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Information Center for Mathematical Science

### 논문검색

Information Center for Mathematical Science

#### 논문검색

Journal of Differential Equations
( Vol. 149 NO.2 / (1998))
On Moments of Negative Eigenvalues for the Pauli Operator

Pages. 292-327
Abstract This paper concerns the three-dimensional Pauli operator \$\$Bbb P = (sigma cdot({f p}- {f A}(x)))^2 + V(x)\$\$ with a nonhomogeneous megnetic field \${f B} = { m curl} f A\$. The following Lieb-Thirring type inequality for the moment of negative eigenvalues is established as \$\$sum_{lambda_j<0}|lambda_j| leqslant C_1 int_{Bbb R^3} |V(x)|^{5/2}_{-} dx + C_2int_{Bbb R^3} [b_p(x)]^{3/2} |V(x)|_{-} dx,\$\$ where \$q>3/2\$ and \$b_p(x)\$ is the \$L^p\$ average of \$|f B|\$ over certain cube centered at \$x\$ with a side scaling like \$|{f B}|^{-1/2}\$. We also show that, if \$f B\$ has a constant direction, \$\$sum_{lambda_j<0}|lambda_j|^gamma leqslant C_{1,gamma} int_{Bbb R^3} |V(x)|^{gamma+3/2}_{-} dx + C_{2,gamma}int_{Bbb R^3} b_p(x)|V(x)|^{gamma+1/2}_{-} dx,\$\$ where \$gamma > 1/2\$ and \$p > 1\$. 1. Introduction 2. Some preliminaries 3. A partition of unity 4. The proof of theorem 1.1 5. Fields with constant directions