407 research outputs found
B\"acklund--Darboux transformations in Sato's Grassmannian
We define B\"acklund--Darboux transformations in Sato's Grassmannian. They
can be regarded as Darboux transformations on maximal algebras of commuting
ordinary differential operators. We describe the action of these
transformations on related objects: wave functions, tau-functions and spectral
algebras.
This paper is the second of a series of papers (hep-th/9510211,
q-alg/9602011, q-alg/9602012) on the bispectral problem.Comment: 13 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figure
Bispectral algebras of commuting ordinary differential operators
We develop a systematic way for constructing bispectral algebras of commuting
ordinary differential operators of any rank . It combines and unifies the
ideas of Duistermaat-Gr\"unbaum and Wilson. Our construction is completely
algorithmic and enables us to obtain all previously known classes or individual
examples of bispectral operators. The method also provides new broad families
of bispectral algebras which may help to penetrate deeper into the problem.Comment: 46 pages, LaTeX2e, uses amsfonts.sty and latexsym.sty, no figures,
rearrangement of the introduction, skipping Conjecture 0.2 of the first
version, to appear in Communications in Mathematical Physic
Highest Weight Modules of W1+β, Darboux Transformations and the Bispectral Problem
This paper is a survey of our recent results on the bispectral
problem. We describe a new method for constructing bispectral algebras
of any rank and illustrate the method by a series of new examples as well
as by all previously known ones. Next we exhibit a close connection of
the bispectral problem to the representation theory of W1+ββalgerba. This
connection allows us to explain and generalise to any rank the result of Magri
and Zubelli on the symmetries of the manifold of the bispectral operators of
rank and order two
Electron Beam Induced Capacitance
A model of signal formation in the Electron Beam Induced Capacitance (EBICap) mode of the Scanning Electron Microscopy (SEM) is proposed. In the frame of this model the possibilities of this technique are analyzed. It is shown that EBICap is suitable to obtain a local depletion region width and for mapping of this parameter. Experimental results demonstrating the potentialities of EBICap are presented
Density of States and Conductivity of Granular Metal or Array of Quantum Dots
The conductivity of a granular metal or an array of quantum dots usually has
the temperature dependence associated with variable range hopping within the
soft Coulomb gap of density of states. This is difficult to explain because
neutral dots have a hard charging gap at the Fermi level. We show that
uncontrolled or intentional doping of the insulator around dots by donors leads
to random charging of dots and finite bare density of states at the Fermi
level. Then Coulomb interactions between electrons of distant dots results in
the a soft Coulomb gap. We show that in a sparse array of dots the bare density
of states oscillates as a function of concentration of donors and causes
periodic changes in the temperature dependence of conductivity. In a dense
array of dots the bare density of states is totally smeared if there are
several donors per dot in the insulator.Comment: 13 pages, 15 figures. Some misprints are fixed. Some figures are
dropped. Some small changes are given to improve the organizatio
On the spectrum and support theory of a finite tensor category
Finite tensor categories (FTCs) are important generalizations of the
categories of finite dimensional modules of finite dimensional Hopf algebras,
which play a key role in many areas of mathematics and mathematical physics.
There are two fundamentally different support theories for them: a
cohomological one and a universal one based on the noncommutative Balmer
spectra of their stable (triangulated) categories .
In this paper we introduce the key notion of the categorical center
of the cohomology ring
of an FTC, . This enables us to put
forward a complete and detailed program for determining the exact relationship
between the two support theories, based on of
the cohomology ring of an FTC, .
More specifically, we construct a continuous map from the noncommutative
Balmer spectrum of an FTC, , to the of the categorical
center , and prove that this map is surjective
under a weaker finite generation assumption for than the one
conjectured by Etingof-Ostrik. Under stronger assumptions, we prove that (i)
the map is homeomorphism and (ii) the two-sided thick ideals of are classified by the specialization closed subsets of .
We conjecture that both results hold for all FTCs. Many examples are
presented that demonstrate how in important cases arises as a fixed point subring of and how
the two-sided thick ideals of are determined in a uniform
fashion. The majority of our results are proved in the greater generality of
monoidal triangulated categories.Comment: Appendix B has been revised from the prior version after considering
comments from Greg Stevenso
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