On norm resolvent convergence of Schrödinger operators with δ'-like potentials

For a function V:{\ R}\rightarrow {\ R} that is integrable and compactly supported, we prove the norm resolvent convergence, as ɛ → 0, of a family S_ɛ of one-dimensional Schrödinger operators on the line of the form S_\varepsilon := -\frac{\mathrm{d}^2}{\mathrm{d}x^2} + \frac{1}{\varepsilon ^2}V\Bigl (\frac{x}{\varepsilon }\Bigr ).

If the potential V satisfies the conditions \int _{\ R}V(\xi ) \,d\xi =0,\qquad \int _{\ R}\xi V(\xi ) \,d\xi =-1,

then the functions ɛ^-2V(x/ɛ) converge in the sense of distributions as ɛ → 0 to δ'(x), and the limit S₀ of S_ɛ 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 S₀ 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 S₀ essentially depends on V: although the above results are true generically, in the exceptional (or 'resonant') case, the limit S₀ 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 S₀.