Strong-electric-field eigenvalue asymptotics for the Iwatsuka model Available to Purchase

We consider the two-dimensional Schrödinger operator, $Hg(b)=−∂2∕∂x2+[(1∕−1)(∂∕∂y)−b(x)]2−gV(x,y)$⁠, where $V$ is a non-negative scalar potential decaying at infinity like $(1+∣x∣+∣y∣)−m$⁠, and $(0,b(x))$ is a magnetic vector potential. Here, $b$ is of the form $b(x)=∫0xB(t)dt$ and the magnetic field $B$ is assumed to be positive, bounded, and monotonically increasing on $R$ (the Iwatsuka model). Following the argument as in Refs. 15, 16, and 17 [Raikov, G. D., Lett. Math. Phys., 21, 41–49 (1991); Raikov, G. D, Commun. Math. Phys., 155, 415–428 (1993); Raikov, G. D. Asymptotic Anal., 16, 87–89 (1998)], we obtain the asymptotics of the number of discrete spectra of $Hg(b)$ crossing a real number $λ$ in the gap of the essential spectrum as the coupling constant $g$ tends to $±∞$⁠, respectively.