Weakly coupled bound state of 2-D Schrödinger operator with potential-measure

The weakly coupled fractional one - dimensional Schrödinger operator with index 1 < α ≤ 2

We study fundamental properties of the fractional, one-dimensional Weyl operator P̂α densely defined on the Hilbert space H = L2(R, dx) and determine the asymptotic behaviour of both the free Green’s function and its variation with respect to energy for bound states. In the sequel we specify the Birman-Schwinger representation for the Schrödinger operator KαP̂ − g|V̂ | and extract the finite-rank ...

Ground State for the Schrödinger Operator with the Weighted Hardy Potential

Weakly Coupled Schrödinger Operators on Regular Metric Trees

Spectral properties of the Schrödinger operator Aλ = −∆+λV on regular metric trees are studied. It is shown that as λ goes to zero the behavior of the negative eigenvalues of Aλ depends on the global structure of the tree. Mathematics Subject Classification: 34L40, 34B24, 34B45.

Persistent currents for 2 D Schrödinger operator with a strong δ - interaction on a loop

One of the most often studied features of mesoscopic systems are the persistent currents in rings threaded by a magnetic flux – let us mention, e.g., [CGR, CWB] and scores of other theoretical and experimental papers where they were discussed. For a charged particle (an electron) confined to a loop Γ the effect is manifested by the dependence of the corresponding eigenvalues λn on the flux φ th...

D ec 2 00 7 DYNAMICAL RESONANCES AND SSF SINGULARITIES FOR A MAGNETIC SCHRÖDINGER OPERATOR

We consider the Hamiltonian H of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator H has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb H by appropriate scalar potentials V and investigate the transformation of these embedded eigenvalues into resonances. First,...