27 research outputs found
Quantum graphs as holonomic constraints
We consider the dynamics on a quantum graph as the limit of the dynamics
generated by a one-particle Hamiltonian in R^2 with a potential having a deep
strict minimum on the graph, when the width of the well shrinks to zero. For a
generic graph we prove convergence outside the vertices to the free dynamics on
the edges. For a simple model of a graph with two edges and one vertex, we
prove convergence of the dynamics to the one generated by the Laplacian with
Dirichlet boundary conditions in the vertex.Comment: 28 pages, 3 figure
Eigenfunctions at the threshold energies of magnetic Dirac operators
Discussed are modes and resonances of Dirac operators with
vector potentials . Asymptotic
limits of modes at infinity are derived when ,
, provided that has modes. In wider classes of vector
potentials, sparseness of the vector potentials which give rise to the
modes of are established. It is proved that no has
resonances if , .Comment: 25 pages, New results are adde
ON THE APPROXIMATION NUMBERS OF SOBOLEV EMBEDDINGS ON SINGULAR DOMAINS AND TREES
Upper and lower bounds are determined for a function which counts the approximation numbers of the Sobolev embedding W 1,p (�)/C ↩ → L p (�)/C, for a wide class of domains � of finite volume in R n and 1 < p < ∞. Results on the distribution of the eigenvalues of the Neumann Laplacian in L 2 (�) are special consequences
Coupling in the singular limit of thin quantum waveguides
We analyze the problem of approximating a smooth quantum waveguide with a
quantum graph. We consider a planar curve with compactly supported curvature
and a strip of constant width around the curve. We rescale the curvature and
the width in such a way that the strip can be approximated by a singular limit
curve, consisting of one vertex and two infinite, straight edges, i.e. a broken
line. We discuss the convergence of the Laplacian, with Dirichlet boundary
conditions on the strip, in a suitable sense and we obtain two possible limits:
the Laplacian on the line with Dirichlet boundary conditions in the origin and
a non trivial family of point perturbations of the Laplacian on the line. The
first case generically occurs and corresponds to the decoupling of the two
components of the limit curve, while in the second case a coupling takes place.
We present also two families of curves which give rise to coupling.Comment: 20 pages, no figures. Minor misprint corrected, two references adde