Dirac operators on hypersurfaces as large mass limits (Q1984846)

Let \(n \geq 2\), let \(N := 2^{[(n+1)/2]}\), and let \(\alpha_1, \dots, \alpha_{n+1}\) be a family of anticommuting \(N \times N\) matrices with \(\alpha_j^2=I_N\), \(j \in \{ 1, \dots N\}\), where \(I_N\) is the \(N \times N\) identity matrix. Moreover, let \(\Omega \subset \mathbb{R}^n\) be a bounded domain with smooth boundary and let \(\nu = (\nu_1, \dots, \nu_n)\) be the unit normal vector field at \(\partial \Omega\) pointing outwards of \(\Omega\). In the present paper, the authors study the discrete spectrum for two types of self-adjoint Dirac operators: for \(m \in \mathbb{R}\), the Dirac operator \(A_m\) in \(L^2(\Omega; \mathbb{C}^N)\) with infinite mass boundary conditions defined by \[ A_m u = -i \sum_{j=1}^n \alpha_j \frac{\partial u}{\partial x_j} + m \alpha_{n+1} u, \quad \text{dom}\, A_m = \left\{ u \in H^1(\Omega; \mathbb{C}^N): u = -i \alpha_{n+1} \sum_{j=1}^n \alpha_j \nu_j u \text{ on } \partial \Omega \right\}, \] and for \(m, M \in \mathbb{R}\), the Dirac operator \(B_{m,M}\) in \(L^2(\mathbb{R}^n; \mathbb{C}^N)\) given by \[ B_{m,M} u = -i \sum_{j=1}^n \alpha_j \frac{\partial u}{\partial x_j} + [m \mathbf{1}_\Omega u + M \mathbf{1}_{\mathbb{R}^n \setminus \Omega}] \alpha_{n+1} u, \quad \text{dom}\, B_{m,M} = H^1(\mathbb{R}^n; \mathbb{C}^N), \] are considered. In the last formulae, \(H^1(\Omega; \mathbb{C}^N)\) and \(H^1(\mathbb{R}^n; \mathbb{C}^N)\) denote the standard Sobolev spaces of first order of vector-valued functions with \(N\) components and \(\mathbf{1}_O\) is the characteristic function for a measurable set \(O\). In the main results of the paper, the authors prove several statements on the asymptotic behavior of the eigenvalues of \(A_m\) and \(B_{m,M}\). For this reason, quadratic forms for \(A_m^2\) and \(B_{m,M}^2\) are derived, which are, with the help of the min-max principle and the monotone convergence of quadratic forms, suitable to analyze the discrete eigenvalues of \(A_m\) and \(B_{m,M}\). In the following, denote by \(\mathcal{D}\) the intrinsic Dirac operator on \(\partial \Omega\). In their first main result, the authors prove that the discrete eigenvalues of \(A_m\) converge to corresponding eigenvalues of \(\mathcal{D}\), as \(m \rightarrow -\infty\). This shows that the spectrum of the Dirac operator with infinite mass boundary conditions is closely related to the spectrum of \(\mathcal{D}\), and the fact that the effective operator \(\mathcal{D}\) is an intrinsic Dirac operator is an interesting observation which is a main novelty of this paper. Concerning the relation of \(A_m\) and \(B_{m,M}\) it is shown that all discrete eigenvalues of \(B_{m,M}\) converge to eigenvalues of \(A_m\), as \(M\rightarrow \infty\). In particular, this shows that the discrete eigenvalues of the Dirac operator \(A_m\) with infinite mass boundary conditions are close to the eigenvalues of the Dirac operator \(B_{m,M}\) with a large mass term in \(\mathbb{R}^n \setminus \Omega\). Finally, the authors prove, independently from the above mentioned results, that the discrete eigenvalues of \(B_{m,M}\) converge in the asymptotic regime \(m \rightarrow -\infty\) and \(M \rightarrow +\infty\) with \(m/M\rightarrow 0\) to corresponding eigenvalues of \(\mathcal{D}\).