2009/11/30 by Yu. D. Golovaty, Yu D Golovaty, R. O. Hryniv +1 · 36 citations
Mathematics · Physics and Astronomy · #Boundary value problem #Convergence (economics) #Dirac operator #Limit (mathematics) #Limiting #Nonlinear Partial Differential Equations #Norm (philosophy) #Numerical methods in inverse problems #Operator (biology) #Resolvent #Resolvent formalism #Schrödinger's cat #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP #msc:34L25 #msc:47E05 #msc:81Q10
paper · pdf · doi:10.1088/1751-8113/43/15/155204
published in Journal of Physics A Mathematical and Theoretical 43(15), 155204 (Institute of Physics) · 16 pages, 2 figures. The proof of Lemma 2.1 was corrected.The main results of the paper are unchanged. Other minor changes were made
openalex publication_date 2010/03/25 · arxiv created 2010/11/27 · arxiv updated 2015/03/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
For a function that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schrödinger operators on the line of the form If the potential V satisfies the conditions then the functions ε −2 V ( x /ε) converge in the sense of distributions as ε → 0 to δ'( x ), and the limit S 0 of S ε might be considered as a 'physically motivated' interpretation of the one-dimensional Schrödinger operator with a potential δ'. In 1985, Šeba claimed that the limit operator S 0 is the direct sum of the free Schrödinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with the potential δ'. In this paper, we show that in fact S 0 essentially depends on V : although the above results are true generically, in the exceptional (or 'resonant') case, the limit S 0 is non-trivial and is determined by the properties of an auxiliary Sturm–Liouville spectral problem associated with V . We then set V (ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances α k ∞ k = − ∞ for which a partial transmission of the wave package occurs for S 0 .