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Publications of Alexander Watson    :chronological  alphabetical  combined listing:

%% Papers Published   
   Author = {Lu, J and Watson, AB and Weinstein, MI},
   Title = {Dirac operators and domain walls},
   Journal = {Siam Journal on Mathematical Analysis},
   Volume = {52},
   Number = {2},
   Pages = {1115-1145},
   Year = {2020},
   Month = {January},
   url = {},
   Abstract = {Copyright © by SIAM. We study the eigenvalue problem for a
             one-dimensional Dirac operator with a spatially varying
             "mass" term. It is well-known that when the mass function
             has the form of a kink, or domain wall, transitioning
             between strictly positive and strictly negative asymptotic
             mass, ±κ∞, at ±∞, the Dirac operator has a simple
             eigenvalue of zero energy (geometric multiplicity equal to
             one) within a gap in the continuous spectrum, with
             corresponding exponentially localized zero mode. We consider
             the eigenvalue problem for the one-dimensional Dirac
             operator with mass function defined by "gluing" together n
             domain wall-type transitions, assuming that the distance
             between transitions, 2δ, is sufficiently large, focusing on
             the illustrative cases n = 2 and 3. When n = 2 we prove that
             the Dirac operator has two real simple eigenvalues of
             opposite sign and of order e-2|κ∞|δ. The associated
             eigenfunctions are, up to L2 error of order e-2|κ∞|δ,
             linear combinations of shifted copies of the single domain
             wall zero mode. For the case n = 3, we prove the Dirac
             operator has two nonzero simple eigenvalues as in the two
             domain wall case and a simple eigenvalue at energy zero. The
             associated eigenfunctions of these eigenvalues can again, up
             to small error, be expressed as linear combinations of
             shifted copies of the single domain wall zero mode. When n >
             3 no new technical difficulty arises and the result is
             similar. Our methods are based on a Lyapunov-Schmidt
             reduction/ Schur complement strategy, which maps the Dirac
             operator eigenvalue problem for eigenstates with near-zero
             energies to the problem of determining the kernel of an n×n
             matrix reduction, which depends nonlinearly on the
             eigenvalue parameter. The class of Dirac operators we
             consider controls the bifurcation of topologically protected
             "edge states" from Dirac points (linear band crossings) for
             classes of Schrödinger operators with domain wall modulated
             periodic potentials in one and two space dimensions. The
             present results may be used to construct a rich class of
             defect modes in periodic structures modulated by multiple
             domain walls.},
   Doi = {10.1137/19M127416X},
   Key = {fds349661}

   Author = {Watson, A and Weinstein, MI},
   Title = {Wavepackets in Inhomogeneous Periodic Media: Propagation
             Through a One-Dimensional Band Crossing},
   Journal = {Communications in Mathematical Physics},
   Volume = {363},
   Number = {2},
   Pages = {655-698},
   Publisher = {Springer Nature America, Inc},
   Year = {2018},
   Month = {October},
   url = {},
   Abstract = {© 2018, Springer-Verlag GmbH Germany, part of Springer
             Nature. We consider a model of an electron in a crystal
             moving under the influence of an external electric
             field:Schrödinger’s equation in one spatial dimension
             with a potential which is the sum of a periodic function V
             and a smooth function W. We assume that the period of V is
             much shorter than the scale of variation of W and denote the
             ratio of these scales by ϵ. We consider the dynamics of
             semiclassical wavepacket asymptotic (in the limit ϵ↓ 0)
             solutions which are spectrally localized near to a crossing
             of two Bloch band dispersion functions of the periodic
             operator -12∂z2+V(z). We show that the dynamics is
             qualitatively different from the case where bands are
             well-separated: at the time the wavepacket is incident on
             the band crossing, a second wavepacket is ‘excited’
             which has opposite group velocity to the incident
             wavepacket. We then show that our result is consistent with
             the solution of a ‘Landau–Zener’-type
   Doi = {10.1007/s00220-018-3213-x},
   Key = {fds338113}

   Author = {Watson, AB and Lu, J and Weinstein, MI},
   Title = {Wavepackets in inhomogeneous periodic media: Effective
             particle-field dynamics and Berry curvature},
   Journal = {Journal of Mathematical Physics},
   Volume = {58},
   Number = {2},
   Pages = {021503-021503},
   Publisher = {AIP Publishing},
   Year = {2017},
   Month = {February},
   url = {},
   Abstract = {© Published by AIP Publishing. We consider a model of an
             electron in a crystal moving under the influence of an
             external electric field: Schrödinger's equation with a
             potential which is the sum of a periodic function and a
             general smooth function. We identify two dimensionless
             parameters: (re-scaled) Planck's constant and the ratio of
             the lattice spacing to the scale of variation of the
             external potential. We consider the special case where both
             parameters are equal and denote this parameter ∈. In the
             limit ∈ ↓ 0, we prove the existence of solutions known
             as semiclassical wavepackets which are asymptotic up to
             "Ehrenfest time" t ln 1/∈. To leading order, the center of
             mass and average quasimomentum of these solutions evolve
             along trajectories generated by the classical Hamiltonian
             given by the sum of the Bloch band energy and the external
             potential. We then derive all corrections to the evolution
             of these observables proportional to ∈. The corrections
             depend on the gauge-invariant Berry curvature of the Bloch
             band and a coupling to the evolution of the wave-packet
             envelope, which satisfies Schrödinger's equation with a
             time-dependent harmonic oscillator Hamiltonian. This
             infinite dimensional coupled "particle-field" system may be
             derived from an "extended" ∈-dependent Hamiltonian. It is
             known that such coupling of observables (discrete
             particle-like degrees of freedom) to the wave-envelope
             (continuum field-like degrees of freedom) can have a
             significant impact on the overall dynamics.},
   Doi = {10.1063/1.4976200},
   Key = {fds328439}
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