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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 \(H^\pm(B)\psi=0\).

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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