The self-adjointness conditions for a higher order differential operator with an operator coefficient

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian $\mathbb{L}$ to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing $\mathbb{L}$ are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

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Combined influence of axial electron temperature and exponential plasma density ramp on the self-focusing of a chirped laser in plasma

Zeitschrift für Naturforschung A ◽

10.1515/zna-2020-0021 ◽

2020 ◽

Vol 75 (7) ◽

pp. 671-675

Author(s):

Niti Kant ◽

Vishal Thakur

Keyword(s):

Electron Temperature ◽

Pulse Laser ◽

Plasma Density ◽

Beam Width ◽

Spot Size ◽

Higher Order ◽

The Self ◽

Chirped Pulse ◽

Self Focusing ◽

Paraxial Ray

AbstractAn analysis of the self-focusing of highly intense chirped pulse laser under exponential plasma density ramp with higher order value of axial electron temperature has been done. Beam width parameter is derived by using paraxial ray approximation and then solved numerically. It is seen that self-focusing of chirped pulse laser is intensely affected by the higher order values of axial electron temperature. Further, influence of exponential plasma density ramp is studied and it is concluded that self-focusing of laser enhances and occurs earlier. On the other hand defocusing of beam reduces to the great extent. It is noticed that the laser spot size reduces significantly under joint influence of the density ramp and the axial electron temperature. Present analysis may be useful for the analysis of quantum dots, the laser induced fusion and etc.

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A partially diffusive cholera model based on a general second-order differential operator

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2021.125181 ◽

2021 ◽

pp. 125181

Author(s):

Kazuo Yamazaki ◽

Chayu Yang ◽

Jin Wang

Keyword(s):

Differential Operator ◽

Second Order ◽

Order Differential Operator ◽

Model Based

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Gap opening in two-dimensional periodic systems

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500809 ◽

2019 ◽

Vol 23 (01) ◽

pp. 1950080

Author(s):

D. I. Borisov ◽

P. Exner

Keyword(s):

Differential Operator ◽

Finite Number ◽

Distinctive Feature ◽

Perturbation Parameter ◽

Periodic Systems ◽

Order Differential Operator ◽

Two Dimensional ◽

Gap Opening ◽

Gap Control ◽

Waveguide System

We present a new method of gap control in two-dimensional periodic systems with the perturbation consisting of a second-order differential operator and a family of narrow potential “walls” separating the period cells in one direction. We show that under appropriate assumptions one can open gaps around points determined by dispersion curves of the associated “waveguide” system, in general any finite number of them, and to control their widths in terms of the perturbation parameter. Moreover, a distinctive feature of those gaps is that their edge values are attained by the corresponding band functions at internal points of the Brillouin zone.

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