62 research outputs found
Source identities and kernel functions for deformed (quantum) Ruijsenaars models
We consider the relativistic generalization of the quantum
Calogero-Sutherland models due to Ruijsenaars, comprising the rational,
hyperbolic, trigonometric and elliptic cases. For each of these cases, we find
an exact common eigenfunction for a generalization of Ruijsenaars analytic
difference operators that gives, as special cases, many different kernel
functions; in particular, we find kernel functions for Chalykh-
Feigin-Veselov-Sergeev-type deformations of such difference operators which
generalize known kernel functions for the Ruijsenaars models. We also discuss
possible applications of our results.Comment: 24 page
Exact solutions of two complementary 1D quantum many-body systems on the half-line
We consider two particular 1D quantum many-body systems with local
interactions related to the root system . Both models describe identical
particles moving on the half-line with non-trivial boundary conditions at the
origin, and they are in many ways complementary to each other. We discuss the
Bethe Ansatz solution for the first model where the interaction potentials are
delta-functions, and we find that this provides an exact solution not only in
the boson case but even for the generalized model where the particles are
distinguishable. In the second model the particles have particular momentum
dependent interactions, and we find that it is non-trivial and exactly solvable
by Bethe Ansatz only in case the particles are fermions. This latter model has
a natural physical interpretation as the non-relativistic limit of the massive
Thirring model on the half-line. We establish a duality relation between the
bosonic delta-interaction model and the fermionic model with local momentum
dependent interactions. We also elaborate on the physical interpretation of
these models. In our discussion the Yang-Baxter relations and the Reflection
equation play a central role.Comment: 15 pages, a mistake corrected changing one of our conclusion
Quasi-invariant Hermite polynomials and Lassalle-Nekrasov correspondence
Lassalle and Nekrasov discovered in the 1990s a surprising correspondence
between the rational Calogero-Moser system with a harmonic term and its
trigonometric version. We present a conceptual explanation of this
correspondence using the rational Cherednik algebra and establish its
quasi-invariant extension.
More specifically, we consider configurations of real
hyperplanes with multiplicities admitting the rational Baker-Akhiezer function
and use this to introduce a new class of non-symmetric polynomials, which we
call -Hermite polynomials. These polynomials form a linear basis in
the space of -quasi-invariants, which is an eigenbasis for the
corresponding generalised rational Calogero-Moser operator with harmonic term.
In the case of the Coxeter configuration of type this leads to a
quasi-invariant version of the Lassalle-Nekrasov correspondence and its higher
order analogues.Comment: 32 page
An Explicit Formula for Symmetric Polynomials Related to the Eigenfunctions of Calogero-Sutherland Models
We review a recent construction of an explicit analytic series representation for symmetric polynomials which up to a groundstate factor are eigenfunctions of Calogero-Sutherland type models. We also indicate a generalisation of this result to polynomials which give the eigenfunctions of so-called 'deformed' Calogero-Sutherland type models
Baker-Akhiezer functions and generalised Macdonald-Mehta integrals
For the rational Baker-Akhiezer functions associated with special
arrangements of hyperplanes with multiplicities we establish an integral
identity, which may be viewed as a generalisation of the self-duality property
of the usual Gaussian function with respect to the Fourier transformation. We
show that the value of properly normalised Baker-Akhiezer function at the
origin can be given by an integral of Macdonald-Mehta type and explicitly
compute these integrals for all known Baker-Akhiezer arrangements. We use the
Dotsenko-Fateev integrals to extend this calculation to all deformed root
systems, related to the non-exceptional basic classical Lie superalgebras.Comment: 26 pages; slightly revised version with minor correction
Stochastic B\"acklund transformations
How does one introduce randomness into a classical dynamical system in order
to produce something which is related to the `corresponding' quantum system? We
consider this question from a probabilistic point of view, in the context of
some integrable Hamiltonian systems
Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles
As is well-known, there exists a four parameter family of local interactions
in 1D. We interpret these parameters as coupling constants of delta-type
interactions which include different kinds of momentum dependent terms, and we
determine all cases leading to many-body systems of distinguishable particles
which are exactly solvable by the coordinate Bethe Ansatz. We find two such
families of systems, one with two independent coupling constants deforming the
well-known delta interaction model to non-identical particles, and the other
with a particular one-parameter combination of the delta- and (so-called)
delta-prime interaction. We also find that the model of non-identical particles
gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the
other model we write down explicit formulas for all eigenfunctions.Comment: 23 pages v2: references adde
Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems
We obtain kernel functions associated with the quantum relativistic Toda
systems, both for the periodic version and for the nonperiodic version with its
dual. This involves taking limits of previously known results concerning kernel
functions for the elliptic and hyperbolic relativistic Calogero-Moser systems.
We show that the special kernel functions at issue admit a limit that yields
generating functions of B\"acklund transformations for the classical
relativistic Calogero-Moser and Toda systems. We also obtain the
nonrelativistic counterparts of our results, which tie in with previous results
in the literature.Comment: 76 page
Singular factorizations, self-adjoint extensions, and applications to quantum many-body physics
We study self-adjoint operators defined by factorizing second order
differential operators in first order ones. We discuss examples where such
factorizations introduce singular interactions into simple quantum mechanical
models like the harmonic oscillator or the free particle on the circle. The
generalization of these examples to the many-body case yields quantum models of
distinguishable and interacting particles in one dimensions which can be solved
explicitly and by simple means. Our considerations lead us to a simple method
to construct exactly solvable quantum many-body systems of Calogero-Sutherland
type.Comment: 17 pages, LaTe
Fit for purpose? Pattern cutting and seams in wearables development
This paper describes how a group of practitioners and researchers are working across disciplines at Nottingham Trent University in the area of Technical Textiles. It introduces strands of ongoing enquiry centred around the development and application of stretch sensors on the body, focusing on how textile and fashion knowledge are being reflexively revealed in the collaborative development of seamful wearable concepts, and on the tensions between design philosophies as revealed by definitions of purpose. We discuss the current research direction of the Aeolia project, which seeks to exploit the literal gaps found in pattern cutting for fitted stretch garments towards experiential forms and potential interactions. Normative goals of fitness for purpose and seamlessness are interrogated and the potential for more integrated design processes, which may at first appear ‘upside down’, is discussed
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