102 research outputs found
Kirchhoff's Rule for Quantum Wires
In this article we formulate and discuss one particle quantum scattering
theory on an arbitrary finite graph with open ends and where we define the
Hamiltonian to be (minus) the Laplace operator with general boundary conditions
at the vertices. This results in a scattering theory with channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy is explicitly given in
terms of the boundary conditions and the lengths of the internal lines. It is
shown to be unitary, which may be viewed as the quantum version of Kirchhoff's
law. We exhibit covariance and symmetry properties. It is symmetric if the
boundary conditions are real. Also there is a duality transformation on the set
of boundary conditions and the lengths of the internal lines such that the low
energy behaviour of one theory gives the high energy behaviour of the
transformed theory. Finally we provide a composition rule by which the on-shell
S-matrix of a graph is factorizable in terms of the S-matrices of its
subgraphs. All proofs only use known facts from the theory of self-adjoint
extensions, standard linear algebra, complex function theory and elementary
arguments from the theory of Hermitean symplectic forms.Comment: 40 page
Time decay of scaling critical electromagnetic Schr\"odinger flows
We obtain a representation formula for solutions to Schr\"odinger equations
with a class of homogeneous, scaling-critical electromagnetic potentials. As a
consequence, we prove the sharp time decay estimate for
the 3D-inverse square and the 2D-Aharonov-Bohm potentials.Comment: 32 pages, 1 figur
A Fourier integrator for the cubic nonlinear Schr\"{o}dinger equation with rough initial data
Standard numerical integrators suffer from an order reduction when applied to
nonlinear Schr\"{o}dinger equations with low-regularity initial data. For
example, standard Strang splitting requires the boundedness of the solution in
in order to be second-order convergent in , i.e., it requires
the boundedness of four additional derivatives of the solution. We present a
new type of integrator that is based on the variation-of-constants formula and
makes use of certain resonance based approximations in Fourier space. The
latter can be efficiently evaluated by fast Fourier methods. For second-order
convergence, the new integrator requires two additional derivatives of the
solution in one space dimension, and three derivatives in higher space
dimensions. Numerical examples illustrating our convergence results are
included. These examples demonstrate the clear advantage of the Fourier
integrator over standard Strang splitting for initial data with low regularity
Stochastic PDEs, Regularity Structures, and Interacting Particle Systems
These lecture notes grew out of a series of lectures given by the second
named author in short courses in Toulouse, Matsumoto, and Darmstadt. The main
aim is to explain some aspects of the theory of "Regularity structures"
developed recently by Hairer in arXiv:1303.5113 . This theory gives a way to
study well-posedness for a class of stochastic PDEs that could not be treated
previously. Prominent examples include the KPZ equation as well as the dynamic
model. Such equations can be expanded into formal perturbative
expansions. Roughly speaking the theory of regularity structures provides a way
to truncate this expansion after finitely many terms and to solve a fixed point
problem for the "remainder". The key ingredient is a new notion of "regularity"
which is based on the terms of this expansion.Comment: Fixed typo
Stability of stationary equivariant wave maps from the hyperbolic plane
In this paper we initiate the study of equivariant wave maps from 2d hyperbolic space, H², into rotationally symmetric surfaces. This problem exhibits markedly different phenomena than its Euclidean counterpart due to the exponential volume growth of concentric geodesic spheres on the domain. In particular, when the target is S², we find a family of equivariant harmonic maps H²→ S², indexed by a parameter that measures how far the image of each harmonic map wraps around the sphere. These maps have energies taking all values between zero and the energy of the unique corotational Euclidean harmonic map, Q[subscript euc], from R² to S², given by stereographic projection. We prove that the harmonic maps are asymptotically stable for values of the parameter smaller than a threshold that is large enough to allow for maps that wrap more than halfway around the sphere. Indeed, we prove Strichartz estimates for the operator obtained by linearizing around such a harmonic map. However, for harmonic maps with energies approaching the Euclidean energy of Q[subscript euc], asymptotic stability via a perturbative argument based on Strichartz estimates is precluded by the existence of gap eigenvalues in the spectrum of the linearized operator. When the target is H², we find a continuous family of asymptotically stable equivariant harmonic maps H² → H² with arbitrarily small and arbitrarily large energies. This stands in sharp contrast to the corresponding problem on Euclidean space, where all finite energy solutions scatter to zero as time tends to infinity.National Science Foundation (U.S.) (Grant DMS-1302782
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