A 3D-Schrödinger operator under magnetic steps with semiclassical applications

Semiclassical reduction for magnetic Schrödinger operator with periodic zero-range potentials and applications

The two-dimensional Schrödinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields and a weak coupling we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.

Semiclassical resonances for a two-level Schrödinger operator with a conical intersection

We study the resonant set of a two-level Schrödinger operator with a linear conical intersection. This model operator can be decomposed into a direct sum of first order systems on the real half-line. For these ordinary differential systems we locally construct exact WKB solutions, which are connected to global solutions, amongst which are resonant states. The main results are a generalized Bohr...

On the semiclassical 3D Neumann Laplacian with variable magnetic field

Semiclassical resolvent estimates for Schrödinger operators with Coulomb singularities

Consider the Schrödinger operator with semiclassical parameter h, in the limit where h goes to zero. When the involved long-range potential is smooth, it is well known that the boundary values of the operator’s resolvent at a positive energy λ are bounded by O(h−1) if and only if the associated Hamilton flow is non-trapping at energy λ. In the present paper, we extend this result to the case wh...

On the spectrum and eigenfunctions of the Schrödinger operator with Aharonov-Bohm magnetic field

We explicitly compute the spectrum and eigenfunctions of the magnetic Schrödinger operator H( A,V) = (i∇ + A)2 + V in L2(R2), with Aharonov-Bohm vector potential, A(x1,x2)= α(−x2,x1)/|x|2, and either quadratic or Coulomb scalar potential V . We also determine sharp constants in the CLR inequality, both dependent on the fractional part of α and both greater than unity. In the case of quadratic p...