In the last 30 days

• Open/close Yangians, S-matrices and AdS/CFT

  Alessandro Torrielli 2011 J. Phys. A: Math. Theor. 44 263001 Tag this article

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  This review is meant to be an account of the properties of the infinite-dimensional quantum group (specifically, Yangian) symmetry lying behind the integrability of the AdS/CFT spectral problem. In passing, the chance is taken to give a concise anthology of basic facts concerning Yangians and integrable systems, and to store a series of remarks, observations and proofs the author has collected in a 5 year span of research on the subject. We hope this exercise will be useful for future attempts to study Yangians in field and string theories, with or without supersymmetry.

• Open/close The Schrödinger equation with position-dependent mass

  J P Killingbeck 2011 J. Phys. A: Math. Theor. 44 285208 Tag this article

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  It is shown that the traditional Hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a Schrödinger equation involving a position-dependent mass. Some numerical applications are given to demonstrate the effectiveness of the method.

• Open/close Local conservation laws of second-order evolution equations

  Roman O Popovych and Anatoly M Samoilenko 2008 J. Phys. A: Math. Theor. 41 362002 Tag this article

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  Generalizing results by Bryant and Griffiths (1995 Duke Math. J. 78 531), we completely describe local conservation laws of second-order (1 + 1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.

• Open/close The Darboux transformation of the derivative nonlinear Schrödinger equation

  Shuwei Xu et al 2011 J. Phys. A: Math. Theor. 44 305203 Tag this article

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  The n-fold Darboux transformation (DT) is a 2 × 2 matrix for the Kaup–Newell (KN) system. In this paper, each element of this matrix is expressed by a ratio of the ( n + 1) × ( n + 1) determinant and n × n determinant of eigenfunctions. Using these formulae, the expressions of the q ^{[ n]} and r ^{[ n]} in the KN system are generated by the n-fold DT. Further, under the reduction condition, the rogue wave, rational traveling solution, dark soliton, bright soliton, breather solution and periodic solution of the derivative nonlinear Schrödinger equation are given explicitly by different seed solutions. In particular, the rogue wave and rational traveling solution are two kinds of new solutions. The complete classification of these solutions generated by one-fold DT is given.

• Open/close A compact hamiltonian with the same asymptotic mean spectral density as the Riemann zeros

  M V Berry and J P Keating 2011 J. Phys. A: Math. Theor. 44 285203 Tag this article

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  For the classical hamiltonian ( x + 1/ x)( p + 1/ p), with position x and conjugate momentum p, all orbits are bounded. After a symmetrization, the corresponding quantum integral equation possesses a family of self-adjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specifying the boundary condition at the origin, with a discrete spectrum of real energies E. On the cylinder {−∞ < E < ∞, 0 ≤ α < 2π}, there is a single eigencurve in the form of a helix winding clockwise. The rise between successive windings gets sharper as the scaled Planck's constant decreases. This behaviour can be understood semiclassically. The first two terms of the asymptotic eigenvalue density are the same as those for the density of heights of the Riemann zeros.

• Open/close Work distribution for the driven harmonic oscillator with time-dependent strength: exact solution and slow driving

  Thomas Speck 2011 J. Phys. A: Math. Theor. 44 305001 Tag this article

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  We study the work distribution of a single particle moving in a harmonic potential with time-dependent strength. This simple system has a non-Gaussian work distribution with exponential tails. The time evolution of the corresponding moment generating function is given by two coupled ordinary differential equations that are solved numerically. Based on this result we study the behavior of the work distribution in the limit of slow but finite driving and show that it approaches a Gaussian distribution arbitrarily well.

• Open/close The Berry–Keating Hamiltonian and the local Riemann hypothesis

  Mark Srednicki 2011 J. Phys. A: Math. Theor. 44 305202 Tag this article

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  The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or p-adic configuration space) have a real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry–Keating Hamiltonian projected onto the subspace of oscillator eigenfunctions of a lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert–Pólya conjecture. The p-adic case is also discussed.

• Open/close New quasi-exactly solvable periodic potentials

  Qiong-Tao Xie 2011 J. Phys. A: Math. Theor. 44 285302 Tag this article

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  We present two new classes of quasi-exactly solvable periodic potentials. It is shown that their solutions are given in terms of a known special function. When the parameters in the two periodic potentials satisfy a certain condition, some of the Bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first Brillouin zone. Two fundamental sets of solutions have been constructed to calculate all the Bloch waves and energy bands.

• Open/close Soliton solutions of a Calogero model in a harmonic potential

  Alexander G Abanov et al 2011 J. Phys. A: Math. Theor. 44 295203 Tag this article

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  A classical Calogero model in an external harmonic potential is known to be integrable for any number of particles. We consider here reductions which play a role of the 'soliton' solutions of the model. We obtain these solutions both for the model with finite number of particles and in a hydrodynamic limit. In the latter limit, the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite-dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background.

• Open/close Quantum tunneling effect of a driven inverted harmonic oscillator

  Guang-Jie Guo et al 2011 J. Phys. A: Math. Theor. 44 305301 Tag this article

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  Using the algebraic method, we derive the exact wavefunction of a driven time-dependent inverted harmonic oscillator (TDIHO). With the Gaussian wave-packet as the initial state, the general expressions of the transmission probability and dwell time of the driven TDIHO are obtained. For a weak driving force, the dwell time can be separated into two terms. The first term is for the unperturbed part, and the second one is that from the perturbation, which is the effect from the driving force. Furthermore, with two examples, the effect of the driving force on the dwell time is explicitly indicated in detail.