Energy of Surface States for 3D Magnetic Schrödinger Operators

Abstract

We establish a semi-classical formula for the sum of eigenvalues of a magnetic Schrödinger operator in a three-dimensional domain with compact smooth boundary and Neumann boundary conditions. The eigenvalues we consider have eigenfunctions localized near the boundary of the domain, hence they correspond to surface states. Using relevant coordinates that straighten out the boundary, the leading order term of the energy is described in terms of the eigenvalues of model operators in the half-axis and the half-plane.

Appendices

Appendix 1: Proof of Lemma 7.2

Using (7.16), (7.8) and (7.9), we obtain that for some constant $c_{1}>0$

(9.1)

Similarly, using (7.9) and (7.17), we have for some constant $c_{2}>0$

$$\begin{aligned}&(1-c_{2}(\ell +T))\int _{Q_{0,\ell ,T}}\!\!\!\!|g(y_{0})|^{1/2}|\widetilde{u}|^{2}dy \le \left\| u\right\| _{L^{2}({\mathcal {V}}_{x_{0}})}^{2}\nonumber \\&\quad \le (1+c_{2}(\ell +T))\int _{Q_{0,\ell ,T}}\!\!\!\!|g(y_{0})|^{1/2}|\widetilde{u}|^{2}dy. \end{aligned}$$

(9.2)

By the Cauchy–Schwarz inequality, we get using (7.11) that there exists a constant $c_{3}>0$ such that

(9.3)

for any $\varepsilon >0$. Next, we perform the change of variables $z=(z_{1},z_{2},z_{3})= \Big ({{\lambda ^{1/2}_{1}}}y_{1},{{\lambda ^{1/2}_{2}}}y_{2},y_{3}\Big )$. We thus infer using (7.19) the following quadratic form in the $(z_{1},z_{2},z_{3})$ variables

(9.4)

where $\mathbf{F}=(\mathbf{F}_{1}, \mathbf{F}_{2},\mathbf{F}_{3})$ is the magnetic potential given by

$$\begin{aligned} \mathbf{F}_{1}(z)=\lambda _{1}^{-1/2}\breve{ \mathbf{A}}^\mathrm{lin}_{1}(z),\quad \mathbf{F}_{2}(z)=\lambda _{2}^{-1/2}\breve{ \mathbf{A}}^\mathrm{lin}_{2}(z)\quad \mathbf{F}_{3}(z)=\breve{\mathbf{A}}^\mathrm{lin}_{3}(z). \end{aligned}$$

Also, we have

$$\begin{aligned} \displaystyle {\int _{Q_{0,\ell ,T}}^{}} |\widetilde{u}|^{2}\,|g(y_{0})|^{1/2}dy= \displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}} |\breve{{u}}|^{2}\,dz \end{aligned}$$

(9.5)

Substituting this into (9.3) yields

$$\begin{aligned} (1-c_{2}(\ell +T)) \displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}} |\breve{{u}}|^{2}\,dz \le \left\| u\right\| _{L^{2}({\mathcal {V}}_{x_{0}})}^{2} \le (1+c_{2}(\ell +T)) \displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}} |\breve{{u}}|^{2}\,dz. \end{aligned}$$

(9.6)

Let $\beta =(\beta _{1},\beta _{2},\beta _{3})=\mathrm{curl}_{z}(\mathbf{F}(z))$ and note that the coefficients of $\beta$ and $\alpha$ (see (7.14)) are related by

$$\begin{aligned} \beta _{1}= {\lambda ^{-1/2}_{2}}\alpha _{1},\quad \beta _{2}= {\lambda ^{-1/2}_{1}}\alpha _{2},\quad \beta _{3}=( \lambda _{1}\lambda _{2})^{-1/2}\alpha _{3}. \end{aligned}$$

The relation (7.13) gives that

$$\begin{aligned} |\beta |=|\mathrm{curl}_{z}(\mathbf{F}(z)) |=(\beta _{1}^{2}+\beta _{2}^{2}+\beta _{3}^{2})^{1/2} = |\widetilde{\mathbf{B}}|(y_{0}) \end{aligned}$$

(9.7)

We thus perform a gauge transformation so that there exists a function $\phi _{0}\in C^{\infty }(\widetilde{Q}_{0,\ell ,T})$ such that

$$\begin{aligned} \mathbf{F}(z)=b_{0}\mathbf{F}_{\theta _{0}}(z)+\nabla \phi _{0},\quad b_{0}=|\widetilde{\mathbf{B}}(y_{0})|, \end{aligned}$$

(9.8)

where, for $\theta \in [0,\pi /2], \mathbf{F}_{\theta }$ is the magnetic field from (3.4) and

$$\begin{aligned} {\theta _{0}}:=\widetilde{\theta }(y_{0})=\arcsin \left( \dfrac{|\beta _{3}|}{|\beta |}\right) . \end{aligned}$$

(9.9)

We emphasize here that (9.9) is compatible with the definition of $\theta (x)$ given in (1.10), i.e., $\widetilde{\theta }(y_{0})=\theta (\Phi _{x_{0}}^{-1}(y_{0}))$. Combining (9.3), (9.4) and (9.8), we obtain, using (9.5),

(9.10)

for any $\varepsilon >0$. Choose $\varepsilon \ge \ell +T$. Inserting (9.10) into (9.1), we obtain that for some constant $c_{4}>0$

$$\begin{aligned}&(1-c_{4}\varepsilon ) \displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}}|(-ih\nabla _{z}+b_{0}\mathbf{F}_{\theta _{0}})e^{i\phi _{0}/h}\,\breve{{u}}|^{2}\,dz - c_{4}(\ell ^{2}+T^{2})^{2} \varepsilon ^{-1}\displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}}|\breve{{u}}|^{2}dz \nonumber \\&\quad \le {\mathcal {Q}}_{h}(u) \le (1+c_{4}\varepsilon )\displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}}|(-ih\nabla _{z}+b_{0}\mathbf{F}_{\theta _{0}})e^{i\phi _{0}/h}\,\breve{{u}}|^{2}\,dz\nonumber \\&\qquad +\, c_{4}(\ell ^{2}+T^{2})^{2} \varepsilon ^{-1}\displaystyle {\int _{\widetilde{Q}_{0,\ell ,T}}^{}} |\breve{{u}}|^{2}\,dz. \end{aligned}$$

(9.11)

Recall (9.6) and choose $C=\max \{c_{2},c_{4}\}$, thereby establishing (7.22) and (7.23).

Appendix 2: Proof of Lemma 8.2

According to Lemma 8.1, the lemma follows if we can prove a lower bound on the right-hand side of (8.5). We start by estimating ${\mathcal {Q}}_{h}(\psi _{1}f_{j})$. Using the IMS decomposition formula, it follows that

$$\begin{aligned} {\mathcal {Q}}_{h}(\psi _{1}f_{j})= \displaystyle {\sum _{l\in {J}}^{}}\left( {\mathcal {Q}}_{h}(\chi _{l}\psi _{1}f_{j})-h^{2}\left\| |\nabla \chi _{l}|\psi _{1}f_{j}\right\| _{L^{2}(\Omega )}^{2}\right) . \end{aligned}$$

(10.1)

Using (8.11), and implementing (8.10), we get

$$\begin{aligned}&{\mathcal {Q}}_{h}(\psi _{1}f_{j})-(\Lambda h+C_{1}h^{2}\varsigma ^{-2})\left\| \psi _{1}f_{j}\right\| _{L^{2}(\Omega )}^{2} \nonumber \\&\quad \ge \displaystyle {\sum _{l\in J}^{}}\left( {\mathcal {Q}}_{h}(\psi _{1}\chi _{l}f_{j})-\big (\Lambda h+ (C_{1}+C_{2}) h^{2}\varsigma ^{-2}\big ) \left\| \psi _{1}\chi _{l}f_{j}\right\| _{L^{2}(\Omega )}^{2}\right) , \end{aligned}$$

(10.2)

where we used that $\varsigma ^{-2}\gg 1$ (see (8.47) below).

Applying the IMS formula once again, we then find, using that $a\ll 1$,

$$\begin{aligned}&{\mathcal {Q}}_{h}(\psi _{1}\chi _{l}f_{j}) = \sum _{m\in {\mathcal {I}}_{l}} \Big \{ {\mathcal {Q}}_{h} (\varphi _{m,l}\psi _{1}\chi _{l}f_{j}) -h^{2} \left\| |\nabla \varphi _{m,l}| \psi _{1}\chi _{l}f_{j}\right\| _{L^{2}(\Omega )}^{2}\Big \}\nonumber \\&\quad \ge \sum _{m\in {\mathcal {I}}_{l}} \Big \{ {\mathcal {Q}}_{h}(\varphi _{m,l}\psi _{1}\chi _{l}f_{j}) - (C_{1}+C_{2}+C_{3}^{\prime })h^{2}(a\varsigma )^{-2}\left\| \varphi _{m,l} \psi _{1}\chi _{l}f_{j}\right\| _{L^{2}(\Omega )}^{2}\Big \}\,.\nonumber \\ \end{aligned}$$

(10.3)

The last inequality follows from (8.14) and $C^{\prime }_{3}:=C_{3}\sup _{l\in J} \Vert D\Phi _{l}\Vert ^{2}$. Inserting this into (10.2), it follows that

$$\begin{aligned}&{\mathcal {Q}}_{h}(\psi _{1}f_{j})-(\Lambda h+C_{1}h^{2}\varsigma ^{-2})\left\| \psi _{1}f_{j}\right\| _{L^{2}(\Omega )}^{2}\nonumber \\&\quad \ge \displaystyle {\sum _{l\in J}^{}}\sum _{m\in {\mathcal {I}}_{l}}\left( {\mathcal {Q}}_{h}(\varphi _{m,l} \psi _{1}\chi _{l}f_{j}) \right. \nonumber \\&\qquad \left. -\big (\Lambda h+ (C_{1}+C_{2}+C_{3}^{\prime }) h^{2}(a\varsigma )^{-2}\big ) \left\| \varphi _{m,l}\psi _{1}\chi _{l}f_{j}\right\| _{L^{2}(\Omega )}^{2}\right) \,. \end{aligned}$$

(10.4)

Applying Lemma 7.2 with $y_{0}$ replaced by $y_{m,l}, u=\varphi _{m,l}\psi _{1}\chi _{l}f_{j}, \ell =(1+a)\varsigma , T=\varsigma$, we then deduce that there exists a function $\phi _{m,l}:=\phi _{y_{m,l}}\in C^{\infty }(\widetilde{Q}^{m,l}_{(1+a)\varsigma })$ such that, for all $\varepsilon \in (0,1]$ satisfying $\varepsilon \gg \varsigma$, one has, using $a\ll 1$,

$$\begin{aligned}&{\mathcal {Q}}_{h}(\psi _{1}f_{j})-(\Lambda h+C_{1}h^{2}\varsigma ^{-2})\left\| \psi _{1}f_{j}\right\| _{L^{2}(\Omega )}^{2} \nonumber \\&\quad \ge (1-C\varepsilon ) \sum _{l\in J}\sum _{m\in {\mathcal {I}}_{l}} \displaystyle {\int _{\widetilde{Q}^{m,l}_{(1+a)\varsigma }}^{}}|(-ih\nabla _{z}+b_{m,l}\mathbf{F}_{\theta _{m,l}})\,e^{i\phi _{m,l}/h}\breve{\varphi } _{m,l}\breve{\psi }_{1}\breve{\chi }_{l}\breve{f}_{j}|^{2} dz \nonumber \\&\qquad -\big ((\Lambda h+ (C_{1}+C_{2}+C_{3}^{\prime })h^{2}(a\varsigma )^{-2})(1+3C\varsigma )\nonumber \\&\qquad +\, 25C\varsigma ^{4}\varepsilon ^{-1}\big ) \sum _{l\in J}\sum _{m\in {\mathcal {I}}_{l}} \displaystyle {\int _{{\widetilde{Q}^{m,l}_{(1+a)\varsigma }}}^{}} |\breve{\varphi }_{m,l}\breve{\psi }_{1}\breve{\chi }_{l}\breve{f}_{j}|^{2}\,dz , \end{aligned}$$

(10.5)

where $C$ is the constant from Lemma 7.2. Put $\widetilde{C}=\max \big \{C_{1}+C_{2}+C^{\prime }_{3},25 C\big \}$. Inserting (10.5) into (8.5) yields the desired estimate of the lemma.