Spectral theory of one-dimensional Dirac operator in singular case for piecewise homogeneous semiaxis (Q2768810)

The author obtains a bounded in \(D^{+}=(0,\infty)\times I_{n}^{+},\;I_{n}^{+}=\bigcup_{k=1}^{n+1}(l_{k-1},l_{k}),\;l_0\geq 0,\;l_{n+1}=\infty\) solution of the system of equations with the Dirac operator NEWLINE\[NEWLINEi{\partial u^{j}\over\partial t}+\left(\begin{matrix} 0&1\\ -1&0 \end{matrix}\right){\partial u^{j}\over\partial x}+\left(\begin{matrix} p^{j}(x)&q^{j}(x)\\ q^{j}(x)&-p^{j}(x)\end{matrix}\right)u^{j}(t,x)=0,\;(t,x)\in D^{+},\;j=1,\ldots,n+1,NEWLINE\]NEWLINE with the initial condition \(u^{j}(t,x)|_{t=0}=f^{j}(x),\;x\in I_{n}^{+},\;j=1,\ldots,n+1\), the boundary conditions \((\alpha_{11}^0,\beta_{11}^0)u^1|_{x=l_0}=0,\) and the conjunction conditions \(\left(\begin{smallmatrix} \alpha_{11}^{k}&\beta_{11}^{k}\\ \alpha_{21}^{k}&\beta_{21}^{k}\end{smallmatrix}\right)u^{k}|_{x=l_{k}}=\left(\begin{smallmatrix} \alpha_{12}^{k}&\beta_{12}^{k}\\ \alpha_{22}^{k}&\beta_{22}^{k}\end{smallmatrix}\right)u^{k+1}|_{x=l_{k}}\), \(k=1,\ldots,n\). Here \(u^{j}=(u^{j}_1,u^{j}_2)\) is an unknown vector-function; \(f^{j}=(f^{j}_1,f^{j}_2)\) is a given vector-function; \(\alpha_{mi}^{k}, \beta_{mi}^{k}\) are known numbers. Theorems on expansion by eigenfunctions of the Dirac operator and the fundamental identity of the integral transform of the Dirac operator are proved.