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On the semiclassical 3D Neumann Laplacian with variable magnetic field. (English) Zbl 1200.35202

Let \(\Omega\) be an open bounded subset of \(\mathbb R^3\) with a smooth boundary \(\partial\Omega\). For a vector potential \(\mathbf A\in C^{\infty}(\overline\Omega,\mathbb R^3)\) and \(h>0\), it is introduced \(\beta=\nabla\times\mathbf A\) and the quadratic form defined for all \(\psi\in H^1(\Omega,\mathbb C)\) by \(q^h_{\mathbf A}(\psi)=\int_{\Omega}|(ih\nabla+\mathbf A)\psi|^2dx\). One of the author’s interests is the lowest eigenvalue \(\lambda_1^h(\mathbf A)\) of the associated self-adjoint operator, i.e., the Neumann realization of \((ih\nabla+\mathbf A)^2\) on \(\Omega\) denoted by \(P^h_{\mathbf A}\). Let us formulate the main theorem of the paper. First, we give some preliminaries.
For \(x\in\partial\Omega\), it is introduced \(\hat\beta(x)=\sigma(\theta(x))\|\beta(x)\|\), where \(\theta(x)\) is defined by \(\|\beta(x)\|\sin\theta(x)=\beta\cdot\nu(x)\) with \(\nu(x)\) the inward pointing normal at \(x\) and \(\sigma(\theta(x))\) denotes the bottom of the spectrum of an operator which is explicitly defined in the paper. The author assumes that \(\hat\beta\) admits a non-degenerate minimum at \(x_0\) and that at this point \(0<\theta(x_0)<\frac{\pi}2\).
Further, denote by \(\boldsymbol\Theta_{\beta}\) the Hessian matrix at \(x_0\), namely, \[ \boldsymbol\Theta_{\beta}(r,s)=\frac 12\partial^2_r\hat\beta(x_0)r^2+\frac 12 \partial^2_{rs}\hat\beta(x_0)(sr+rs)+\frac 12\partial^2_s\hat\beta(x_0)s^2 \] and introduce the operator \(\tilde{\boldsymbol\Theta}_{\beta}=\boldsymbol\Theta_{\beta} (D_{\tau},\frac{\tau}{\sin\theta})\). Under the above restrictions, the author proves the main theorem which states that there exists \(d\in\mathbb R\) such that for all \(n\in\mathbb N^*\) there exists \(D_n>0\) and \(h_n>0\) such that, for \(0<h\leq h_n\), there exists at least one eigenvalue \(\mu_n\) of \(P^h_{\mathbf A}\) such that \[ |\mu_n-[\hat\beta(x_0)h+C^{\beta,K}(x_0)\|\beta(x_0)\|^{\frac 12} h^{\frac 32}+(\gamma_n(\tilde{\boldsymbol\Theta}_{\beta})+d)h^2]|\leq D_nh^{\frac 52}, \] where \(\gamma_n(\tilde{\boldsymbol\Theta}_{\beta})\) is the \(n\)-th eigenvalue of \(\tilde{\boldsymbol\Theta}_{\beta}\) and \(C^{\beta,K}(x_0)\) is a quantity defined in the paper.
The theorem admits the following corollary: under the same restrictions as above, there exist \(d\in\mathbb R\), \(D_1>0\), and \(h_1>0\) such that, for \(0<h\leq h_1\), \[ \lambda_1^h(\mathbf A)\leq \hat\beta(x_0)h+C^{\beta,K}(x_0)\|\beta(x_0)\|^{\frac 12}h^{\frac 32}+(\gamma_1(\tilde{\boldsymbol\Theta}_{\beta})+d)h^2+D_1h^{\frac 52}. \]

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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