Eigenvalue asymptotics for the Schrödinger operator with steplike magnetic field and slowly decreasing electric potential (Q1425473)

The author considers the operator \[ H_V=-\partial ^2 /\partial x_1 ^2 +((1/\imath)\partial /\partial x_1 -b(x_1))^2 +V(x_1,x_2) \equiv H_0+V(x_1,x_2) \] in \(L^2(\mathbb{R}^2)\), where \(b(x_1)=\int _0 ^{x_1} B(t)\,dt\) and \(B(t)\) has limits \(B_{\pm}>0, \; B_-<B_+\), as \(t\rightarrow \pm \infty\). Moreover, \(\partial B(t)/\partial t\) and \(B(t)-B_{\pm}\) have a power decay as \(t\rightarrow \pm \infty\). Then \(H_0\) is essentially self-adjoint and its spectrum has a band structure. Assuming that \(V\) and both partial derivatives \(\partial V/\partial x_i, i=1,2\), have a power decay with an exponent less than \(1\), the author studies the asymptotic distribution of the eigenvalues of \(H_V\) near the edges of the spectral gaps. The classical Weyl-type asymptotic formula holds true under some additional assumptions on \(V\).