Hill Operator Research Articles

Stability of periodic waves for the defocusing fractional cubic nonlinear Schrödinger equation

In this paper, we determine the spectral instability of periodic odd waves for the defocusing fractional cubic nonlinear Schrödinger equation. Our approach is based on periodic perturbations that have the same period as the standing wave solution, and we construct real periodic waves by minimizing a suitable constrained problem. The odd solution generates three negative simple eigenvalues for the associated linearized operator, and we obtain all this spectral information by using tools related to the oscillation theorem for fractional Hill operators. Newton’s iteration method is presented to generate the odd periodic standing wave solutions and numerical results have been used to apply the spectral stability theory via Krein signature as established in Kapitula et al. (2004) and Kapitula et al. (2005).

A micromechanical constitutive model of high-temperature shape memory alloys

In this work, a new micromechanical constitutive model is constructed for high-temperature shape memory alloys. Different from the classical constitutive models, the microstructural patterns, heterogeneities of stress and strain fields at the microscopic scale are addressed. In the proposed model, three different inelastic deformation mechanisms, i.e., reversible martensitic transformation, irreversible transformation-induced plasticity, viscoplasticity, as well as their complex interactions, are considered. In addition to the well-known austenite and martensite phases, a new austenite-martensite interface phase is introduced to address the high local stress induced by the martensitic transformation. The modified Mori-Tanaka's homogenization scheme and Hill's interfacial operator are employed to determine the non-uniform stress fields of the three phases. To satisfy the thermodynamic compatibility of the proposed model, the driving forces and evolution equations of the three inelastic deformation mechanisms are derived based on the constructed Helmholtz free energy and the fundamental laws of irreversible thermodynamics. Inheritances of the plastic deformation induced by moving austenite-martensite (A-M) interface during repeated martensitic transformation and its reverse are incorporated. Temperature evolution caused by internal heat production and heat exchange with the ambient medium is also addressed. To validate the capability of the new constructed model, the predicted results of creep behavior and stress-assisted two-way shape memory effect of NiTiPd high-temperature shape memory alloys are compared with the experimental ones. The proposed model provides a theoretical tool for simultaneously analyzing the deformation behaviors at the macroscopic scale and underlying mechanisms at the microscopic scale.

Orbital stability of periodic standing waves for the cubic fractional nonlinear Schrödinger equation

In this paper, the existence and orbital stability of the periodic standing wave solutions for the nonlinear fractional Schrödinger (fNLS) equation with cubic nonlinearity is studied. The existence is determined by using a minimizing constrained problem in the complex setting and it is showed that the corresponding real solution is always positive. The orbital stability is proved by combining some tools regarding the oscillation theorem for fractional Hill operators and the Vakhitov-Kolokolov condition, well known for Schrödinger equations. We then perform a numerical approach to generate the periodic standing wave solutions of the fNLS equation by using the Petviashvili's iteration method. We also investigate the Vakhitov-Kolokolov condition numerically which cannot be obtained analytically for some values of the order of the fractional derivative.

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients II

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To obtain this asymptotic expansion as t→∞, methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of a non-positive Hill operator with periodic coefficients.

Margenau–Hill operator valued measures and joint measurability

We employ the Margenau–Hill (MH) correspondence rule for associating classical functions with quantum operators to construct quasi-probability mass functions. Using this we obtain the fuzzy one parameter quasi measurement operator (QMO) characterizing the incompatibility of noncommuting spin observables of qubits, qutrits and 2-qubit systems. Positivity of the fuzzy MH-QMO places upper bounds on the associated unsharpness parameter. This serves as a sufficient condition for measurement incompatibility of spin observables. We assess the amount of unsharpness required for joint measurability (compatibility) of the noncommuting qubit, qutrit and 2-qubit observables. We show that the degree of compatibility of a pair of orthogonal qubit observables agrees perfectly with the necessary and sufficient conditions for joint measurability. Furthermore, we obtain analytical upper bounds on the unsharpness parameter specifying the range of joint measurability of spin components of qutrits and pairs of orthogonal spin observables of a 2-qubit system. Our results indicate that the measurement incompatibility of spin observables increases with Hilbert space dimension.

Asymptotic Behavior of Solutions of the Cauchy Problem for a Hyperbolic Equation with Periodic Coefficients (Case: H0 > 0)

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To obtain an asymptotic expansion as t→∞, the basic methods of the spectral theory of differential operators are used, as well as the properties of the spectrum of the Hill operator with periodic coefficients in the case when the operator is positive: H0>0.

On the Lipschitz stability of inverse nodal problem for Dirac system

Inverse nodal problem on Dirac operator is determination problem of the parameters in the boundary conditions, number m and potential function V by using a set of nodal points of a component of two component vector eigenfunctions as the given spectral data. In this study, we solve a stability problem using nodal set of vector eigenfunctions and show that the space of all V functions is homeomorphic to the partition set of all space of asymptotically equivalent nodal sequences induced by an equivalence relation. Moreover, we give a reconstruction formula for the potential function as a limit of a sequence of functions and associated nodal data of one component of vector eigenfunction. Our technique depends on the explicit asymptotic expressions of the nodal parameters and, it is basically similar to [1, 2] which is given for Sturm-Liouville and Hill's operators, respectively.

On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS

We study the periodic cubic derivative non-linear Schr\"odinger equation (dNLS) and the (focussing) quintic non-linear Schr\"odinger equation (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior, when posed on the line ${\mathbb R}$. We describe the (three parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide a criteria for stability, depending on the sign of a scalar quantity. The proof relies on an instability index count, which in turn critically depends on a detailed spectral analysis of a self-adjoint matrix Hill operator. We exhibit a region in parameter space, which produces spectrally stable waves. We also provide an explicit description of the stability of all bell-shaped traveling waves for the quintic NLS, which turns out to be a two parameter subfamily of the one exhibited for DNLS. We give a complete description of their stability - as it turns out some are spectrally stable, while other are spectrally unstable, with respect to co-periodic perturbations.

Schrödinger operators periodic in octants

We consider Schrödinger operators with periodic potentials in the positive quadrant on the plane with Dirichlet boundary conditions. We show that for any integer N and any interval I there exists a periodic potential such that the Schrödinger operator has N eigenvalues counted with multiplicity in this interval and there is no other spectrum in the interval. Furthermore, to the right and to the left of it there is a essential spectrum. Moreover, we prove similar results for Schrödinger operators for a product of an orthant and Euclidean space. The proof is based on the inverse spectral theory for Hill operators on the real line.

Inverse spectral problems for Hill-type operators with frozen argument

The paper deals with nonlocal differential operators possessing a term with frozen (fixed) argument appearing, in particular, in modelling various physical systems with feedback. The presence of a feedback means that the external affect on the system depends on its current state. If this state is taken into account only at some fixed physical point, then mathematically this corresponds to an operator with frozen argument. In the present paper, we consider the operator $$Ly\equiv -y^{\prime \prime }(x)+q(x)y(a),$$ $$y^{(\nu )}(0)=\gamma y^{(\nu )}(1),$$ $$\nu =0,1,$$ where $$\gamma \in {\mathbb C}{\setminus }\{0\}.$$ The operator L is a nonlocal analog of the classical Hill operator describing various processes in cyclic or periodic media. We study two inverse problems of recovering the complex-valued square-integrable potential q(x) from some spectral information about L. The first problem involves only single spectrum as the input data. We obtain complete characterization of the spectrum and prove that its specification determines q(x) uniquely if and only if $$\gamma \ne \pm 1.$$ For the rest (periodic and antiperiodic) cases, we describe classes of iso-spectral potentials and provide restrictions under which the uniqueness holds. The second inverse problem deals with recovering q(x) from the two spectra related to $$\gamma =\pm 1.$$ We obtain necessary and sufficient conditions for its solvability and establish that uniqueness holds if and only if $$a=0,1.$$ For $$a\in (0,1),$$ we describe classes of iso-bispectral potentials and give restrictions under which the uniqueness resumes. Algorithms for solving both inverse problems are provided. In the appendix, we prove Riesz-basisness of an auxiliary two-sided sequence of sines.

Hill's operators with the potentials analytically dependent on energy

We consider Schrödinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded in the right half-plane, then the high energy spectrum is real, and the corresponding asymptotics are determined. Moreover, the Dirichlet and Neumann problems are considered. These results are used to analyze the good Boussinesq equation.

Spectral triangles of non-selfadjoint Hill and Dirac operators

This is a survey of results from the last 10 to 12 years about the structure of the spectra of Hill–Schrödinger and Dirac operators. Let be a Hill operator or a one-dimensional Dirac operator on the interval . If is considered with Dirichlet, periodic, or antiperiodic boundary conditions, then the corresponding spectra are discrete and, for sufficiently large , close to in the Hill case or close to in the Dirac case (). There is one Dirichlet eigenvalue and two periodic (if is even) or antiperiodic (if is odd) eigenvalues and (counted with multiplicity). Asymptotic estimates are given for the spectral gaps and the deviations in terms of the Fourier coefficients of the potentials. Moreover, precise asymptotic expressions for and are found for special potentials that are trigonometric polynomials. Bibliography: 45 titles.

Corrigendum to: A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

Corrigendum to: A Riemann–Hilbert Problem Approach to Infinite Gap Hill's Operators and the Korteweg–de Vries Equation

The spectrum of the Hamiltonian with a PT-symmetric periodic optical potential

We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with potential [Formula: see text], where [Formula: see text] We prove that the second critical point [Formula: see text], after which the real parts of the first and second bands disappear, is a number between [Formula: see text] and [Formula: see text]. Moreover, we prove that [Formula: see text] is the degeneration point for the first periodic eigenvalue. Besides, we give a scheme by which one can find arbitrary precise value of the second critical point as well as the [Formula: see text]th critical points after which the real parts of the [Formula: see text]th and [Formula: see text]th bands disappear, where [Formula: see text]

On the wellposedness of the KdV equation on the space of pseudomeasures

In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudomeasures, also referred to as the Fourier Lebesgue space $$\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})$$ , where $$\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})$$ is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space $$\mathscr {F}\ell ^{s,\infty }(\mathbb {T},\mathbb {R})$$ with $$-1/2 < s \le 0$$ and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space $$H^{-1}(\mathbb {T},\mathbb {R})$$ to be in $$\mathscr {F}\ell ^{\infty }(\mathbb {T},\mathbb {R})$$ in terms of asymptotic behavior of spectral quantities of the Hill operator $$-\partial _{x}^{2} + q$$ . In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

Perturbation of the Hill operator by narrow potentials

We consider a perturbation of a periodic second order differential operator, defined on the real axis, which is a special case of the Hill operator. The perturbation is realized by a sum of two complex-valued potentials with compact supports. The potentials depend on two small parameters. One of them describes the lengths of the supports of the potentials and the reciprocal to the second one corresponds to the maximum values of the potentials. We obtain a sufficient condition, under fulfillment of which, the eigenvalues arise from the edges of non-degenerate lacunas of continuous spectrum, and construct their asymptotics. We also give a sufficient condition under which the eigenvalues do not arise.

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

AbstractWe consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we rederive the formulas for the classical periodic traveling waves, while for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. In both cases, we show spectral stability, for all values of the parameters. This is achieved by studying the nonstandard eigenvalue problems in the form , where is a Hill operator.

The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential

The paper is concerned with the spectral properties of second- order differential operators defined by periodic and quasi-periodic boundary conditions. We obtain asymptotic formulae for the eigenvalues, derive estimates of projections, give estimates for the equiconvergence of spectral decompositions, find sufficient conditions for operators to be spectral, and write down an asymptotic expansion for the semigroup of operators generated by the negative of the differential operator under consideration. Our estimates involve coefficients of the Fourier potential. The main results of the paper are obtained by using the method of similar operators. Bibliography: 34 titles.

Essential spectral singularities and the spectral expansion for the Hill operator

In this paper we investigate the spectral expansion for the one-dimensional Schrodinger operator with a periodic complex-valued potential. For this we consider in detail the spectral singularities and introduce new concepts as essential spectral singularities and singular quasimomenta.

Spectral analysis of wave propagation on branching strings

In this paper the classical Hill problem with complex potentials are extended to the star graph. The definition of the Hill operator on such graph is discussed. The operator is defined with complex, periodic potentials and using special boundary conditions connecting values of the functions at the vertices. An explicit description of the resolvent is given and the spectrum is described exactly, the inverse problem with respect to the reflection coefficients is solved.