Half Line Research Articles

One-dimensional Schrödinger operator with decaying white noise potential

In this paper, we consider a random one-dimensional Schrödinger operator Hω on the half line which has, as its potential term, Gaussian white noise multiplied by a decaying factor. Although the potential term is not an ordinary function, but a distribution, it is possible to realize Hω as a symmetric operator in L2([0, ∞); dt) as was pointed out by the present author [Minami, Lect. Notes Math. 1299, 298 (1986)], and it will be shown that Hω is actually self-adjoint with probability one. When the white noise in Hω is replaced by random functions of a specific type, [Kotani and Ushiroya, Commun. Math. Phys. 115, 247 (1988)] made a precise analysis of the positive part of the spectrum. According to them, if the decaying factor is not square integrable, the positive part of the spectrum typically consists of singular continuous and dense pure-point parts, which are separated by a threshold number. On the other hand, the positive part of the spectrum is purely absolutely continuous when the decaying factor is square integrable. In this work, we shall focus on the case of square integrable decaying factor, and prove the absolute continuity of the positive part of the spectrum of Hω. We shall further prove that the negative part of the spectrum of Hω is discrete, with no accumulation points other than 0.

Discrete Scale Invariance and U(2) Family of Two-Body Contact Interactions in One Dimension

Because of the absence of indistinguishability constraint, interparticle interactions between nonidentical particles have in general much more variety than those between identical particles. In particular, it is known that there exists a U(2) family of two-body contact interactions between nonidentical particles in one spatial dimension. This paper studies breakdown of continuous scale invariance to discrete scale invariance under this U(2) family of two-body contact interactions in two-body problems of nonidentical particles on the half line. We show that, in contrast to the corresponding identical-particle problem, there exist two distinct channels that admit geometric sequences of two-body bound states.

The f↔f˜ Correspondence and Its Applications in Quantum Information Geometry

Due to the classifying theorems by Petz and Kubo–Ando, we know that there are bijective correspondences between Quantum Fisher Information(s), operator means, and the class of symmetric, normalized operator monotone functions on the positive half line; this last class is usually denoted as Fop. This class of operator monotone function has a significant structure, which is worthy of study; indeed, any step in understanding Fop, besides being interesting per se, immediately translates into a property of the classes of operator means and therefore of Quantum Fisher Information(s). In recent years, the f↔f correspondence has been introduced, which associates a non-regular element of Fop to any regular element of the same set. In terms of operator means, this amounts to associating a mean with multiplicative character to a mean that has an additive character. In this paper, we survey a number of different settings where this technique has proven useful in Quantum Information Geometry. In Sections 1–4, all the needed background is provided. In Sections 5–14, we describe the main applications of the f↔f˜ correspondence.

Existence results for singular strongly non-linear integro-differential BVPs on the half line

This work is devoted to the study of singular strongly non-linear integro-differential equations of the type (Φ(k(t)v′(t)))′=ft,∫0tv(s)ds,v(t),v′(t),a.e.onR0+:=[0,+∞[,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (\\Phi (k(t)v'(t)))'=f\\left( t,\\int _0^t v(s)\\, \ extrm{d}s,v(t),v'(t) \\right) , \ ext{ a.e. } \ ext{ on } {\\mathbb {R}}^{+}_0 := [0, + \\infty [, \\end{aligned}$$\\end{document}where f is a Carathéodory function, Φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi $$\\end{document} is a strictly increasing homeomorphism, and k is a non-negative integrable function, which is allowed to vanish on a set of zero Lebesgue measure, such that 1/k∈Llocp(R0+)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/k \\in L^p_\ extrm{loc}({\\mathbb {R}}^{+}_0)$$\\end{document} for a certain p>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p>1$$\\end{document}. By considering a suitable set of assumptions, including a Nagumo–Wintner growth condition, we prove existence and non-existence results for boundary value problems associated with the non-linear integro-differential equation of our interest in the sub-critical regime on the real half line.

Well-posedness for the Schrödinger-KdV system on the half-line

In this paper we obtain improved local well–posedness results for the Schrödinger-KdV system on the half-line. We employ the Laplace–Fourier method in conjunction with the restricted norm method of Bourgain appropriately modified in order to accommodate the bounded operators of the half–line problem. Our result extends the previous local results in [5], [6] and [18] matching the results that Wu, [25], obtained for the real line system. We also demonstrate the uniqueness for the full range of locally well–posed solutions. In addition we obtain global well–posedness on the half–line for the energy solutions with zero boundary data, along with polynomial–in–time bounds for higher order Sobolev norms for the Schrödinger part.

Direct and inverse resonance problems for the massless Dirac operator on the half line

For the direct problem, we give the asymptotic distribution of the resonances in the complex plane. For the inverse problem, we prove several uniqueness theorems of recovering the potential on the whole interval from partial resonances and the information of the potential on a subinterval.

Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension n⩾3. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space Rn. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear systems corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear systems on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear systems to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

We use the unified transform method (UTM) to solve initial‐boundary value problems for the fractional advection–diffusion‐type equation (FADE) on the real half line. We generalize this equation using the modified definition of the Atangana–Baleanu fractional derivative of order in order to satisfy the initial condition. A solution methodology is proposed when the UTM is implemented in fractional differential equations with boundary conditions of Dirichlet and Robin type, in particular when using the modified definitions of fractional operators with non‐singular kernel. In addition, an interface problem is stated and solved in the adjacent domains and , where perfect contact continuity conditions are imposed. The exact solutions obtained include as a particular case, the diffusion and advection–diffusion equations with integer‐order derivatives. Finally, representative curves for the solution are shown by varying the fractional order.

Efficient and Accurate Spectral Method for Solving Fractional Differential Equations on the Half Line Using Orthogonal Generalized Rational Jacobi Functions

Efficient and Accurate Spectral Method for Solving Fractional Differential Equations on the Half Line Using Orthogonal Generalized Rational Jacobi Functions

New class of exact coherent states: Enhanced quantization of motion on the half line

New class of exact coherent states: Enhanced quantization of motion on the half line

A New Hyperjerk System With a Half Line Equilibrium: Multistability, Period Doubling Reversals, Antimonotonocity, Electronic Circuit, FPGA Design, and an Application to Image Encryption

A New Hyperjerk System With a Half Line Equilibrium: Multistability, Period Doubling Reversals, Antimonotonocity, Electronic Circuit, FPGA Design, and an Application to Image Encryption

On the solvability and the dimension of the solution set of abstract linear boundary value problems and applications to ODE'S

A solvability theory for linear abstract boundary value problems, that is of linear operator equations subject to an additional (“boundary”) condition given by a linear operator, is developed using a new approach based on normal solvability of the induced map or a pseudoinverse of the boundary map when restricted to the null space of the homogeneous problem. Uniqueness and dimension of their solution sets is also established. Applications to BVP's for finite-dimensional as well as infinite-dimensional systems of ordinary differential equations defined on the half line or the whole line are given. Systems having exponential dichotomy, or satisfying Riccati type inequalities are studied, as well as systems with asymptotically hyperbolic linear part and systems between pairs of admissible Banach spaces.

Harmonic Oscillator with a Step and/or a Ramp

We discuss the one-dimensional Schrödinger equation for a harmonic oscillator with a finite step at the origin and/or an external field described by a ramp function. The first half of this paper is a partial review of our recent work. The latter half is devoted to an extension of the problem, i.e., imposing an external linear field on the negative half line. The solvability of the problem via the Hermite polynomials is discussed. We demonstrate that a harmonic oscillator with a step and a ramp can have one eigenstate whose wavefunction is expressed in terms of Hermite polynomials of different orders. Explicit examples are also provided at appropriate places in the text.

Spectrum of the Cesàro-Hardy operator in rearrangement invariant spaces

In this paper, we investigate the spectrum of the Cesàro-Hardy operator in rearrangement invariant spaces over a finite interval and a half line, thereby extending Boyd’s and Leibowitz’s results for Lp(1 < p ⩽ 8) spaces. In particular, when our rearrangement invariant space is the Lorentz space Lp,q , a full description of the spectrum and fine spectra is presented.

Asymptotic stability of travelling wave to a hyperbolic-elliptic coupled system of the radiating gas on half line

Asymptotic stability of travelling wave to a hyperbolic-elliptic coupled system of the radiating gas on half line

Stationary solutions to the one-dimensional full compressible Navier-Stokes-Korteweg equations in the half line

This paper is concerned with the stationary solutions to the one-dimensional full (non-isentropic) compressible Navier-Stokes-Korteweg equations with far field states and boundary data in the half line. When the fluids enter into and blow out the region through the boundary, respectively, the unique existence of the stationary solutions to the one-dimensional full compressible Navier-Stokes-Korteweg equations in the half line is shown provided that the boundary strength is small enough. Moreover, we also give the spatial decay rates of the stationary solutions. The main ingredient of the proof is the manifold theory and the center manifold theorem that take the accurate analysis of the cubic characteristic equations.

Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line

Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line

Effects of inoculation of corn silage with Saccharomyces cerevisiae on silage fermentation characteristics, nutrient digestibility, mycoflora and aflatoxin production

In the present study, fresh whole corn plants along with half milk kernels line were treated with live and hydrolysed yeast with different concentrations before ensiling and kept in airtight bags and then into mini silos in order to achieve anaerobic conditions for proper fermentation. The buckets were opened after different time intervals to characterise the material, quick acidification, dry matter recovery, and aerobic stability of silage respectively. Moreover, mycoflora and aflatoxin contamination were also analysed. The overall result reported that the silage quality was improved by the application of live and hydrolysed yeast. The best result was reported by the application of live yeast (T2: 10 g/kg) which significantly improved the fermentative, proximate, and digestibility parameters and reduced the mycoflora and aflatoxin contamination. Our results present promising new options for the use of natural compounds that may help to improve silage quality and reduce aflatoxin contamination.

Discretization of inverse scattering on a half line

AbstractWe solve inverse scattering problem for Schrödinger operators with compactly supported potentials on the half line. We discretize S‐matrix: We take the value of the S‐matrix on some infinite sequence of positive real numbers. Using this sequence obtained from S‐matrix, we recover uniquely the potential by a new explicit formula, without the Gelfand–Levitan–Marchenko equation.

Periodic Gabor frames on positive half line

In this paper, we introduce the concept of periodic Gabor frames on positive half line. Firstly, we establish a necessary and sufficient condition for a periodic Gabor system to be a Gabor frame. Then, we present some equivalent characterizations of Parseval Gabor frames on positive half line by means of some fundamental equations in time domain.