Resonances at the Threshold for Pauli Operators in Dimension Two

Abstract

It is well-known that, due to the interaction between the spin and the magnetic field, the two-dimensional Pauli operator has an eigenvalue 0 at the threshold of its essential spectrum. We show that when perturbed by an effectively positive perturbation, V, coupled with a small parameter $\varepsilon$, these eigenvalues become resonances. Moreover, we derive explicit expressions for the leading terms of their imaginary parts in the limit $\varepsilon \searrow 0$. These show, in particular, that the dependence of the imaginary part of the resonances on $\varepsilon$ is determined by the flux of the magnetic field. The cases of non-degenerate and degenerate zero eigenvalue are treated separately. We also discuss applications of our main results to particles with anomalous magnetic moments.

Appendices

Appendix A

Proof of Propositions 2.3 and 2.5

Although not explicitly stated there, the existence of the operator $S_0$ in (2.16) follows from the proof of [14, Theorem 5.6]. The self-adjointness of $S_0$ follows by considering the expansion (2.16) for $z=-x$ with $x>0$. Then $(P_-(A) -z)^{-1}= (P_-(A) +x)^{-1}$ is self-adjoint. On the other hand, $\omega (1+{\alpha '}) (-x)^{{\alpha '}-1}$ and $\zeta ({\alpha '}) (-x)^{-{\alpha '}}$ are real numbers, see Eq. (2.15). Hence, the first three operators on the right-hand side of (2.16) are self-adjoint. Therefore $S_0$ is self-adjoint too. In the same way, one proves the existence and self-adjointness of $\mathrm{T_0}$ in Proposition 2.5. $\square$

Appendix B

The following technical result is not new, see [11, Eq. (3.56)]. For the sake of completeness, we give a short proof.

Lemma B.1

Let $a>1$. Then

$$\begin{aligned} \sup _{s>0} \, \sup _{R >a}\, \Big | \int _{-R}^R \frac{e^{-i s \frac{y}{R}}}{y+i}\, dy\, \Big | \, < \, \infty . \end{aligned}$$

Proof

By the residue theorem,

$$\begin{aligned} \int _{-R}^R \frac{e^{-i s \frac{y}{R}}}{y+i}\, dy\ = -2\pi i \, e^{-\frac{s}{R}} - \int _\Gamma \frac{e^{-i s \frac{z}{R}}}{z+i}dz \end{aligned}$$

where $\Gamma$ is the semi-circle of radius R in the lower complex half-plane centered in the origin and directed from (R, 0) to $(-R,0)$. The first term on the hand side is obviously bounded uniformly in $s>0$ and $R>a$. As for the second term, parametrizing $\Gamma$ in the usual way we find

$$\begin{aligned} \Big | \int _\Gamma \frac{e^{-i s \frac{z}{R}}}{z+i}dz\, \Big |&= R\ \Big | \int _0^\pi \frac{e^{-i s \cos \theta -s \sin \theta }e^{i\theta }}{R \cos \theta -i R\sin \theta +i}\, d\theta \, \Big | \le \frac{\pi R}{R-1}\le \frac{\pi a}{a-1}\, , \end{aligned}$$

since $|R \cos \theta -i R\sin \theta +i | \ge R-1$. $\square$

Appendix C

The following result is a straightforward generalization of [12, Proposition 1.1], see also [4, Proposition 1.3].

Lemma C.1

Let $1<\alpha \le 2$. Let $f,g: \mathbb {R}_+ \rightarrow \mathbb {R}$ be positive decreasing function such that $f(\varepsilon ) \rightarrow 0, \ g(\varepsilon )\rightarrow 0$ as $\varepsilon \rightarrow 0$. Assume that there exist $\lambda ^j_\varepsilon = x^j_\varepsilon -i\gamma ^j_\varepsilon$ such that for sufficiently small $\varepsilon$,

$$\begin{aligned} \sup _{t>0} \big | \mathcal {\big \langle }\psi _0, \, e^{-it P_{\scriptscriptstyle -}(A,\varepsilon )}\, \psi _0 \mathcal {\big \rangle }- e^{-it\lambda ^j_\varepsilon } \big | \, \le \, f(\varepsilon ), \quad j=1,2, \end{aligned}$$

(C.1)

with $\psi _0$ given in (3.2). If

$$\begin{aligned} 0 \le \gamma ^1_\varepsilon \le g(\varepsilon ), \end{aligned}$$

(C.2)

for all $\varepsilon$ small enough, then there exists $\varepsilon _0>0$ and a constant C such that

$$\begin{aligned} | \lambda _\varepsilon ^1 -\lambda _\varepsilon ^2 | \le C\, f(\varepsilon ) g(\varepsilon ) \quad \forall \ \varepsilon \in [0, \varepsilon _0]. \end{aligned}$$

(C.3)

Proof

By (C.1),

$$\begin{aligned} \sup _{t>0} \big | e^{-it\lambda ^1_\varepsilon } - e^{-it\lambda ^2_\varepsilon } \big | \, \le \, 2 f(\varepsilon ) \ \end{aligned}$$

holds for $\varepsilon$ small enough. This in combination with (C.2) implies

$$\begin{aligned} \big |1- e^{-it(\lambda ^2_\varepsilon -\lambda ^1_\varepsilon )} \big | \, \le \, 2 f(\varepsilon ) \, e^{t g(\varepsilon )} \quad \forall \ t>0. \end{aligned}$$

(C.4)

Now let

$$\begin{aligned} z= 1- e^{-it(\lambda ^2_\varepsilon -\lambda ^1_\varepsilon )} =: a+ ib, \quad a,b\in \mathbb {R}. \end{aligned}$$

Taking $\varepsilon _0$ small enough, we can make sure that $|z| \le \frac{1}{2}$ for all $0\le \varepsilon \le \varepsilon _0$ and $t\le \frac{1}{g(\varepsilon )}$. Hence using the principal branch of the complex logarithm, and the estimate

$$\begin{aligned} |\log (1-z) |\le & {} \left| \arctan \Big ( \frac{b}{1-a} \Big ) \right| +\big | \log |1-z| \big |\nonumber \\\le & {} \frac{|b|}{1-|a|} + 2 |z| \le 4 |z| \quad \forall \, z: |z| \le \frac{1}{2}, \end{aligned}$$

(C.5)

we get

$$\begin{aligned} t | \lambda _\varepsilon ^1 -\lambda _\varepsilon ^2 | =\big | \log \big (1-(1- e^{-it(\lambda ^2_\varepsilon -\lambda ^1_\varepsilon )} ) \big )\big | \, \le \, 4 \big |1- e^{-it(\lambda ^2_\varepsilon -\lambda ^1_\varepsilon )} \big |. \end{aligned}$$

The claim thus follows from (C.4) upon setting $t= \frac{1}{g(\varepsilon )}$. $\square$