Isospectral Operators Research Articles

Exact solvable family of discrete Schrödinger operators with long-range hoppings

It is shown that one-dimensional discrete Schrödinger operators, with a uniform electric field and long-range hoppings, form an isospectral family with discrete and simple spectrum (and isospectral operators with eigenfunctions having different decay properties). Explicit relations between hopping decay rates and eigenvectors decay rates are obtained. Small perturbations of exponentially decay hopping operators preserve dynamical localization.

Hermitizable, isospectral complex second-order differential operators

The first aim of the paper is to study the Hermitizability of second-order differential operators, and then the corresponding isospectral operators. The explicit criteria for the Hermitizable or isospectral properties are presented. The second aim of the paper is to study a non-Hermitian model, which is now well known. In a regular sense, the model does not belong to the class of Hermitizable operators studied in this paper, but we will use the theory developed in the past years, to present an alternative and illustrated proof of the discreteness of its spectrum. The harmonic function plays a critical role in the study of spectrum. Two constructions of the function are presented. The required conclusion for the discrete spectrum is proved by some comparison technique.

Criteria for two spectral problems of 1D operators

This paper introduces shortly recent progress on the following three problems: The principal eigenvalue in four cases, isospectral operators, and discrete spectrum, for birth-death processes and one-dimensional diffusion processes. Unified basic estimates of principal eigenvalues in various situations are obtained, the ratio of their upper and lower bounds is no more than 4. Short criteria for discrete spectrum are presented in dimension one.

Isospectrality and heat content

We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schr\"{o}dinger operators acting in $L^2[0,1]$ with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.

Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. I: General classification of reducibility and analysis of the third-order algebra

We study possible factorizations of supersymmetric (SUSY) transformations in the one-dimensional quantum mechanics into chains of elementary Darboux transformations with nonsingular coefficients. A classification of irreducible (almost) isospectral transformations and of related SUSY algebras is presented. A detailed analysis of SUSY algebras and isospectral operators is performed for the third-order case. Bibliography: 29 titles.

On the complete integrability of completely integrable systems

The question of complete integrability of evolution equations associated to $n\times n$ first order isospectral operators is investigated using the inverse scattering method. It is shown that for $n>2$, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle variables. This construction depends on the analysis of a natural 2-form and symplectic foliation for the groups GL(n) and SU(n).}

Iso-spectral operators: Some model examples with discontinuous coefficients

Some explicit examples are presented of operators with different discontinuous coefficient functions which possess eigenvalue spectra identical to each other and to the corresponding operator with continuous coefficients. Cases covered include, for the second-order (Sturm-Liouville) problem, the vibrating stepped-density string and the circularly symmetric vibrations of a stepped, inverse-square density annular membrane. Fourth-order systems dealt with correspond to vibrating beams with various end-conditions.