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## Publications of Alexander Watson    :chronological  alphabetical  combined listing:

```%% Papers Published
@article{fds349661,
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 = {http://dx.doi.org/10.1137/19M127416X},
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}
}

@article{fds338113,
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 = {http://dx.doi.org/10.1007/s00220-018-3213-x},
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
model.},
Doi = {10.1007/s00220-018-3213-x},
Key = {fds338113}
}

@article{fds328439,
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 = {http://dx.doi.org/10.1063/1.4976200},
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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