Schrödinger operators with interactions on unbounded hypersurfaces. (6th June 2018) - Rabinovich, Vladimir
Abstract : We consider Schrödinger operators H = −Δ + W + W s on R n with regular potentials W ∈ L ∞ ( R n ) and singular potentials W s ∈ D ′ ( R n ) with supports on unbounded enough smooth hypersurfaces Γ. In particular, we consider singular potentials that are linear combinations of δ −functions on Γ and its normal derivatives. We consider extensions of H as symmetric operators in L 2 ( R n ) with domain C 0 ∞ ( R n ⃥ Γ ) to self‐adjoint operators H in L 2 ( R n ) . These extensions are realized as operators of transmission problems for −Δ + W in the space H 2 ( R n ⃥ Γ ) with some transmission conditions on Γ. Applying this approach, we obtain an effective description of essential spectra of the described Schrödinger operators.