Semiunitary Transforrnation and Isospectral Hamiltonians in Arbitrary Dimensional Space

By means of the complex Clifford algebra, a new realization of multi-dimensional semiunitary transformation is put forward and then applied to studying the isospectrality of nonrelativistic Hamiltonians of multi-dimensional quantum mechanical systems, in which the generalized Pauli coupling interaction and spin-orbit coupling interaction appear naturally. Moreover, it is shown that the semiunitary operators, together with the Hamiltonian of quantum mechanical system, satisfy the polynomially-deformed angular momentum algebra.

Lorentz violation induces isospectrality breaking in Einstein-bumblebee gravity theory

In this study,we investigate the quasinormal modes(QNMs)of a Lorentz-violating spacetime,factoring in a cosmological constant,within the framework of Einstein-bumblebee gravity.Our findings reveal that the interaction of spacetime with an anisotropic bumblebee field imparts distinct contributions to the axial and polar sectors of the vector perturbations.This subsequently breaks the isospectrality typically observed in vector modes.Numerical evidence strongly indicates isospectral breaking in the vector modes of Einstein-bumblebee black holes:a pronounced breakage in the real part of the frequencies,while the imaginary component seems less affected.This isospectral breaking indicates the existence of two different waveforms in the Ringdown phase of the black hole,which provides a potential signal of quantum gravity observable in current experiments.

Isospectral Operators

For a large class of integral operators or second-order differential operators,their isospectral(or cospectral)operators are constructed explicitly in terms of htransform(duality).This provides us a simple way to extend the known knowledge on the spectrum(or the estimation of the principal eigenvalue)from a smaller class of operators to a much larger one.In particular,an open problem about the positivity of the principal eigenvalue for birth–death processes is solved in the paper.

COMMUTATOR REPRESENTATION OF ISOSPECTRAL EQUATIONS

1. The isospectral evolution equations for the eigenvalue problems Ly=λy and yx=Uy have the Lax form and the zero curvature form, respectively:L1=[V, L]; U1-Vx+[U, V]=0. It is a fundamental problem to search for the conditions, under which the soliton equations u1=X（u） in the vector field form are equivalent to them. In the present paper, a

Hermitizable, isospectral complex second-order differential operators

The first aim of the paper is to study the Hermitizability of secondorder 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.

Note on Isospectral Riemannian Manifolds

1 Introduction Let （M, g） be a compact and connected Riemannian manifold. The Laplace operator △ on functions on M has a discrete spectrum Spec（M, g）= {0=λ0【λ1≤λ2≤…}. We say that two Riemannian manifolds （M, g）

DARBOUX TRANSFORMATIONS CONNECTED WITH HIGHER ORDER ISOSPECTRAL PROBLEMS

The background of the present investigation is the Darboux transformation (DT),which is nowadays an effective method in generating solutions of partial differential equations (PDE) (see [1—5] and the references therein).

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation.Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.

A NEW HIERARCHY OF LAX AND LIOUVILLE INTEGRABLE EVOLUTION EQUATIONS ASSOCIATED WITH AN ISOSPECTRAL PROBLEM IN THE LOOP ALGEBRA■

In this paper, an isospectral problem with five potentials is investigated in loop algebra A2 such that a new hierarchy of evolution equations with five arbitrary functions is obtained. And then by fixing the five arbitrary functions to be certain flmctions and using the trace identity, the generalized Hamiltonian structure of the hierarchy of evolution equations is given, it is shown that this hierarchy of equations is Liouville integrable. Finally some special cases of the isospectral problem are also given.

ON A PAIR OF NON-ISOMETRIC ISOSPECTRAL DOMAINS WITH FRACTAL BOUNDARIES AND THE WEYL-BERRY CONJECTURE

This paper is divided into two parts. In the first part the authors extend Kac's classical problem to the fractal case, i.e., to ask: Must two isospectral planar domains with fractal boundaries be isometric?It is demonstrated that the answer to this question is no, by constructing a pair of disjoint isospectral planar domains whose boundaries have the same interior Bouligand-Minkowski dimension but are not isometric. In the second part of this paper the authors give the exact two-term asymptotics for the Dirichlet counting functions associated with the examples given here and obtain sharp two sided estimates for the second term of the counting functions. The first result in the second part of the paper shows that the coefficient of the second term is an oscillatory function of λ, which implies that the Weyl-Berry conjecture, for the examples given here, is false. The second result implies that the weaker form of the Weyl-Berry conjecture, for these examples, is true. This in turn means that the interior Bouligand-Minkowski dimension of the examples is a spectral invariant.

Discrete integrable system and its integrable coupling

This paper derives new discrete integrable system based on discrete isospectral problem. It shows that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

The Double Integrable Couplings of Tu Hierarchy

Two types of Lie algebras are presented,from which two integrable couplings associated with the Tuisospectral problem are obtained,respectively.One of them possesses the Hamiltonian structure generated by a linearisomorphism and the quadratic-form identity.An approach for working out the double integrable couplings of the sameintegrable system is presented in the paper.

Liouville Integrable System and Associated Integrable Coupling

Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.

Equivalent transformation between the matrices for expanding integrable model of the hierarchy of evolution equation

A direct method for obtaining the expanding integrable models of the hierarchies of evolution equations was proposed. By using the equivalent transformation between the matrices, a new isospectral problem was directly established according to the known isospectral problem, which can be used to obtain the expanding integrable model of the known hierarchy.

Analytical Solutions of the 1D Dirac Equation Using the Tridiagonal Representation Approach

This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.

A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator.The proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the problem.These improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded domains.The authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger operators.As an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

On spectrum of Hermitizable tridiagonal matrices

This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices.As an illustration of the application of the author’s recent results on Hermitizable matrices,an explicit criterion for discrete spectrum of the matrices is presented,with a slight and technical restriction.The problem is well known,but from the author’s knowledge,it has been largely opened for quite a long time.It is important in various application,in quantum mechanics for instance.The main tool to solve the problem is the isospectral technique developed a few years ago.Two alternative constructions of the isospectral operator are presented;they are helpful in theoretical analysis and in numerical computations,respectively.Some illustrated examples are included.

Computing top eigenpairs of Hermitizable matrix

The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.

Non-isometric Riemannian G-Manifolds with Equal Equivariant Spectra

In this paper,the author examines the two methods that people used to systematically construct isospectral non-isometric Riemannian manifolds,the Sunada–Pesce–Sutton method and the torus action method,and shows that both methods can be used to produce equivariantly isospectral non-isometric Riemannian G-manifolds.The author also shows that the Milnor’s isospectral pair is not equivariantly isospectral.