Eigenvalue Asymptotics for a Schr\\"odinger Operator with Non-Constant\n Magnetic Field Along One Direction

We consider the discrete spectrum of the two-dimensional Hamiltonian\n$H=H_0+V$, where $H_0$ is a Schr\\"odinger operator with a non-constant magnetic\nfield $B$ that depends only on one of the spatial variables, and $V$ is an\nelectric potential that decays at infinity. We study the accumulation rate of\nthe eigenvalues of H in the gaps of its essential spectrum. First, under some\ngeneral conditions on $B$ and $V$, we introduce effective Hamiltonians that\ngovern the main asymptotic term of the eigenvalue counting function. Further,\nwe use the effective Hamiltonians to find the asymptotic behavior of the\neigenvalues in the case where the potential V is a power-like decaying function\nand in the case where it is a compactly supported function, showing a\nsemiclassical behavior of the eigenvalues in the first case and a\nnon-semiclassical behavior in the second one. We also provide a criterion for\nthe finiteness of the number of eigenvalues in the gaps of the essential\nspectrum of $H$\n