43 research outputs found
Convergence of nodal sets in the adiabatic limit
We study the nodal sets of non-degenerate eigenfunctions of the Laplacian on
fibre bundles in the adiabatic limit. This limit consists in
considering a family of Riemannian metrics, that are close to
Riemannian submersions, for which the ratio of the diameter of the fibres to
that of the base is given by .
We assume to be compact and allow for fibres with boundary, under the
condition that the ground state eigenvalue of the Dirichlet-Laplacian on
is independent of the base point. We prove for that the
nodal set of the Dirichlet-eigenfunction converges to the pre-image
under of the nodal set of a function on that is determined as
the solution to an effective equation. In particular this implies that the
nodal set meets the boundary for small enough and shows that many
known results on this question, obtained for some types of domains, also hold
on a large class of manifolds with boundary. For the special case of a closed
manifold fibred over the circle we obtain finer estimates and prove
that every connected component of the nodal set of is smoothly
isotopic to the typical fibre of .Comment: revised version, Annals of Global Analysis and Geometry, 201
A many-body RAGE theorem
We prove a generalized version of the RAGE theorem for N-body quantum
systems. The result states that only bound states of systems with particles persist in the long time average. The limit is formulated by means
of an appropriate weak topology for many-body systems, which was introduced by
the second author in a previous work, and is based on reduced density matrices.
This topology is connected to the weak-* topology of states on the algebras of
canonical commutation or anti-commutation relations, and we give a formulation
of our main result in this setting.Comment: Final version to appear in Comm. Math. Phys.; specifically:
Communications in Mathematical Physics, Springer Verlag (Germany), 2015, in
pres
The adiabatic limit of the connection Laplacian
We study the behaviour of Laplace-type operators H on a complex vector bundle
E M in the adiabatic limit of the base space. This space is a
fibre bundle M B with compact fibres and the limit corresponds to
blowing up directions perpendicular to the fibres by a factor 1/.
Under a gap condition on the fibre-wise eigenvalues we prove existence of
effective operators that provide asymptotics to any order in for H
(with Dirichlet boundary conditions), on an appropriate almost-invariant
subspace of L(E).Comment: To appear in the Journal of Geometric Analysi
Effective Hamiltonians for Thin Dirichlet Tubes with Varying Cross-Section
We show how to translate recent results on effective Hamiltonians for quantum
systems constrained to a submanifold by a sharply peaked potential to quantum
systems on thin Dirichlet tubes. While the structure of the problem and the
form of the effective Hamiltonian stays the same, the difficulties in the
proofs are different.Comment: 6 pages, 1 figur
Generalised Quantum Waveguides
We study general quantum waveguides and establish explicit effective
Hamiltonians for the Laplacian on these spaces. A conventional quantum
waveguide is an -tubular neighbourhood of a curve in
and the object of interest is the Dirichlet Laplacian on this
tube in the asymptotic limit . We generalise this by
considering fibre bundles over a -dimensional submanifold
with fibres diffeomorphic to ,
whose total space is embedded into an -neighbourhood of . From
this point of view takes the role of the curve and that of the
disc-shaped cross-section of a conventional quantum waveguide. Our approach
allows, among other things, for waveguides whose cross-sections are
deformed along and also the study of the Laplacian on the boundaries of
such waveguides. By applying recent results on the adiabatic limit of
Schr\"odinger operators on fibre bundles we show, in particular, that for small
energies the dynamics and the spectrum of the Laplacian on are reflected by
the adiabatic approximation associated to the ground state band of the normal
Laplacian. We give explicit formulas for the according effective operator on
in various scenarios, thereby improving and extending many of the
known results on quantum waveguides and quantum layers in
Particle Creation at a Point Source by Means of Interior-Boundary Conditions
We consider a way of defining quantum Hamiltonians involving particle
creation and annihilation based on an interior-boundary condition (IBC) on the
wave function, where the wave function is the particle-position representation
of a vector in Fock space, and the IBC relates (essentially) the values of the
wave function at any two configurations that differ only by the creation of a
particle. Here we prove, for a model of particle creation at one or more point
sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian
can indeed be rigorously defined in this way without the need for any
ultraviolet regularization, and that it is self-adjoint. We prove further that
introducing an ultraviolet cut-off (thus smearing out particles over a positive
radius) and applying a certain known renormalization procedure (taking the
limit of removing the cut-off while subtracting a constant that tends to
infinity) yields, up to addition of a finite constant, the Hamiltonian defined
by the IBC.Comment: 41 page