56 research outputs found
On the Bethe Ansatz for the Jaynes-Cummings-Gaudin model
We investigate the quantum Jaynes-Cummings model - a particular case of the
Gaudin model with one of the spins being infinite. Starting from the Bethe
equations we derive Baxter's equation and from it a closed set of equations for
the eigenvalues of the commuting Hamiltonians. A scalar product in the
separated variables representation is found for which the commuting
Hamiltonians are Hermitian. In the semi classical limit the Bethe roots
accumulate on very specific curves in the complex plane. We give the equation
of these curves. They build up a system of cuts modeling the spectral curve as
a two sheeted cover of the complex plane. Finally, we extend some of these
results to the XXX Heisenberg spin chain.Comment: 16 page
Bethe eigenvectors of higher transfer matrices
We consider the XXX-type and Gaudin quantum integrable models associated with
the Lie algebra . The models are defined on a tensor product irreducible
-modules. For each model, there exist one-parameter families of
commuting operators on the tensor product, called the transfer matrices. We
show that the Bethe vectors for these models, given by the algebraic nested
Bethe ansatz are eigenvectors of higher transfer matrices and compute the
corresponding eigenvalues.Comment: 48 pages, amstex.tex (ver 2.2), misprints correcte
Manin matrices and Talalaev's formula
We study special class of matrices with noncommutative entries and
demonstrate their various applications in integrable systems theory. They
appeared in Yu. Manin's works in 87-92 as linear homomorphisms between
polynomial rings; more explicitly they read: 1) elements in the same column
commute; 2) commutators of the cross terms are equal: (e.g. ). We claim
that such matrices behave almost as well as matrices with commutative elements.
Namely theorems of linear algebra (e.g., a natural definition of the
determinant, the Cayley-Hamilton theorem, the Newton identities and so on and
so forth) holds true for them.
On the other hand, we remark that such matrices are somewhat ubiquitous in
the theory of quantum integrability. For instance, Manin matrices (and their
q-analogs) include matrices satisfying the Yang-Baxter relation "RTT=TTR" and
the so--called Cartier-Foata matrices. Also, they enter Talalaev's
hep-th/0404153 remarkable formulas: ,
det(1-e^{-\p}T_{Yangian}(z)) for the "quantum spectral curve", etc. We show
that theorems of linear algebra, after being established for such matrices,
have various applications to quantum integrable systems and Lie algebras, e.g
in the construction of new generators in (and, in general,
in the construction of quantum conservation laws), in the
Knizhnik-Zamolodchikov equation, and in the problem of Wick ordering. We also
discuss applications to the separation of variables problem, new Capelli
identities and the Langlands correspondence.Comment: 40 pages, V2: exposition reorganized, some proofs added, misprints
e.g. in Newton id-s fixed, normal ordering convention turned to standard one,
refs. adde
Creation of the territory of the advancing socio-economic development as a way to diversify the economy of a single-industry city
In the scientific work the definition of the concept of “single-industry city” is given, its criteria and characteristics are highlighted.The necessity of state support in the development of single-industry towns is substantiated. It is proved that one of the main ways to assist in the diversification of the economy of single-industry towns is the creation of territories of advanced socio-economic development in them, which are characterized by special conditions and have their own characteristic
Integrable Models From Twisted Half Loop Algebras
This paper is devoted to the construction of new integrable quantum
mechanical models based on certain subalgebras of the half loop algebra of
gl(N). Various results about these subalgebras are proven by presenting them in
the notation of the St Petersburg school. These results are then used to
demonstrate the integrability, and find the symmetries, of two types of
physical system: twisted Gaudin magnets, and Calogero-type models of particles
on several half-lines meeting at a point.Comment: 22 pages, 1 figure, Introduction improved, References adde
Black Silicon with high density and high aspect ratio nanowhiskers
Physical properties of black Silicon (b-Si) formed on Si wafers by reactive
ion etching in chlorine plasma are reported in an attempt to clarify the
formation mechanism and the origin of the observed optical and electrical
phenomena which are promising for a variety of applications. The b-Si
consisting of high density and high aspect ratio sub-micron length whiskers or
pillars with tip diameters of well under 3 nm exhibits strong photoluminescence
(PL) both in visible and infrared, which are interpreted in conjunction with
defects, confinement effects and near band-edge emission. Structural analysis
indicate that the whiskers are all crystalline and encapsulated by a thin Si
oxide layer. Infrared vibrational spectrum of Si-O-Si bondings in terms of
transverse-optic (TO) and longitudinal-optic (LO) phonons indicates that
disorder induced LO-TO optical mode coupling can be an effective tool in
assessing structural quality of the b-Si. The same phonons are likely coupled
to electrons in visible region PL transitions. Field emission properties of
these nanoscopic features are demonstrated indicating the influence of the tip
shape on the emission. Overall properties are discussed in terms of surface
morphology of the nano whiskers
Benchmarking of Uranium-238 Evaluations against Spherical Transmission and (n,xn)-Reaction Experimental Data
Abstract. The double differential cross sections for the U(n,xn) reaction at 14 MeV and neutron leakage spectra from the uranium sphere of 24 cm outer and 8 cm inner diameters with the central T-D and 252 Cf neutron sources measured at the Institute of Physics and Power Engineering were used for benchmarking the evaluated cross sections from ENDF-B6, JEFF-3.0, and "Maslov" libraries and preliminary versions of JEFF-3.1 and ENDF-B7 evaluations for 238 U
Spaces of quasi-exponentials and representations of gl_N
We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k
L_{\lambda^{(s)}})_\lambda, the weight subspace of weight of the
tensor product of k polynomial irreducible gl_N-modules with highest weights
\lambda^{(1)},...,\lambda^{(k)}, respectively. The Bethe algebra depends on N
complex numbers K=(K_1,...,K_N). Under the assumption that K_1,...,K_N are
distinct, we prove that the image of B_K in the endomorphisms of
(\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic to the algebra of
functions on the intersection of k suitable Schubert cycles in the Grassmannian
of N-dimensional spaces of quasi-exponentials with exponents K. We also prove
that the B_K-module (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic
to the coregular representation of that algebra of functions. We present a
Bethe ansatz construction identifying the eigenvectors of the Bethe algebra
with points of that intersection of Schubert cycles.Comment: Latex, 29 page
Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
Gaudin algebras form a family of maximal commutative subalgebras in the
tensor product of copies of the universal enveloping algebra U(\g) of a
semisimple Lie algebra \g. This family is parameterized by collections of
pairwise distinct complex numbers . We obtain some new commutative
subalgebras in U(\g)^{\otimes n} as limit cases of Gaudin subalgebras. These
commutative subalgebras turn to be related to the hamiltonians of bending flows
and to the Gelfand--Tsetlin bases. We use this to prove the simplicity of
spectrum in the Gaudin model for some new cases.Comment: 11 pages, references adde
- …