satisfies a form of universality: the statistics only depends on the local spectral density, and of the type of random perturbation, but it is independent of the precise law of the perturbation. This local statistics can be described in terms of the Gaussian Analytic Function, a classical ensemble of random entire functions. [ABSTRACT FROM AUTHOR]

Using adequate strategies, we localize the spectrum of a class of differential operators. We consider the conditioning of the pseudo-spectrum for a family of nonselfadjoint convection-diffusion operators defined on an unbounded open set of ℝ n . [ABSTRACT FROM AUTHOR]

In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a C 1 -conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes. [ABSTRACT FROM AUTHOR]

The spectral properties of non-self-adjoint extensions A [ B ] of a symmetric operator in a Hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding Weyl functions. These extensions are given in terms of abstract boundary conditions involving an (in general non-symmetric) boundary operator B . In the abstract part of this paper, sufficient conditions for sectoriality and m-sectoriality as well as sufficient conditions for A [ B ] to have a non-empty resolvent set are provided in terms of the parameter B and the Weyl function. Special attention is paid to Weyl functions that decay along the negative real line or inside some sector in the complex plane, and spectral enclosures for A [ B ] are proved in this situation. The abstract results are applied to elliptic differential operators with local and non-local Robin boundary conditions on unbounded domains, to Schrödinger operators with δ -potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line, and to quantum graphs with non-self-adjoint vertex couplings. [ABSTRACT FROM AUTHOR]

The Kato Theorem on similarity for sequences of projections in a Hilbert space is extended to the case when both sequences consist of nonselfadjoint projections. Passing to subspaces, this leads to stability theorems for Riesz bases of subspaces, at least one of which is finite dimensional, and for arbitrary vector Riesz bases. The following is proved as an application. If {φn}∞ n=1 is a Riesz basis and |θn| ≤ C for large n, where the constant C depends only on {φn}∞ n=1, then {φn + θnφn+1}∞ n=1 also forms a Riesz basis. [ABSTRACT FROM AUTHOR]

Parseval equality is proved for Sturm-Liouville equation with block-triangular, increasing at infinity operator potential. [ABSTRACT FROM AUTHOR]

In a previous paper we generalized the theory of W ⁎ -modules to the setting of modules over nonselfadjoint dual operator algebras, obtaining the class of weak ⁎ -rigged modules. At that time we promised a forthcoming paper devoted to other aspects of the theory. We fulfill this promise in the present work and its sequel “Rigged modules II”, giving many new results about weak ⁎ -rigged modules and their tensor products. We also discuss the Picard group of weak* closed subalgebras of a commutative algebra. For example, we compute the weak Picard group of H ∞ ( D ) , and prove that for a weak* closed function algebra A , the weak Picard group is a semidirect product of the automorphism group of A , and the subgroup consisting of symmetric equivalence bimodules. [ABSTRACT FROM AUTHOR]

This paper is concerned with the classification of non-self-adjoint second-order difference equations. The relationship between the number of summable solutions of non-self-adjoint difference equations and that of the discrete linear Hamiltonian system is discussed. A classification for non-self-adjoint second-order difference equations is established, which is independent of the choice of the rotated half plane and the fixed point. [ABSTRACT FROM AUTHOR]

In this work, we use regularized determinants to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schrödinger operator ( − i ∇ − A ) 2 − b in R 3 , with constant magnetic field of strength b > 0 . The distribution of the above discrete spectrum near the Landau levels 2 b q , q ∈ N , is more interesting since they play the role of thresholds of the spectrum of the free operator. First, we obtain sharp upper bounds on the number of complex eigenvalues near the Landau levels. Under appropriate hypothesis, we then prove the presence of an infinite number of complex eigenvalues near each Landau level 2 b q , q ∈ N , and the existence of sectors free of complex eigenvalues. We also prove that the eigenvalues are localized in certain sectors adjoining the Landau levels. In particular, we provide an adequate answer to the open problem from [34] about the existence of complex eigenvalues accumulating near the Landau levels. Furthermore, we prove that the Landau levels are the only possible accumulation points of the complex eigenvalues. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space L 2 ( R ) ⊕ L 2 ( R ) induced by matrix differential expressions of the form (0.1) ( τ 11 ( ⋅ , D ) τ 12 ( ⋅ , D ) τ 21 ( ⋅ , D ) τ 22 ( ⋅ , D ) ) , where τ 11 , τ 12 , τ 21 , τ 22 are respectively m -th, n -th, k -th and 0 order ordinary differential expressions with m = n + k being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. It turns out that the points of the essential spectrum either have a local origin, which can be traced to points where the ellipticity in the sense of Douglis and Nirenberg breaks down, or they are caused by singularity at infinity. [ABSTRACT FROM AUTHOR]

In this paper, Sturmian comparison theory is developed for the pair of second-order differential equations; first of which is the nonlinear differential equations of the form [ABSTRACT FROM AUTHOR]

We examine Nica-Pimsner algebras associated with semigroup actions of ℤn+ on a C*-algebra A by *-endomorphisms. We give necessary and sufficient conditions on the dynamics for exactness and nuclearity of the Nica-Pimsner algebras. Furthermore, we parameterize the KMS states at finite temperature by the tracial states on A. A parametrization is also shown for KMS states at zero temperature (resp. ground states) by the tracial states on A (resp. states on A). Finally, we give a formula for obtaining tracial states on the Nica-Pimsner algebras. [ABSTRACT FROM AUTHOR]

In this paper, we determine the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP) -y"+q(x)y=λ2y, x∊R+ = [0,∞] (α0+ α1λ+2αλ2y'(0) - (β0 + β1λ+ β2λ2y(0)=0 where q is a complex-valued function,αi,βi ∊C, i=0,1,2 and λ is a eigenparameter, and introduce the convergence properties of principal functions. [ABSTRACT FROM AUTHOR]

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained. [ABSTRACT FROM AUTHOR]

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated. A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions. [ABSTRACT FROM AUTHOR]

We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds. We then prove that the birth of point eigenvalues with nonzero real part (the ones which lead to linear instability) from the essential spectrum is only possible from the embedded eigenvalues or thresholds, and therefore can not take place beyond the embedded thresholds. We also prove that “in the nonrelativistic limit” ω → m , the point eigenvalues can only accumulate to 0 and ± 2 m i . [ABSTRACT FROM AUTHOR]

Under study are some commuting rank 2 differential operators with polynomial coefficients. We prove that, for every spectral curve of the form w = z + c z 2+ c z+ c with arbitrary coefficients c , there exist commuting nonselfadjoint operators of orders 4 and 6 with polynomial coefficients of arbitrary degree. [ABSTRACT FROM AUTHOR]

Let $( Pu ) ( t ) =- \frac{d}{dt} ( \omega^{2} ( t ) q ( t ) \frac{du ( t )}{dt} )$ be a degenerate non-self-adjoint operator defined on the space $H_{\ell} = L^{2} (0,1)^{\ell}$ with Dirichlet-type boundary conditions, where $\omega(t)\in C^{1} (0,1)$ is a positive function with further assumptions that will be specified later, and $q(t)\in C^{2} ( [ 0,1 ], \operatorname{End} C^{\ell} )$ is a matrix function. In this article, some spectral characteristics of the operator P are considered. We estimate the resolvent of P and then prove the limit argument theorem. Finally, we find a formula for the distribution of eigenvalues of the operator P acting on $H_{\ell}$ . [ABSTRACT FROM AUTHOR]

The transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory. In this paper, for the Helmholtz transmission eigenvalue problem, we give a weak formulation which is a nonselfadjoint linear eigenvalue problem. Based on the weak formulation, we first discuss the non-conforming finite element approximation, and prove the error estimates of the discrete eigenvalues obtained by the Adini element, the Morley–Zienkiewicz element, the modified-Zienkiewicz element et al. And we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem. [ABSTRACT FROM AUTHOR]

Bagarello, Inoue, and Trapani [J. Math. Phys. 55, 033501 (2014)] investigated some operators defined by the Riesz bases. These operators connect with quasi- Hermitian quantum mechanics, and its relatives. In this paper, we introduce a notion of generalized Riesz bases which is a generalization of Riesz bases and investigate some operators defined by the generalized Riesz bases by changing the frameworks of the operators defined in the work of Bagarello, Inoue, and Trapani. [ABSTRACT FROM AUTHOR]

In this paper we give a new proof of a result by S. Reich and A.J. Zaslavski (S. Reich and A.J. Zaslavski, A fixed point theorem for Matkowski contractions, Fixed Point Theory, 8(2007), No. 2, 303-307). Some new fixed point theorems for nonself generalized contractions are also given. [ABSTRACT FROM AUTHOR]

We consider a class of modified Schrödinger operators where the semiclassical Laplacian is perturbed with h-dependent interface conditions occurring at the boundaries of the potential's support. Under positivity assumptions on the potential, we show that this modification produces a small perturbation on the dynamics as h → 0, independently from the time scale. In the case of a time dependent potential, this yields uniform-in-h stability estimates for products of instantaneous propagators. Then, following a standard approach, the non-autonomous dynamical system is defined as a limit of stepwise propagators and its small-h expansion is provided under suitable regularity assumptions on the potential's variations. [ABSTRACT FROM AUTHOR]

Metallic lithium is used to demonstrate the possibilities of applying non-self-adjoint operators for quantitative description of orbital excitations of electrons in crystals. It is shown that, the nonequilibrium distribution function can be calculated when solving the spectral problem; therefore, the kinetic properties of a material can also be described with the unified band theory. [ABSTRACT FROM AUTHOR]

We study the structure of weakly-closed nonself-adjoint algebras arising from representations of single vertex 2-graphs. These are the algebras generated by 2 isometric tuples which satisfy a certain commutation relation. We show that these algebras have a lower-triangular 3×3 form. The left-hand side of this matrix decomposition is a slice of the enveloping von Neumann algebra generated by the 2-graph algebra. We further give necessary and sufficient conditions for these algebras themselves to be von Neumann algebras. The paper concludes with further study of atomic representations. [ABSTRACT FROM AUTHOR]

Nonselfadjoint second-order differential pencils on a finite interval with nonseparated boundary conditions are studied. We establish some important properties of spectral characteristics and investigate inverse problems of recovering the operator from its spectral data. For these inverse problems, we prove the corresponding uniqueness theorems and provide procedures for constructing their solutions. [ABSTRACT FROM AUTHOR]

It is shown in the limit-circle case that system of root functions of the non-self-adjoint maximal dissipative (accumulative) Bessel operator and its perturbation Sturm-Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the maximal dissipative (accumulative) Bessel operators is investigated, and it is proved that system of root functions form a basis (Riesz and Bari bases) in the same Hilbert space. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The current work is concerned with eigenfunction expansion of the transient response of general one-dimensional non-self-adjoint distributed dynamic systems. The main goal of the current work is to propose an alternative eigenfunction expansion approach for transient analysis of general one-dimensional non-self-adjoint systems which avoids complex-valued eigenvalue problems. With the implicit filter scheme proposed here one splits the solution into two additive components, namely, a filtered solution for which all boundary conditions are homogeneous and a carefully selected filter. Moreover, the eigenvalue problem associated with the differential equation for the filtered solution is self-adjoint and, therefore, all eigenvalues are real-valued and the eigenfunctions satisfy the classical orthogonality relations. The unknown time-dependent coefficients of the filter are then solved simultaneously with the modal co-ordinates of the filtered solution in order to satisfy the original boundary conditions. The proposed approach is demonstrated on a distributed system composed of a cantilever beam with end mass, viscous damper and linear spring. [ABSTRACT FROM AUTHOR]

In this paper, solutions of different nonlinear differential equations are obtained using the connection between the theory of solitons and the theory of commuting nonselfadjoint operators, established by M.S. Livšic and Y. Avishai in [15] and based on the Marchenko method for solving nonlinear differential equations. n -tuples of commuting nonselfadjoint bounded linear operators, when one of them belongs to a larger class of nondissipative operators, are used. Preliminary results are obtained with the help of the properties of this larger class of operators—couplings of dissipative and antidissipative operators with real absolutely continuous spectra (whose triangular model and corresponding nondissipative curves have been introduced and investigated by the authors in [5] ). The preliminary results concern the application of this connection for the considered n -tuples of commuting operators. A generalized open system, corresponding to collective motions, is introduced. Solutions of the nonlinear Schrödinger equation, the Heisenberg equation, the Sine-Gordon equation, the Davey–Stewartson equation are obtained, using the generalized open systems and the matrix wave equations for appropriate pairs and triplets of commuting nonselfadjoint operators. [ABSTRACT FROM AUTHOR]

In this paper we investigate the one-dimensional Schrodinger operator L ( q ) with complex-valued periodic potential q when q ∈ L 1 [ 0 , 1 ] and q n = 0 for n = 0 , − 1 , − 2 , . . . , where q n are the Fourier coefficients of q with respect to the system { e i 2 π n x } . We prove that the Bloch eigenvalues are ( 2 π n + t ) 2 for n ∈ Z , t ∈ C and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator. [ABSTRACT FROM AUTHOR]

We study a system of nonlinear differential equations used as basic for the construction of triangular models for commutative systems of linear nonself-adjoint bounded operators. [ABSTRACT FROM AUTHOR]

Using the method of similar operators, we study the spectral properties of a fourth-order differential operator with two types of classical boundary conditions. We obtain asymptotics for the spectrum and estimates for the spectral decompositions of this operator. We construct the semigroup generated by the opposite fourth-order differential operator. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with the perturbed operatorwhereis a closed densely defined linear operator on a separable Hilbert spaceXwith domainwhileare linear operators onXwith the same domainand satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions assuring the existence of a Riesz basis of eigenvectors of the operator. The obtained results are of importance for application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid and to a nonself-adjoint Gribov operator in Bargmann space. [ABSTRACT FROM AUTHOR]

We consider a pencil of non-self-adjoint matrix Sturm-Liouville operators on the half line and study the inverse problem of constructing this pencil by its Weyl matrix. A uniqueness theorem is proved, and a constructive algorithm for the solution is obtained. [ABSTRACT FROM AUTHOR]

In the work by V. A. Zolotarev, Dokl. Akad. Nauk Arm. SSR, 63, No. 3, 136-140 (1976), a triangular model is constructed for a system of twice-commuting linear bounded completely nonself-adjoint operators { A, A} ([ A, A] = 0, [ A, A] = 0) such that rank ( A)( A) = 1 (2 i( A) = A − A, k = 1, 2) and the spectrum of each operator A, k = 1, 2 , is concentrated at zero. The indicated triangular model has the form of a system of operators of integration over the independent variable in L where the domain Ω = [0, a] × [0, b] is a compact set in ℝ bounded by the lines x = a and y = b and a decreasing smooth curve L connecting the points (0 , b) and ( a, 0) . [ABSTRACT FROM AUTHOR]

Abstract: We consider a new class of non-self-adjoint matrices that arise from an indefinite self-adjoint linear pencil of matrices, and obtain the spectral asymptotics of the spectra as the size of the matrices diverges to infinity. We prove that the spectrum is qualitatively different when a certain parameter c equals 0, and when it is non-zero, and that certain features of the spectrum depend on Diophantine properties of c. [Copyright &y& Elsevier]

Abstract: Let and be respectively perturbations of the unperturbed Schrödinger operators on and on with constant magnetic field of strength , and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H and . In particular, these estimates give a priori information on the distribution of eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge. [Copyright &y& Elsevier]

Thesolvability of the inverse boundary problem with an unknown coefficient dependent on time for the third order pseudoparabolic equation with non-self-adjoint boundary conditions is investigated in the present paper. Here we have introduced the definition of the classical solution of the considered inverse boundary value problem, which is reduced to the system of integral equations by the Fourier method. At first, the existence and uniqueness of the solution of the obtaining system of integral equations is proved by the method of contraction mappings; then the existence and uniqueness of the classical solution of the stated problem is proved. [ABSTRACT FROM AUTHOR]

Abstract: The spectral properties of a class of tridiagonal matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The obtained results extend some results on the direct and inverse spectral problems for periodic Jacobi matrices and for some non-self-adjoint tridiagonal matrices. [Copyright &y& Elsevier]

Abstract: We obtain eigenvalue enclosures and basisness results for eigen- and associated functions of a non-self-adjoint unbounded linear operator pencil in which is uniformly positive and the essential spectrum of the pencil is empty. Both Riesz basisness and Bari basisness results are obtained. The results are applied to a system of singular differential equations arising in the study of Hagen–Poiseuille flow with non-axisymmetric disturbances. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the nonselfadjoint (dissipative) boundary value transmission problems in Weyl’s limit-circle case. At first using the method of operator-theoretic formulation we pass to a new operator. After showing that this new operator is a maximal dissipative operator, we construct a selfadjoint dilation of the maximal dissipative operator. Using the equivalence of the Lax–Phillips scattering function and the Sz.-Nagy-Foiaş characteristic function, we show that all eigenfunctions and associated functions are complete in the space . [Copyright &y& Elsevier]

We study operator algebras associated to integral domains. In particular, with respect to a set of natural identities we look at the possible nonselfadjoint operator algebras which encode the ring struc- ture of an integral domain. We show that these algebras give a new class of examples of semicrossed products by discrete semigroups. We inves- tigate the structure of these algebras together with a particular class of representations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the criteria of the three cases for non-self-adjoint singular Sturm-Liouville difference equations. It has been known that this class of equations was classified into cases I, II and III by R.H. Wilson (Proc. R. Soc. A 401(2005)). In this paper, several criteria of the cases I, II and Ill are established, respectively. [ABSTRACT FROM AUTHOR]

Abstract: We prove that solutions to elliptic equations in two variables in divergence form, possibly non-self-adjoint and with lower order terms, satisfy the strong unique continuation property. [Copyright &y& Elsevier]

Let L denote the operator generated in ℓ2(N, C2) by an+1yn+1 (2) + bnyn(2) + pnyn(1) = λyn(1), an-1 yn-1(1)+bnyn (1)+qnyn(2) = λyn(2), n∈N and the boundary condition (γ0 + γ1λ)y1(2) + ( β0 + β1λ)y0(1) = 0, where (an), (bn),(pn), and (qn), n ∈ N are complex sequences γi, βi ∈ C, i =0,1, and λ is an eigenparameter. In this paper we investigated the principal functions corresponding to the eigenvalues and the spectral singularities of L. [ABSTRACT FROM AUTHOR]

Full description of Riesz basis property for eigenfunctions of boundary value problems for first order differential equations with involutions is given. [ABSTRACT FROM AUTHOR]

We consider the maximal dissipative second-order difference (or discrete Sturm-Liouville) operators acting in the Hilbert space lw² (Z) (Z:= {0, ±1, ±2, . . .}), that is, the extensions of a minimal symmetric operator with defect index (2, 2) (in the Weyl-Hamburger limit-circle cases at ±8). We investigate two classes of maximal dissipative operators with separated boundary conditions, called "dissipative at -8" and "dissipative at 8." In each case, we construct a self-adjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also establish a functional model of the maximal dissipative operator and determine its characteristic function through the Titchmarsh-Weyl function of the self-adjoint operator. We prove the completeness of the system of eigenvectors and associated vectors of the maximal dissipative operators. [ABSTRACT FROM AUTHOR]

The inverse problem for the Schrodinger operator with complex periodic potentials and discontinuous coefficient on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is analysed. The inverse problem is formulated, a uniqueness theorem is proved, a constructive procedure for the solution of the inverse problem is provided. [ABSTRACT FROM PUBLISHER]