scholarly journals A Sharp Lieb-Thirring Inequality for Functional Difference Operators

2021 ◽  
Author(s):  
Ari Laptev ◽  
◽  
Lukas Schimmer ◽  
Keyword(s):  
Essential Spectrum ◽  
Resonance State ◽  
Difference Operators ◽  
Functional Difference ◽  
One Dimensional

We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated to mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

EQUIVALENCE IN GEOMETRIC SCALING OF TRANSPORT PROPERTIES AND WAVEFUNCTIONS

1993 ◽  
Vol 07 (05) ◽  
pp. 1309-1319 ◽  
Author(s):  
CHAITALI BASU ◽  
ABHIJIT MOOKERJEE
Keyword(s):  
Transport Properties ◽  
Anderson Model ◽  
Multifractal Analysis ◽  
Resonance State ◽  
Electron States ◽  
One Dimensional ◽  
Geometric Scaling ◽  
Multifractal Scaling

The transport properties and wavefunction behave identically with respect to multifractal scaling. To establish the above statement, we carried out multifractal analysis on normalised transmittance and normalised wavefunction of two types of electron states, namely the resonance state and the localised state of a one-dimensional Anderson model.

The spectrum of a one-dimensional pseudo-differential operator

1988 ◽  
Vol 104 (3) ◽  
pp. 575-580
Author(s):  
M. W. Wong
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Essential Spectrum ◽  
Sufficient Condition ◽  
Pseudo Differential Operator ◽  
One Dimensional ◽  
Lowest Eigenvalue

AbstractWe describe the spectrum of a self-adjoint pseudo-differential operator on L2 (– ∞, ∞). We show that the essential spectrum coincides with the interval ([1, ∞) and give a lower bound for the lowest eigenvalue in (– ∞, 1). A sufficient condition for the existence of an eigenvalue in (– ∞, 1) is also given.

scholarly journals Inverses of SBP-SAT Finite Difference Operators Approximating the First and Second Derivative

2021 ◽  
Vol 89 (2) ◽  
Author(s):  
Sofia Eriksson
Keyword(s):  
Finite Difference ◽  
Penalty Method ◽  
Arbitrary Order ◽  
Difference Operators ◽  
Advection Equation ◽  
Second Derivative ◽  
Order Of Accuracy ◽  
One Dimensional ◽  
Simultaneous Approximation Term ◽  
Free Parameters

AbstractThe scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matrices, interpreting them as discrete Green’s functions. In this direct way, we also find out precisely which choices of SAT parameters that make the discretization matrices singular. In the second derivative case, it is shown that if the penalty parameters are chosen such that the semi-discrete scheme is dual consistent, the discretization matrix can become singular even when the scheme is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary order of accuracy. For second and fourth order accurate operators, the inverses are provided explicitly.

Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential

1996 ◽  
Vol 126 (5) ◽  
pp. 1087-1096 ◽  
Author(s):  
Karl Michael Schmidt
Keyword(s):  
Dirac Operator ◽  
Essential Spectrum ◽  
Electrostatic Potential ◽  
Simple Proof ◽  
Point Spectrum ◽  
Electrostatic Potentials ◽  
One Dimensional ◽  
Decay Behaviour ◽  
Dense Point ◽  
The One

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.

scholarly journals Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

2011 ◽  
Vol 468 (2138) ◽  
pp. 545-562 ◽  
Author(s):  
Kaori Nagatou ◽  
Michael Plum ◽  
Mitsuhiro T. Nakao
Keyword(s):  
Periodic Function ◽  
Essential Spectrum ◽  
Schrodinger Equation ◽  
Spectral Gaps ◽  
Numerical Approximations ◽  
One Dimensional ◽  
Spectral Bands ◽  
Computer Assistance ◽  
Computational Errors ◽  
Perturbed Periodic

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at . It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

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