## 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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