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Point perturbation-invariant solutions of the Schrödinger equation with a magnetic field. (English. Russian original) Zbl 0898.35081
Math. Notes 60, No. 5, 575-580 (1996); translation from Mat. Zametki 60, No. 5, 768-773 (1996).
Let \(\Omega_\Lambda\) be a unit cell of the lattice \(\Lambda\), which is a parallelogram of the form \(\{x\in \mathbb{R}^2: x=t_1\vec a_1+ t_2\vec a_2\), \(0\leq t_1\), \(t_2<1\}\) spanning a pair of basis vectors of the lattice \((\vec a_1, \vec a_2)\). We denote the area of \(\Omega_\Lambda\) by \(S_\Lambda\), the number of elements in \(\Omega_\Lambda \cap \Gamma\) by \(k\), and the number of quanta of the magnetic flux across a unit cell of \(\Lambda\) by \(\eta\), \(\eta= S_\Lambda \xi\).
The aim of this note is to prove that for any \(k\) and for any level \(\varepsilon_l\), the validity of the inequality \(|\eta |>k\) is a sufficient condition for the existence of Landau-Ando states.
MSC:
35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35Q60 PDEs in connection with optics and electromagnetic theory
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