Uniform spectral asymptotics for a Schrödinger operator on a segment with delta-interaction (Q6585798)

The differential expression \(\widehat {\mathcal{H}}_\varepsilon = -\dfrac {d^2}{dx^2}+V(x)+\alpha \delta (\cdot -\varepsilon )\) with Dirichlet boundary conditions on \([0,1]\), where \(V\) is a complex valued continuous potential, \(\alpha \in \mathbb{C}\setminus \{0\}\), \(\varepsilon \in(0,1)\), has the operator representation \(\mathcal{H}_\varepsilon \) in \(L_2(0,1)\) whose domain consists of all functions \(u\in W_2^1(0,1)\cap W_2^2((0,1)\setminus \{\varepsilon \})\) which satisfy the boundary conditions\N\[\Nu(0) = u(1) = 0,\quad u(\varepsilon + 0) = u(\varepsilon - 0),\quad u'(\varepsilon + 0) - u' (\varepsilon - 0) = \alpha u(\varepsilon).\N\]\NHere \(W_2^j\) denotes the usual Sobolev space. The operator \(\mathcal{H}_\varepsilon \) is considered as a perturbation of the standard Sturm-Liouville operator with Dirichlet-boundary conditions \(\mathcal{H}_0\), which is formally obtained by setting \(\alpha =0\).\N\NThe paper has two main results.\N\NTheorem 2.1 states that \(\mathcal{H}_\varepsilon \) and \(\mathcal{H}_0 \) are \(m\)-sectorial. In particular, there is \(\lambda_0 \in \mathbb{R}\) such that the half-plane \(H=\{\lambda : \text{Re}\, \lambda \le\lambda _0 \}\) is in the resolvent set of all \(\mathcal{H}_\varepsilon \) and \(\mathcal{H}_0\). For each \(\lambda \in H\) there is \(C(\lambda )\) such that \(\|(\mathcal{H}_\varepsilon -\lambda )^{-1} - (\mathcal{H}_0 -\lambda )^{-1}\|\le C(\lambda )\varepsilon ^{\frac32}\), where the norm is an operator norm in \(L(L_2(0,1),W_2^2((0,1)\setminus \{\varepsilon \}))\).\N\NTheorem 2.2 states that the eigenvalues of the operator \(\mathcal{H}_\varepsilon\) have the asymptotic representation\N\[\N\lambda _n^\varepsilon =\pi^2n^2 + 2K_{n,1}(\varepsilon ) + \frac{2K_{n,2} (\varepsilon )}{\pi n} + O(n^{-2} ).\N\]\NThe functions \(K_{n,1}\) and \(K_{n,2}\) are explicitly given and are uniformly bounded. For each \(n\), there is \(C_n\) such that \(|\lambda _n^\varepsilon -\lambda _n^0|\le C_n\varepsilon ^2\) and \(\lambda _n^\varepsilon \) is holomorphic in \(\varepsilon \).