Schroedinger operators with singularly scaled magnetic and electric potentials
The norm resolvent convergence of a family of one-dimensional Schroedinger operators with singular magnetic and electric potentials of small support is studied.
math.SP↗
arXiv subjects
Yuriy Golovaty
Publications and source records attributed to Yuriy Golovaty.
1D Schrödinger operators with short range interactions: two-scale regularization of distributional potentials
For real bounded functions Φand Ψof compact support, we prove the norm resolvent convergence, as εand νtend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{ε, ν}= -D^2+αε^{-2}Φ(ε^{-1}x)+βν^{-1}Ψ(ν^{-1}x), provided the ratio ν/εhas a finite or infinity limit. The limit operator S_0 depends on the shape of Φand Ψas well as on the limit of ratio ν/ε. If the potential αΦpossesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Otherwise S_0 is the direct sum of the Dirichlet half-line Schroedinger operators.
math.SP↗
Schroedinger operators with (αδ'+βδ)-like potentials: norm resolvent convergence and solvable models
For real functions Φand Ψthat are integrable and compactly supported, we prove the norm resolvent convergence, as ε goes to 0, of a family S(ε) of one-dimensional Schroedinger operators on the line of the form S(ε)= -D^2 + αε^{-2} Φ(x/ε) + βε^{-1} Ψ(x/ε). The limit results are shape-dependent: without reference to the convergence of potentials in the sense of distributions the limit operator S(0) exists and strongly depends on the pair (Φ,Ψ). We show that it is impossible to assign just one self-adjoint operator to the pseudo-Hamiltonian -D^2 + αδ'(x) + βδ(x), which is a symbolic notation only for a wide variety of quantum systems with quite different properties.
math.SP↗