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Spectral boundary-value problems for the Dirac system with a singular potential. (English. Russian original) Zbl 1072.35158
St. Petersbg. Math. J. 16, No. 1, 25-57 (2004); translation from Algebra Anal. 16, No. 1, 33-69 (2004).
The boundary-value problem with two real parameters $$\lambda$$ and $$\mu$$ for the Dirac equation $\begin{matrix} (\mathcal{D}+V)w(x)-\lambda w(x)=f(x),\quad x\in\Omega\subset R^3,\quad \mathcal{D}=\sum\limits_1^3\alpha_jD_j+\beta\\ Bu:=i\sigma(\nu(x))v^+(x)-\mu u^+(x)-g(x),\qquad x\in\Gamma=\partial\Omega \end{matrix}$ is considered. Here $$D_j=-i\partial_j,$$ $$w=\left(\begin{matrix} u\\ v\end{matrix}\right),$$ $$\alpha_j=\left(\begin{matrix} 0 & \sigma_j\\ \sigma_j & 0\end{matrix}\right)$$ are Hermitian $$(4\times 4)$$-matrices, $$\sigma_j$$ are the Pauli matrices, $$\beta=\alpha_4$$ is the diagonal $$(4\times 4)$$-matrix with the main diagonal $$(1, 1, -1, -1)$$, $$u^+(x)=u(x)|_\Gamma$$ , $$V(x)$$ is a Hermitian potential, which may have a Coulomb-like singularity. Let the values of the parameters $$\lambda$$ and $$\mu$$ be such that the problem has a unique solution and let $$f=0$$. The problem consists in the finding $$u^+$$ in terms of $$g$$. The $$R$$-matrix is the operator mapping $$g$$ to $$u^+$$. Two approaches to the $$R$$-matrix construction are stated at $$f=0$$, $$g=0$$:
(I) The problem with $$\lambda$$ as a spectral parameter and fixed $$\mu$$.
(II) The problem with $$\mu$$ as a spectral parameter and fixed $$\lambda$$.
The authors investigate spectral properties of both problems and use them for the $$R$$-matrix construction.
##### MSC:
 35Q40 PDEs in connection with quantum mechanics 35P05 General topics in linear spectral theory for PDEs 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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