Asymptotic distribution of negative eigen values for two dimensional Pauli operators with spherically symmetric magnetic fields
Year: 1998
Type: article
Abstract: The magnetic field $b$ is represented as the commutator $b=i [\Pi_{2}, \Pi_{1}]$ and hence $ H\pm$ can be rewritten as (1.2) $H_{\pm}=(\Pi_{1}\pm t\Pi_{2})^{*}(\Pi_{1}\pm i\Pi_{2})$ .This implies that $H_{\pm}\geq 0$ is non-negative.If, in particular, $b(x)>c>0$ is positive, then $H$ -becomes strictly positive, while it is known ([1]) that $H+has$ zero as an eigenvalue with infinite multiplicities.Hence the operator $H+has$ the origin as the bottom of its essential spectrum.We consider the two dimensional Pauli operator$ falls off at infinity, then the operator $H(V)$ has an infinite number of negative discrete eigenvalues accumulating the origin.The aim of the present work is to study the asymptotic distribution near the origin of such negative eigenvalues.This problem has been already discussed by $[8, 10]$ when $b(x)=b$ is a constant magnetic field.We here deal with the case of nonconstant magnetic field $b(x)=b(r),$ $r=|x|$ , with spherical symmetry.We shall formulate the obtained result precisely.Let $\langle x\rangle=(1+|x|^{2})^{1/2}$ .We first make several assumptions on the magnetic field $b(x)$ and the electric potential $V(x)$ .The magnetic field $b(x):R^{2}\rightarrow R$ is assumed to fulfill the fol- lowing three assumptions.is also assumed to satisfy the following three assumptions.There exists $d>0$ such that:Under these assumptions, the operator $H(V)$ formally defined by (1.3) admits a unique self-adjoint realization in the space $L^{2}=L^{2}(R^{2})$ with natural domain $\mathscr{D}=\{u\in L^{2} : H(V)u\in L^{2}\}$ .We denote by the same notation $H(V)$ this self-
Source: Tsukuba Journal of Mathematics
Authors: Akira Iwatsuka, Hideo Tamura
Institutions: Okayama University, Kyoto Institute of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1532898669