(Osamu Ogurisu) V. V. Semenov [1] :2 $\mu$ 1/2 ; $N-1$ $N$ $\mu$ $Q$ $ \mu Q $ ( $2(N-1)$ Corollary $3.5_{\text{ }}$ Remark 3

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1 Title 異常磁気能率を伴うディラック方程式 ( 量子情報理論と開放系 ) Author(s) 小栗栖, 修 Citation 数理解析研究所講究録 (1997), 982: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 (Osamu Ogurisu) V. V. Semenov [1] :2 $\mu$ 1/2 ; $N-1$ $N$ $\mu$ $Q$ $ \mu Q $ ( $2(N-1)$ Corollary $3.5_{\text{ }}$ Remark 36 Remark 37 ) \langle 1 semenov Y. Aharonov A. Casher [2] Semenov Aharonov Aharonov-Chasher 1/2 $\tilde{n}-1$ $ q\phi $ N $q$ $\Phi$

3 42 Semenov [1] $H_{D}$ ( ) $2(N-1)$ $H_{D}$ 2 Aharonov-Casher $N-1$ 1/2 ( [3] $)$ $H_{D}$ 2 2 $\mathrm{e}(\mathrm{r})=(e_{1}(\mathrm{r}), E_{2}(\mathrm{r})),$ $\mathrm{r}=(x^{1}, x^{2})\in \mathbb{r}^{2},$ $E_{1}$, $E_{2}\in C^{\infty}(\mathbb{R}^{2};\mathbb{R})$, $\mu\in \mathbb{r}\backslash \{0\}$ 1/2 $p_{j}=-i\partial/\partial_{x^{j},i=}1,2$, $\sigma^{1}=$, $\sigma^{2}=$, $\sigma^{3}=$ $m>0$ $H_{D}$ $[1, 4]$ : $H_{D}=$

4 43 $H_{D}$ $L^{2}(\mathbb{R}^{2};\mathbb{C}^{4})$ $A= \sum_{2j=1},\sigma^{j}(p_{jj}+i\mu E)$ $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ $E_{1},$ $E_{2}$ $C^{\infty}$ Chernoff [5] $H_{D}$ $C_{0}^{\infty}(\mathbb{R}^{2};\mathbb{C}^{4})$ $H_{D}\mathrm{r}c_{0^{\infty}}(\mathbb{R};\mathbb{C}^{4})2$ $H_{D}$ A $=\dot{a}_{1}dx^{1}+a_{2}\dot{d}x^{2}$ $\mathbb{r}^{2}$ - 1 A A 1/2 $q$ $D(q\mathrm{A})$ : $D(q \mathrm{a})=\sum_{j=1,2}\sigma^{j}(pj-qa_{j})$ $D(q\mathrm{A})$ $L^{2}(\mathbb{R}^{2};\mathbb{C}^{2})$ $A_{1},$ $A_{2}\in C^{\infty}(\mathbb{R}^{2};\mathbb{R})$ $D(q\mathrm{A})$ $C_{0}^{\infty}(\mathbb{R}^{2};\mathbb{C}^{2})$ [5] $D(q\mathrm{A})\mathrm{r}C_{0}^{\infty}(\mathbb{R}^{2};\mathbb{C}^{2})$ $D(q\mathrm{A})$ $L^{2}(\mathbb{R}^{2};\mathbb{C}^{4})$ $H_{D}$ $D(\pm q\mathrm{a})$ $U$ /1 $0$ $0$ $0\backslash$ $U=$ $001$

5 44 Theorem A $\mathrm{a}=e_{2}dx^{1}-e_{1}dx^{2}$ $U^{*}H_{D}U=$ (1) Proo $C_{0}^{\infty}(\mathbb{R}^{2};\mathbb{C}^{4})$ (1) $E_{1},$ $E_{2}$ $C^{\infty}$ $H_{D}$ (1) [5] $U$ $C_{0}^{\infty}(\mathbb{R}^{2};\mathbb{C}^{4})$ (1) Remark 2.2 C. R. Hagen [6] 2 $s\in\{\pm 1\}$ Hagen ( $s$ ) Aharonov Casher [7] $H_{NR}$ $H_{D}$ 2 $H_{NR}$

6 45 4 $H_{NR}$ W. Hunziker [8] $H_{D}$ $carrow$ $\infty$ ( $c$ ) : $\lim_{carrow\infty}h_{d}=$ $H_{NR}-- \frac{1}{2m}a*a$ $H_{D}$ Ahronov $H_{NR}$ $c=1$ Remark 2.3 Aharonov Casher [7] $(\mathrm{a}\mathrm{c})$ Aharonov-Bohm (AB) Aharonov-Casher $U$ 2

7 $=1,2$ /2 Theorem 2.1 $H_{D}$ $-$ $H_{D}$ Semenov 1/2 ( [3]) $H_{D}$ $\phi\in C^{\infty}(\mathbb{R}^{2}arrow \mathbb{r})$ $\mathrm{s}.\mathrm{t}$, $E_{j}=-. \frac{\partial\phi}{\partial x^{j}}$, $\dot{\mathcal{j}}$. $\phi$ $\phi$ $\mathrm{a}=e_{2}d_{x}1-e_{1}dx2$ $d\mathrm{a}=\delta\phi dx\wedge d1x2$ -\Delta \mbox{\boldmath $\phi$}/4\mbox{\boldmath $\pi$} Lemma 3.1 (Shigekawa, Proposition 2.5 [3]) $H_{1},$ $H_{2}$ $S$ : $H_{1}arrow H_{2}$ $m$ $H_{1}\oplus H_{2}$ $T=$ $T$ $\sigma(t)=\{\sqrt{m^{2}+s};s\in\sigma(s^{*}s)\}\cup\{-\sqrt{m^{2}+s};s\in\sigma(ss^{*})\}$, $\sigma_{ess}(\tau)=\{\sqrt{m^{2}+s};s\in\sigma_{ess}(s^{*}s)\}\cup\{-\sqrt{m^{2}+s};s\in\sigma ess(ss^{*})\}$

8 47 $\sigma(a)$ $\sigma_{ess}(a)$ $A$ Lemma 3.2 (Deift, [9]) $\sigma(ss^{*})\backslash \{0\}=\sigma(s^{*}s)\backslash \{0\}$, $\sigma_{ess}(ss^{*})\backslash \{\mathrm{o}\}=\sigma ess(s^{*}s)\backslash \{0\}$ $H_{D}$ $\circ$ Corollary 3.3 $\mu=1$ (i) $ \mathrm{r} arrow$. $\infty$ $\Delta\phi(\mathrm{r})arrow 0$ $\sigma(h_{d})=\sigma ess(hd)=(-\infty, -m]\cup[m, \infty)$ (ii) $ \mathrm{r} arrow\infty$ $\Delta\emptyset(\Gamma)arrow 1$ $\sigma_{ess}(h_{d})=\{\pm\sqrt{2k+m^{2}};k\in \mathrm{n}\}\cup\{m\}$ $m$ (iii) $ \mathrm{r} arrow\infty$ $\Delta\phi(\mathrm{r})arrow\infty$ $\sigma(h_{d})$ $m$ $m$ Proof.$\cdot$ (ii) [3] Example 4.1 $\sigma_{ess}(d(\pm \mathrm{a})2)=\{2(k-1);k\in \mathrm{n}\}$

9 $\mathbb{r}$ $\mathrm{k}$ 48 $0$ $\mathrm{k}\mathrm{e}\mathrm{r}d(\pm \mathrm{a})\subset \mathrm{k}\mathrm{e}\mathrm{r}(\sigma^{3}\mp 1)$ Lemma 3.1 $H_{D}=D(\pm \mathrm{a})\pm m\sigma^{3}$ $\sigma_{es}s(d(\pm \mathrm{a})\pm m\sigma^{3})=\{+\sqrt{2k+m^{2}}, -\sqrt{2k+m^{2}};k\in \mathrm{n}\}\cup\{m\}$ $m$ The- orem 2.1 (i) (iii) Remark 3.4 Corollary 33 $H_{D}$ 1/2 (cf. [3]) $H_{D}$ 1, if $x\geq 0$ $\epsilon(x)=\{$ $-1$, if $x<0$ Corollary $\lim_{ \mathrm{r} arrow\infty^{e^{\phi}}}(\mathrm{r})/ \mathrm{r} ^{-\nu}$ 3.50 $\nu\in$ $N$ $N< \mu\nu $ dim ker$(h2-dm^{2})= \max\{2(n-1), 0\}$, $\mathrm{k}\mathrm{e}\mathrm{r}(h_{d^{-}}^{2}m2)=\mathrm{k}\mathrm{e}\mathrm{r}(hd-\epsilon(\mu\nu)m)$

10 49 Proof.$\cdot$ $T$ $\mathrm{k}\mathrm{e}\mathrm{r}t^{2}=\mathrm{k}\mathrm{e}\mathrm{r}t$ [10] Theorem 3.1 Theorem 32 $\dim \mathrm{k}\mathrm{e}\mathrm{r}d(\pm\mu \mathrm{a})=\dim \mathrm{k}\mathrm{e}\mathrm{r}d(\pm\mu \mathrm{a})^{2}=\max\{n-1, \mathrm{o}\}$, $\mathrm{k}\mathrm{e}\mathrm{r}d(\pm\mu \mathrm{a})=\mathrm{k}\mathrm{e}\mathrm{r}d(\pm\mu \mathrm{a})^{2}\subset \mathrm{k}\mathrm{e}\mathrm{r}(\sigma^{3}\mp\epsilon(\mu\nu))$ Theorem 2.1 $UH_{D}^{2}U=D(\mu \mathrm{a})2\oplus D(-\mu \mathrm{a})^{2}+m^{2}$ Remark 3.6 Corollary 3.5. $\subset$ \mbox{\boldmath $\sigma$}(hd) $(-\infty, -m]\cup[m, \infty)$ 1. \epsilon (\mu \nu ) $=1$ $m$ $\max\{2(n-1), \mathrm{o}\}$ 2. \epsilon (\mu \nu ) $=-1$ $m$ $-m$ $\max\{2(n-1), 0\}$ $m$ $-m$ $\sigma(h_{d})=(-\infty, -m]\cup[m, \infty)$ (2) [10] Remark 3.5 $-\Delta\phi/4\pi$ Corollary 35 (2)

11 50 Remark 3.7 Section 1 Semenov [1] $H_{D}$ this particle has $(N-1)$-fold degeneracy of the ground state Remark 3.8 $N$ $\phi(\mathrm{r})$ $N$ -\Delta \mbox{\boldmath $\phi$}/4\mbox{\boldmath $\pi$} 2 1/2 $n$ Aharonov-Casher $n$ ( ) [1] V. V. Semenov. Ground state of spin-1/2 neutral particle with an anomalous magnetic in a two-dimensional electrostatic field. J. Phys. $A:Math$. Gen., $25:\mathrm{L}619-\mathrm{L}621$,1992. [2] Y. Aharonov and A. Casher. Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field. Phys. Rev. $A,$ $19(6):2461-$ 2462, June [3] Ichiro Shigekawa. Spectral properties of Schr\"odinger operators with magnetic fields for a spin 1/2 particle. J. Funct. Anal., 101: , 1991.

12 51 [4] W. J. Pauli. Relativistic field theories of elementary particles. Rev. Mod. Phys., 13: , [5] Paul R. Chernoff. Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal., 12: , [6] C. R. Hagen. Exact equivalence of spin-1/2 Aharonov-Bohm and Aharonov-Casher effects. Phys. Rev. Letters, $64(20):2347^{-}2349$,1990. [7] Y. Aharonov and A. Casher. Topological quantum effects for neutral particles. Phys. Rev. Letters, $53(4):319^{-321}$,1984. [8] W. Hunziker. On the nonrelativistic limit of the Dirac theory. Comm. Math. Phys., 40: , [9] P. A. Deift. Applications of a commutation formula. Duke Math. J., $45(2):267^{-}310$,1978. [10] Osamu Ogurisu. Ground state of a spin 1/2 charged particle in an even-dimensional magnetic field. Lett. Math. Phys., 29: , December 1993.

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(Osamu Ogurisu) V. V. Semenov [1] :2 $\mu$ 1/2 ; $N-1$ $N$ $\mu$ $Q$ $ \mu Q $ ( $2(N-1)$ Corollary $3.5_{\text{ }}$ Remark 3

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