On classification of second-order differential equations with complex coefficients

This paper is concerned with the deficiency index problem of second-order differential equations with complex coefficients. It is known that this class of equations is classified into cases I, II, and III according to the number of linearly independent solutions in suitable weighted square integrable spaces. In this study, the original equation is reformulated into a new formally self-adjoint differential system by introducing a new spectral parameter and the relationship between the classifications of the equation and the system is obtained. Moreover, the exact dependence of cases II and III on the corresponding half planes is given and some criteria of the three cases are established.     

Variations on a theme of Jost and Pais

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set , , n2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in L²(Ω;dⁿx), , to modified Fredholm determinants associated with operators in L²(∂Ω;dⁿ⁻¹σ), n2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed.     

Decomposing the essential spectrum

We use C^*-algebra theory to provide a new method of decomposing the essential spectra of self-adjoint and non-self-adjoint Schrödinger operators in one or more space dimensions.     

Non-self-adjoint Jacobi matrices with a rank-one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an n×n Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy-Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration in the Hilbert space L₂[0,l] is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.     

Titchmarsh-Sims-Weyl theory for complex Hamiltonian systems on Sturmian time scales

We study non-self-adjoint Hamiltonian systems on Sturmian time scales, defining Weyl-Sims sets, which replace the classical Weyl circles, and a matrix-valued M-function on suitable cone-shaped domains in the complex plane. Furthermore, we characterize realizations of the corresponding dynamic operator and its adjoint, and construct their resolvents. Even-order scalar equations and the Orr-Sommerfeld equation on time scales are given as examples illustrating the theory, which are new even for difference equations. These results unify previous discrete and continuous theories to dynamic equations on Sturmian time scales.     

A PRIORI AND A POSTERIORI ERROR ESTIMATES OF A WEAKLY OVER-PENALIZED INTERIOR PENALTY METHOD FOR NON-SELF-ADJOINT AND INDEFINITE PROBLEMS

In this paper, we study a weakly over-penalized interior penalty method for non-self- adjoint and indefinite problems. An optimal a priori error estimate in the energy norm is derived. In addition, we introduce a residual-based a posteriori error estimator, which is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to validate our theoretical analysis.     

Lieb–Thirring type inequalities for non-self-adjoint perturbations of magnetic Schrödinger operators

Let H:=_H0+V$H:=H_0+V$ and _H⊥:=_H0,⊥+V$H_{\perp }:=H_{0,\perp }+V$ be respectively perturbations of the unperturbed Schrödinger operators _H0$H_0$ on L²(R³)$L^2({\mathbb{R}}^3)$ and _H0,⊥$H_{0,\perp }$ on L²(R²)$L^2({\mathbb{R}}^2)$ with constant magnetic field of strength b>0$b>0$, and V a complex relatively compact perturbation. We prove Lieb–Thirring type inequalities on the discrete spectrum of H   and _H⊥$H_{\perp }$. 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.     

Trace formulas for non-self-adjoint periodic Schrödinger operators and some applications

Recently, a trace formula for non-self-adjoint periodic Schrödinger operators in L²(R) associated with Dirichlet eigenvalues was proved in [Differential Integral Equations 14 (2001) 671-700]. Here we prove a corresponding trace formula associated with Neumann eigenvalues. In addition we investigate Dirichlet and Neumann eigenvalues of such operators. In particular, using the Dirichlet and Neumann trace formulas, we provide detailed information on location of the Dirichlet and Neumann eigenvalues for the model operator with the potential Ke2ix, where K∈C.     

Trapped modes in coastal waveguides

Postnova and Craster (Wave Motion, 45, 2008, pp. 565-579) describe a method for determining the frequency of trapped modes in slowly-varying elastic plates, and ocean and quantum waveguides. The purpose of the present note is to show that the accuracy of the frequency estimates for ocean waveguides can be significantly increased by taking into account the fact that, as posed, the ocean waveguide problem is not self-adjoint. For an example where the asymptotic problem has an exact solution, comparison with a numerical solution of the full problem shows that the correction to the asymptotically determined frequency is of order the fourth power of the ratio of the shelf width to the scale for longshore variations in the shelf. An explicit simple formula is also given for the trapped mode frequency of an arbitrarily, but extremely weakly and positively, curving coast.