Approximations of quantum-graph vertex couplings by singularly scaled potentials (Q2848012)

The authors investigate the limit properties of a family of Schrödinger operators of the form NEWLINE\[NEWLINEH_\varepsilon= -d^2/dx^2+ \lambda(\varepsilon) Q(x/\varepsilon)/\varepsilon^2NEWLINE\]NEWLINE acting \(n\)-edge star graphs with the Kirchhoff interface conditions at the vertex. Here the real-valued potential \(Q\) has compact support and \(\lambda(\cdot)\) is analytic around \(\varepsilon=0\) with \(\lambda(0)=1\).NEWLINENEWLINE A Schrödinger operator in \(L^2(\Gamma)\) of the form \(S:=-d^2/dx^2+ Q\) satisfying Kirchhoff conditions has a zero-energy resonance of order \(m\) if there exist \(m\) linearly independent resonant solutions \(\psi_1,\dots,\psi_m\) to the equation NEWLINE\[NEWLINE-\psi''+ Q\psi=0,\tag{1}NEWLINE\]NEWLINE which are bounded on \(\Gamma\). Since every bounded solution of (1) is constant outside the support of \(Q\), it follows that \(\psi_i\) solves the Neumann problem NEWLINE\[NEWLINE-\psi''+ Q\psi=0\quad\text{on }\Omega,\quad \psi\in K(\Omega).\tag{2}NEWLINE\]NEWLINE Lemma: If the problem (2) has \(m\) linearly independent solutions, then one can choose them as real-valued functions \(\psi_1,\dots, \psi_m\) satisfying \(\psi_i(a_j)= \delta_{ij}\), \(i,j= 1,\dots, m\). -- They prove the following two theorems:NEWLINENEWLINE I. Their first main result is Theorem 2.3. The Schrödinger operators \(H\), approach \(H\) as \(\varepsilon\to 0\) in the norm-resolvent topology, and moreover for any fixed \(\zeta\in\mathbb{C}\setminus\mathbb{R}\) there is a constant \(C\) such that NEWLINE\[NEWLINE\|(H_\varepsilon- \zeta)^{-1}- (H-\zeta)^{-1}\|_{B(L^2(\Gamma)}\leq C\varepsilon^{1/2},\;\varepsilon\in (0,1].NEWLINE\]NEWLINE II. The authors denote by \(S_0\) the Schrödinger operator describing a free particle moving on the graph \(\Gamma\), i.e. \(S_0=- d^{2/dx^2}\), \(\text{dom\,}H^2(\Gamma)\cap K(\Gamma)\). Theorem 2.4. For any momentum \(k>0\), the on-shell scattering matrix for the pair \((H_\varepsilon, S_0)\) converges as \(\varepsilon\to 0\) to that of \((H,S_0)\).NEWLINENEWLINE In Theorem 2.4, in the limit \(k\to\infty\), the scattering amplitudes coinside asymptotically with that of \(-d^2/dx^2+ \varepsilon^{-2} Q(\varepsilon^{-1})\).