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Zero modes in a system of Aharonov-Bohm solenoids on the Lobachevsky plane. (English) Zbl 1088.81043
Let $$M$$ be a two-dimensional manifold with a conformal metric $$ds^2=dzd\overline z/\lambda^2(z,\overline z)$$, $$\lambda^2(z,\overline z)>0$$. The area 2-form: $$d \sigma=dx\wedge dy/\lambda^2(z,\overline z)$$, the Lobachevsky planes $$M$$: $${\mathfrak D}=\{z\in\mathbb{C};|z|<1\}$$ with $$\lambda=(1-z\overline z)/2$$, and $$\mathbb{C}^+= \{z\in\mathbb{C};\text{Im}\,z> 0\}$$ with $$\lambda=(z-\overline z)/(2i)$$, are used. Let $$P_x=-i\partial_x -a_x$$, $$P_y=-i\partial_y-a_y$$, $$T_\pm=P_x\pm iP_y$$, $$\lambda^2 T_+ T_-=H^-$$, and $$\lambda^2T_-T_+=H^+$$. The authors define the Pauli operator $$P$$ as $$P\psi=P\cdot^t(\psi_+,\psi_-)=^t(H^+\psi_+,H^-\psi_-)$$. Magnetic field on $$M$$: an exact 2-form $$b=Bd\sigma=\lambda^2 (\partial_xa_y-\partial_ya_x)d \sigma$$. Suppose that $$B(z,\overline z)= (\pi/2)\sum^n_{k=1}\theta_k(1-|a_k|^2)^2 \delta(z-a_k)$$: a finite family of Aharonov-Bohm solenoids with non-zero fluxes. Then $$H^\pm$$ has no zero modes.
Next suppose that $$W(z)$$ is an automorphic form on $${\mathfrak D}$$ (Lobachevsky plane) of weight $$2k$$, $$k\geq 1$$, with respect to a discrete co-compact subgroup $$G$$ of SU(1,1). That is, $$W(z)$$ is a meromorphic function on $${\mathfrak D}$$ satisfying $$W(Az)= A'(z)^{-k}W(z)$$ for $$\forall A\in G$$. Let $$B=\theta\lambda^{-2}\log (|W|)$$. If $$k\theta\geq 1$$, $$H^+(B)$$ has zero modes. If $$0<k\theta<k-1$$, $$H^=(B)$$ has zero modes. That is, there exist $$L^2$$-solutions $$\psi \neq 0$$ to $$H^\pm(B)\psi=0$$.

##### MSC:
 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 58A10 Differential forms in global analysis
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