Schrödinger operators with interactions on unbounded hypersurfaces. (6th June 2018)

• Record Type: Journal Article
• Authors: Rabinovich, Vladimir
• Other Names: Kravchenko Vladislav V. guestEditor.; Porter R. Michael guestEditor.; Torba Sergii M. guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 15(2019)
• Page Start: 4981
• Page End: 4998
• Publication Date: 2018-06-06
• Subjects: essential spectrum -- Shrödinger operators -- singular potentials -- δ−type interactions on unbounded hypersurfaces; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5083
• Languages: English
• ISSNs: 0170-4214

Abstract

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.