Discrete spectrum in the gaps of perturbed pseudorelativistic Hamiltonian

M. Sh. Birman ; A. B. Pushnitskii

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 29, Tome 249 (1997), pp. 102-117

Résumé

Pseudorelativistic Hamiltonian $$ G_{1/2}=\bigl((-i\nabla-\mathbf A)^2+I\bigr)^{1/2}+W, \qquad x\in\mathbb R^d, \quad d\ge 2, $$ is considered under wide conditions on potentials $\mathbf A(\mathbf x)$, $W(x)$. It is assumed that the real point $\lambda$ is regular for $G_{1/2}$. Let $G_{1/2}(\alpha)=G_{1/2}-\alpha V$, where $\alpha>0$, $V(x)\ge 0$, $V\in L_d(\mathbb R^d)$. Denote by $N(\lambda,\alpha)$ the number of eigenvalues of $G_{1/2}(t)$ that cross the point $\lambda$ as $t$ increases from 0 to $\alpha$. The Weyl type asymptotics for $N(\lambda,\alpha)$ as $\alpha\to\infty$ is obtained.