Isospectral Deformations Research Articles

Class of Nonlinearity Control Parameter for Bright Solitons of Non-Autonomous NLSE With Trapping Potential

We have obtained the family of exact soliton solutions, of the non-autonomous nonlinear Schrödinger equation with the harmonic trap, by invoking the isospectral deformation technique, which describe the propagation of bright solitons in the 1-D Bose-Einstein condensate confined by a harmonic potential with a varying-in-time longitudinal trapping frequency. This generalization also enables one to analytically study the effect of the nonlinearity control parameter in managing the amplitude and width of the soliton profile.

Invariants of Isospectral Deformations and Spectral Rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (ℤ/2ℤ)2 group of symmetries.

Infinitesimal Isospectral Deformations of Symmetric Spaces, II: Quotients of the Special Unitary Group of Rank Two

Infinitesimal Isospectral Deformations of Symmetric Spaces, II: Quotients of the Special Unitary Group of Rank Two

A novel class of isospectral deformations in supersymmetric quantum mechanics

We deduce a novel class of three-parameter isospectral deformations in supersymmetric quantum mechanics. It is shown that the only previously known one-parameter class of deformations is a member of this larger class.

Constructing Exactly Solvable Pseudo-hermitian Many-Particle Quantum Systems by Isospectral Deformation

A class of non-Dirac-hermitian many-particle quantum systems admitting entirely real spectra and unitary time-evolution is presented. These quantum models are isospectral with Dirac-hermitian systems and are exactly solvable. The general method involves a realization of the basic canonical commutation relations defining the quantum system in terms of operators those are hermitian with respect to a pre-determined positive definite metric in the Hilbert space. Appropriate combinations of these operators result in a large number of pseudo-hermitian quantum systems admitting entirely real spectra and unitary time evolution. Examples of a pseudo-hermitian rational Calogero model and XXZ spin-chain are considered.

An Isospectral Deformation on an Infranil-Orbifold

AbstractWe construct a Laplace isospectral deformation of metrics on an orbifold quotient of a nilmanifold. Each orbifold in the deformation contains singular points with order two isotropy. Isospectrality is obtained by modifying a generalization of Sunada's theorem due to DeTurck and Gordon.

The Dirac operator on compact quantum groups

For the q-deformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of U𝔤 ⊗ Cl(𝔤). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.

Infinitesimal Isospectral Deformations of Symmetric Spaces: Quotients of The Special Unitary Group

Infinitesimal Isospectral Deformations of Symmetric Spaces: Quotients of The Special Unitary Group

Isospectral Deformations of Eguchi-Hanson Spaces as Nonunital Spectral Triples

We study the isospectral deformations of the Eguchi-Hanson spaces along a torus isometric action in the noncompact noncommutative geometry. We concentrate on locality, smoothness and summability conditions of the nonunital spectral triples, and relate them to the geometric conditions to be noncommutative spin manifolds

Implementation of holonomic quantum gates by an isospectral deformation of an Ising dimer chain

We exactly construct one- and two-qubit holonomic quantum gates in terms of isospectral deformations of an Ising model Hamiltonian. A single logical qubit is constructed out of two spin-$\frac{1}{2}$ particles; the qubit is a dimer. We find that the holonomic gates obtained are discrete but dense in the unitary group. Therefore an approximate gate for a desired one can be constructed with arbitrary accuracy.

Quantum Isometry Groups: Examples and Computations

In this follow-up of [4], where the quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold.

Normalizable states through deformation of Lamé and the associated Lamé potentials

We construct new aperiodic potentials with positive energy bound states starting from the Lamé and the associated Lamé potentials by confining them to the half line. The method used is the isospectral deformation discussed in supersymmetric quantum mechanics. The new potentials have normalizable eigenfunctions, with energies the same as certain levels of the original potentials on the half line. These states are similar to the von Neumann–Wigner states as they have an infinite number of zeros. The new potentials and their eigenfunctions are obtained numerically and their plots are given.

On localization of the spectrum of non-self-adjoint differential operators

In this paper, we consider the problem on localization of the spectrum of non-self-adjoint differential operators on unbounded domains with power coefficients. To find the location of spectrum points in the complex plane, we use isospectral deformations of differential operators and the properties of families of closed operators analytic in the Kato sense.

Solvmanifolds and noncommutative tori with real multiplication

We prove that the Shimizu L-function of a real quadratic field is obtained from a (Lorentzian) spectral triple on a noncommutative torus with real multiplication, as an adiabatic limit of the Dirac operator on a 3-dimensional solvmanifold. The Dirac operator on this 3-dimensional geometry gives, via the Connes-Landi isospectral deformations, a spectral triple for the noncommutative tori obtained by deforming the fiber tori to noncommutative spaces. The 3-dimensional solvmanifold is the homotopy quotient in the sense of Baum--Connes of the noncommutative space obtained as the crossed product of the noncommutative torus by the action of the units of the real quadratic field. This noncommutative space is identified with the twisted group C*-algebra of the fundamental group of the 3-manifold. The twisting can be interpreted as the cocycle arising from a magnetic field, as in the theory of the quantum Hall effect. We prove a twisted index theorem that computes the range of the trace on the K-theory of this noncommutative space and gives an estimate on the gaps in the spectrum of the associated Harper operator.

Coherent state maps related to the bounded positive operators

We show that for any bounded positive operator H with the simple spectrum, one can canonically define two coherent state maps. The algebras generated by annihilation operators defined by these coherent state maps are studied. We describe also how the Toda isospectral deformation of H deforms the corresponding coherent state maps and the related operator algebras.

Inverse Spectral Problems in Rectangular Domains

We consider the Schrödinger operator − Δ + q in domains of the form R = {x ∈ ℝ n : 0 ≤ x i ≤ a i , i = 1,…, n} with either Dirichlet or Neumann boundary conditions on the faces of R, and study the constraints on q imposed by fixing the spectrum of − Δ + q with these boundary conditions. We work in the space of potentials, q, which become real-analytic on ℝ n when they are extended evenly across the coordinate planes and then periodically. Our results have the corollary that there are no continuous isospectral deformations for these operators within that class of potentials. This work is based on new formulas for the trace of the wave group in this setting. In addition to the inverse spectral results these formulas lead to asymptotic expansions for the traces of the wave and heat kernels on rectangular domains.

Families of Noncommutative Instantons

These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations) and give a detailed introduction to gauge theories on a toric four-sphere. This includes a Yang-Mills action functional with associated equations of motion and self-duality equations. We construct a particular class of instanton solutions on a SU(2) bundle with a suitable use of twisted conformal symmetries. In the second part, we describe a different deformation of an instanton bundle over the classical four-sphere by constructing a quantum group SU_q(2) bundle on a sphere which is different from the toric one.

Infinitesimal isospectral deformations of the Grassmannian of 3-planes in R6

Ce memoire a pour cadre la grassmannienne G R n,n des n-plans de R 2n , avec n ≥ 3, et son espace reduit G R n,n' qui est l'espace symetrique irreductible, quotient de G R n,n par l'involution envoyant un n-plan sur son orthogonal. Un de nos principaux resultats est la construction de deformations infinitesimales isospectrales non triviales sur GR 3,3 , obtenant ainsi le premier exemple d'espace symetrique irreductible reduit et non infinitesimalement rigide. Nous donnons aussi un critere d'exactitude pour les formes differentielles de degre 1 sur G R n,n , mettant en jeu la nullite d'une transformee de Radon.

Isospectral deformations of the Lagrangian Grassmannians

Nous étudions la grassmannienne lagrangienne spéciale SU(n)/ SO(n), avec n≥3, et son espace réduit X, qui est l’espace symétrique irréductible de rang n-1 quotient de SU(n)/SO(n) par l’action d’un groupe cyclique d’isometries d’ordre n. Notre résultat principal est la construction de déformations infinitésimales isospectrales non triviales de X. Nous obtenons ainsi les premiers exemples en rang quelconque ≥2 d’espaces symétriques irréductibles réduits et non infinitésimalement rigides. Notre résultat peut être vu comme une généralisation de la construction que nous avions donnée dans un précédent papier pour la grassmannienne réduite des 3-plans de ℝ 6 , espace qui est en fait isométrique à l’espace réduit de SU(4)/SO(4).

Isoscattering deformations for complete manifolds of negative curvature

We construct continuous families of nonisometric metrics on simply connected manifolds of dimension n ≥ 9which have the same scattering phase, the same resolvent resonances, and strictly negative sectional curvatures. This situation contrasts sharply with the case of compact manifolds of negative curvature, where Guillemin/Kazhdan, Min-Oo, and Croke/Sharafutdinov showed that there are no nontrivial isospectral deformations of such metrics.