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Schrödinger operators with interactions on unbounded hypersurfaces
Author(s) -
Rabinovich Vladimir
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5083
Subject(s) - mathematics , operator theory , unbounded operator , domain (mathematical analysis) , schrödinger's cat , mathematical analysis , transmission (telecommunications) , space (punctuation) , pure mathematics , spectral theorem , finite rank operator , banach space , linguistics , philosophy , electrical engineering , engineering
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 domainC 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.