115 research outputs found
On the spectrum of S=1/2 XXX Heisenberg chain with elliptic exchange
It is found that the Hamiltonian of S=1/2 isotropic Heisenberg chain with
sites and elliptic non-nearest-neighbor exchange is diagonalized in each sector
of the Hilbert space with magnetization , , by means of
double quasiperiodic meromorphic solutions to the -particle quantum
Calogero-Moser problem on a line. The spectrum and highest-weight states are
determined by the solutions of the systems of transcendental equations of the
Bethe-ansatz type which arise as restrictions to particle pseudomomenta.Comment: 9 pages, Late
On the rational monodromy-free potentials with sextic growth
We study the rational potentials , with sextic growth at infinity, such
that the corresponding one-dimensional \Sch equation has no monodromy in the
complex domain for all values of the spectral parameter. We investigate in
detail the subclass of such potentials which can be constructed by the Darboux
transformations from the well-known class of quasi-exactly solvable potentials
We show that, in contrast with the
case of quadratic growth, there are monodromy-free potentials which have
quasi-rational eigenfunctions, but which can not be given by this construction.
We discuss the relations between the corresponding algebraic varieties, and
present some elementary solutions of the Calogero-Moser problem in the external
field with sextic potential.Comment: 35 pages, 1 figur
Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models
For any root system and an irreducible representation of
the reflection (Weyl) group generated by , a {\em spin
Calogero-Moser model} can be defined for each of the potentials: rational,
hyperbolic, trigonometric and elliptic. For each member of , to
be called a "site", we associate a vector space whose element
is called a "spin". Its dynamical variables are the canonical coordinates
of a particle in , ( rank of ), and spin
exchange operators () which exchange the
spins at the sites and . Here is the reflection
generated by . For each and a {\em spin exchange
model} can be defined. The Hamiltonian of a spin exchange model is a linear
combination of the spin exchange operators only. It is obtained by "freezing"
the canonical variables at the equilibrium point of the corresponding classical
Calogero-Moser model. For and vector representation it
reduces to the well-known Haldane-Shastry model. Universal Lax pair operators
for both spin Calogero-Moser models and spin exchange models are presented
which enable us to construct as many conserved quantities as the number of
sites for {\em degenerate} potentials.Comment: 18 pages, LaTeX2e, no figure
New Algebraic Quantum Many-body Problems
We develop a systematic procedure for constructing quantum many-body problems
whose spectrum can be partially or totally computed by purely algebraic means.
The exactly-solvable models include rational and hyperbolic potentials related
to root systems, in some cases with an additional external field. The
quasi-exactly solvable models can be considered as deformations of the previous
ones which share their algebraic character.Comment: LaTeX 2e with amstex package, 36 page
Explicit solution of the (quantum) elliptic Calogero-Sutherland model
We derive explicit formulas for the eigenfunctions and eigenvalues of the
elliptic Calogero-Sutherland model as infinite series, to all orders and for
arbitrary particle numbers and coupling parameters. The eigenfunctions obtained
provide an elliptic deformation of the Jack polynomials. We prove in certain
special cases that these series have a finite radius of convergence in the nome
of the elliptic functions, including the two particle (= Lam\'e) case for
non-integer coupling parameters.Comment: v1: 17 pages. The solution is given as series in q but only to low
order. v2: 30 pages. Results significantly extended. v3: 35 pages. Paper
completely revised: the results of v1 and v2 are extended to all order
Dynamical boundary conditions for integrable lattices
Some special solutions to the reflection equation are considered. These
boundary matrices are defined on the common quantum space with the other
operators in the chain. The relations with the Drinfeld twist are discussed.Comment: LaTeX, 12page
Quantum Inozemtsev model, quasi-exact solvability and N-fold supersymmetry
Inozemtsev models are classically integrable multi-particle dynamical systems
related to Calogero-Moser models. Because of the additional q^6 (rational
models) or sin^2(2q) (trigonometric models) potentials, their quantum versions
are not exactly solvable in contrast with Calogero-Moser models. We show that
quantum Inozemtsev models can be deformed to be a widest class of partly
solvable (or quasi-exactly solvable) multi-particle dynamical systems. They
posses N-fold supersymmetry which is equivalent to quasi-exact solvability. A
new method for identifying and solving quasi-exactly solvable systems, the
method of pre-superpotential, is presented.Comment: LaTeX2e 28 pages, no figure
Spectrum of a spin chain with inverse square exchange
The spectrum of a one-dimensional chain of spins positioned at the
static equilibrium positions of the particles in a corresponding classical
Calogero system with an exchange interaction inversely proportional to the
square of their distance is studied. As in the translationally invariant
Haldane--Shastry model the spectrum is found to exhibit a very simple structure
containing highly degenerate ``super-multiplets''. The algebra underlying this
structure is identified and several sets of raising and lowering operators are
given explicitely. On the basis of this algebra and numerical studies we give
the complete spectrum and thermodynamics of the system.Comment: 9 pages, late
Exact Spectrum of SU(n) Spin Chain with Inverse-Square Exchange
The spectrum and partition function of a model consisting of SU(n) spins
positioned at the equilibrium positions of a classical Calogero model and
interacting through inverse-square exchange are derived. The energy levels are
equidistant and have a high degree of degeneracy, with several SU(n) multiplets
belonging to the same energy eigenspace. The partition function takes the form
of a q-deformed polynomial. This leads to a description of the system by means
of an effective parafermionic hamiltonian, and to a classification of the
states in terms of "modules" consisting of base-n strings of integers.Comment: 12 pages, CERN-TH-7040/9
Current Problems of the Tertiary Education Modernization and Reform Practices
The sphere of postgraduate studies is the most governance reform-sensitive element of high school educational system in Russia.The article focuses on the governmental decisions taken to address difficulties in high school modernization sphere. The article posits that the mainstream modernization vector of educational system connected with the upgraded scientific and pedagogical personnel education is performed in accordance with transformational process set forth in Bologna declaration signed by the Russian Federation. The Covid-19 pandemic challenged the postgraduate reform results in Russia. The article provides a thorough and discourse analysis of scientific publications, expert interviews and normative documents. The study concludes that the set of fundamental approaches to further Russian system of postgraduate studies is currently semi-functional.Current reforms seek to overcome a destructive impact of the current postgraduate studies system
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