Search results for "LIOUVILLE"

Landis-type conjecture for the half-Laplacian

2023

In this paper, we study the Landis-type conjecture, i.e., unique continuation property from infinity, of the fractional Schrödinger equation with drift and potential terms. We show that if any solution of the equation decays at a certain exponential rate, then it must be trivial. The main ingredients of our proof are the Caffarelli-Silvestre extension and Armitage’s Liouville-type theorem. peerReviewed

Transkendenttiluvuista

2015

Tämän pro gradu - tutkielman aiheena on transkendenttiluvut. Ne ovat lukuja, jotka eivät voi olla minkään kokonaislukukertoimisen polynomin, joka ei ole nollapolynomi, nollakohtia. Tutkielman tärkeimmät tulokset ovat Liouvillen lause, Lindemann-Weierstrassin lause sekä Gelfond-Schneiderin lause. Näiden kolmen lauseen avulla voidaan todistaa joitakin lukuja transkendenttisiksi. Yleisesti ottaen luvun transkendenttiseksi todistaminen on vaikeaa, eikä yleistä menetelmää tähän tunneta. Nämä kolme lausetta kuitenkin auttavat joissakin tapauksissa, kuten todistamaan e:n ja piin transkendenttisiksi. Tutkielma etenee m ääritelmien ja perusalgebran kautta p äätulosten esittelyyn ja transkendenttisyy…

Types and Multiplicity of Solutions to Sturm–Liouville Boundary Value Problem

2015

We consider the second-order nonlinear boundary value problems (BVPs) with Sturm–Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.

Instability of Equilibrium States for Coupled Heat Reservoirs at Different Temperatures

2007

Abstract We consider quantum systems consisting of a “small” system coupled to two reservoirs (called open systems). We show that such systems have no equilibrium states normal with respect to any state of the decoupled system in which the reservoirs are at different temperatures, provided that either the temperatures or the temperature difference divided by the product of the temperatures are not too small. Our proof involves an elaborate spectral analysis of a general class of generators of the dynamics of open quantum systems, including quantum Liouville operators (“positive temperature Hamiltonians”) which generate the dynamics of the systems under consideration.

Eigenvalue Accumulation for Singular Sturm–Liouville Problems Nonlinear in the Spectral Parameter

1999

Abstract For certain singular Sturm–Liouville equations whose coefficients depend continuously on the spectral parameter λ in an interval Λ it is shown that accumulation/nonaccumulation of eigenvalues at an endpoint ν of Λ is essentially determined by oscillatory properties of the equation at the boundary λ = ν . As applications new results are obtained for the radial Dirac operator and the Klein–Gordon equation. Three other physical applications are also considered.

Step-by-step integration for fractional operators

2018

Abstract In this paper, an approach based on the definition of the Riemann–Liouville fractional operators is proposed in order to provide a different discretisation technique as alternative to the Grunwald–Letnikov operators. The proposed Riemann–Liouville discretisation consists of performing step-by-step integration based upon the discretisation of the function f(t). It has been shown that, as f(t) is discretised as stepwise or piecewise function, the Riemann–Liouville fractional integral and derivative are governing by operators very similar to the Grunwald–Letnikov operators. In order to show the accuracy and capabilities of the proposed Riemann–Liouville discretisation technique and th…

Multiplicity results for Sturm-Liouville boundary value problems

2009

Multiplicity results for Sturm-Liouville boundary value problems are obtained. Proofs are based on variational methods.

The Fučík spectrum for nonlocal BVP with Sturm–Liouville boundary condition

2014

Boundary value problem of the form x''=-μx++λx-, αx(0)+(1-α)x'(0)=0, ∫01 x(s)ds=0 is considered, where μ,λ∈ R and α∈ [0,1]. The explicit formulas for the spectrum of this problem are given and the spectra for some α values are constructed. Special attention is paid to the spectrum behavior at the points close to the coordinate origin.

Irreversibility of the transport equations

1974

Unitary reduction of the Liouville equation relative to a two-level atom coupled to a bimodal lossy cavity

2002

The Liouville equation of a two-level atom coupled to a degenerate bimodal lossy cavity is unitarily and exactly reduced to two uncoupled Liouville equations. The first one describes a dissipative Jaynes-Cummings model and the other one a damped harmonic oscillator. Advantages related to the reduction method are discussed.