23,021 research outputs found

### Signatures of S-wave bound-state formation in finite volume

We discuss formation of an S-wave bound-state in finite volume on the basis
of L\"uscher's phase-shift formula.It is found that although a bound-state pole
condition is fulfilled only in the infinite volume limit, its modification by
the finite size corrections is exponentially suppressed by the spatial extent
$L$ in a finite box $L^3$. We also confirm that the appearance of the S-wave
bound state is accompanied by an abrupt sign change of the S-wave scattering
length even in finite volume through numerical simulations. This distinctive
behavior may help us to discriminate the loosely bound state from the lowest
energy level of the scattering state in finite volume simulations.Comment: 25 pages, 30 figures; v2: typos corrected and two references added,
v3: final version to appear in PR

### Shape Invariant Potentials in "Discrete Quantum Mechanics"

Shape invariance is an important ingredient of many exactly solvable quantum
mechanics. Several examples of shape invariant ``discrete quantum mechanical
systems" are introduced and discussed in some detail. They arise in the problem
of describing the equilibrium positions of Ruijsenaars-Schneider type systems,
which are "discrete" counterparts of Calogero and Sutherland systems, the
celebrated exactly solvable multi-particle dynamics. Deformed Hermite and
Laguerre polynomials are the typical examples of the eigenfunctions of the
above shape invariant discrete quantum mechanical systems.Comment: 15 pages, 1 figure. Contribution to a special issue of Journal of
Nonlinear Mathematical Physics in honour of Francesco Calogero on the
occasion of his seventieth birthda

### Equilibrium Positions and Eigenfunctions of Shape Invariant (`Discrete') Quantum Mechanics

Certain aspects of the integrability/solvability of the
Calogero-Sutherland-Moser systems and the Ruijsenaars-Schneider-van Diejen
systems with rational and trigonometric potentials are reviewed. The
equilibrium positions of classical multi-particle systems and the
eigenfunctions of single-particle quantum mechanics are described by the same
orthogonal polynomials: the Hermite, Laguerre, Jacobi, continuous Hahn, Wilson
and Askey-Wilson polynomials. The Hamiltonians of these single-particle quantum
mechanical systems have two remarkable properties, factorization and shape
invariance.Comment: 30 pages, 1 figure. Contribution to proceedings of RIMS workshop
"Elliptic Integrable Systems" (RIMS, Nov. 2004

### Calogero-Sutherland-Moser Systems, Ruijsenaars-Schneider-van Diejen Systems and Orthogonal Polynomials

The equilibrium positions of the multi-particle classical
Calogero-Sutherland-Moser (CSM) systems with rational/trigonometric potentials
associated with the classical root systems are described by the classical
orthogonal polynomials; the Hermite, Laguerre and Jacobi polynomials. The
eigenfunctions of the corresponding single-particle quantum CSM systems are
also expressed in terms of the same orthogonal polynomials. We show that this
interesting property is inherited by the Ruijsenaars-Schneider-van Diejen
(RSvD) systems, which are integrable deformation of the CSM systems; the
equilibrium positions of the multi-particle classical RSvD systems and the
eigenfunctions of the corresponding single-particle quantum RSvD systems are
described by the same orthogonal polynomials, the continuous Hahn (special
case), Wilson and Askey-Wilson polynomials. They belong to the Askey-scheme of
the basic hypergeometric orthogonal polynomials and are deformation of the
Hermite, Laguerre and Jacobi polynomials, respectively. The Hamiltonians of
these single-particle quantum mechanical systems have two remarkable
properties, factorization and shape invariance.Comment: 16 pages, 1 figur

### Quantum & Classical Eigenfunctions in Calogero & Sutherland Systems

An interesting observation was reported by Corrigan-Sasaki that all the
frequencies of small oscillations around equilibrium are " quantised" for
Calogero and Sutherland (C-S) systems, typical integrable multi-particle
dynamics. We present an analytic proof by applying recent results of
Loris-Sasaki. Explicit forms of `classical' and quantum eigenfunctions are
presented for C-S systems based on any root systems.Comment: LaTeX2e 37 pages, references added, typo corrected, a few paragraphs
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### Pseudospin and Deformation-induced Gauge Field in Graphene

The basic properties of $\pi$-electrons near the Fermi level in graphene are
reviewed from a point of view of the pseudospin and a gauge field coupling to
the pseudospin. The applications of the gauge field to the electron-phonon
interaction and to the edge states are reported.Comment: 27 pages, 7 figure

### Calogero-Moser Models III: Elliptic Potentials and Twisting

Universal Lax pairs of the root type with spectral parameter and independent
coupling constants for twisted non-simply laced Calogero-Moser models are
constructed. Together with the Lax pairs for the simply laced models and
untwisted non-simply laced models presented in two previous papers, this
completes the derivation of universal Lax pairs for all of the Calogero-Moser
models based on root systems. As for the twisted models based on B_n, C_n and
BC_nroot systems, a new type of potential term with independent coupling
constants can be added without destroying integrability. They are called
extended twisted models. All of the Lax pairs for the twisted models presented
here are new, except for the one for the F_4 model based on the short roots.
The Lax pairs for the twisted G_2 model have some novel features. Derivation of
various functions, twisted and untwisted, appearing in the Lax pairs for
elliptic potentials with the spectral parameter is provided. The origin of the
spectral parameter is also naturally explained. The Lax pairs with spectral
parameter, twisted and untwisted, for the hyperbolic, the trigonometric and the
rational potential models are obtained as degenerate limits of those for the
elliptic potential models.Comment: LaTeX2e with amsfonts.sty, 36 pages, no figure

### Commuting Charges of the Quantum Korteweg-deVries and Boussinesq Theories from the Reduction of W(infinity) and W(1+infinity) Algebras

Integrability of the quantum Boussinesq equation for c=-2 is demonstrated by
giving a recursive algorithm for generating explicit expressions for the
infinite number of commuting charges based on a reduction of the W(infinity)
algebra. These charges exist for all spins $s \geq 2$. Likewise, reductions of
the W(infinity/2) and W((1+infinity)/2) algebras yield the commuting quantum
charges for the quantum KdV equation at c=-2 and c=1/2, respectively.Comment: 11 pages, RevTe

### Non-Linear Sigma Models on a Half Plane

In the context of integrable field theory with boundary, the integrable
non-linear sigma models in two dimensions, for example, the $O(N)$, the
principal chiral, the ${\rm CP}^{N-1}$ and the complex Grassmannian sigma
models are discussed on a half plane. In contrast to the well known cases of
sine-Gordon, non-linear Schr\"odinger and affine Toda field theories, these
non-linear sigma models in two dimensions are not classically integrable if
restricted on a half plane. It is shown that the infinite set of non-local
charges characterising the integrability on the whole plane is not conserved
for the free (Neumann) boundary condition. If we require that these non-local
charges to be conserved, then the solutions become trivial.Comment: 25 pages, latex, no figure

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