Klein's paradox and the relativistic point interaction
The author treats the one-dimensional Dirac operator \[ H_0(m)=- i(d/dx)\otimes \sigma_ 1+m\otimes \sigma_3\] with mass \(m\) defined in the two compact spinor Hilbert space \(H=L^2(\mathbb R)\otimes C^2\), where \(\sigma_1\) and \(\sigma_3\) are Pauli matrices. Let \[ H_{\alpha}(m)=H_0(m)+\alpha \delta(x),\quad \tilde H_{\beta}(m)=H_0(m)+\beta \delta(x)\otimes \sigma_3,\] \[ H_{\varepsilon}(m)=H_0(m)+(1/\varepsilon)V(x/\varepsilon),\quad \tilde H_{\varepsilon}(m)=H_0(m)+(1/\varepsilon)V(x/\varepsilon)\otimes \sigma_3.\] When the formula \(\int_{\mathbb R}\delta(x)\cdot f(x)\,dx=\{f(0_+)+f(0_- )\}/2\) and the integration by parts are used, the equations \(H_{\alpha}(m)f(x)=Ef(x),\) and \(\tilde H_{\beta}(m)f(x)=Ef(x)\) yield the boundary condition defining \(H_{\alpha}(m)\) and \(\tilde H_{\beta}(m)\). However two relations \(\lim_{\varepsilon \to 0}H_{\varepsilon}(m)=H_{\alpha}(m),\) and \(\lim_{\varepsilon \to 0}\tilde H_{\varepsilon}(m)=\tilde H_{\beta}(m)\) for \(\alpha =\beta =\int V(x)dx\) do not hold. He compares \((H_{\varepsilon}(m)-E)^{-1}\) and \((H_{\alpha}(m)-E)^{-1},\) and shows that the operators \(H_{\varepsilon}(m)\) and \(\tilde H_{\varepsilon}(m)\) converge to the point interaction Hamiltonians \(H_{\alpha}(m)\) and \(\tilde H_{\beta}(m)\), with the coupling constants \(\alpha\), \(\beta\) being nonlinear functionals of the \(V\). That is, let \(V\in C^{\infty}_0(\mathbb R)\), \(u(x)=| V(x)|^{1/2},\) \(v(x)=| V(x)|^{1/2} \operatorname{sgn}V(x)\), \(K\) be an integral operator with a kernel \(K(x,y)=(i/2)u(x)\operatorname{sgn}(x-y)v(y),\) and \((\cdot; \cdot)_{L^2(\mathbb R)}\) denote the scalar product in \(L^2(\mathbb R)\). His results are \[ \text{NR } \lim_{\varepsilon \to 0}H_{\varepsilon}(m)=H_{\alpha}(m)\quad \text{with } \alpha = (v;(1-K^ 2)^{-1} u)_{L^2(\mathbb R)},\] and \[ \text{NR } \lim_{\varepsilon \to 0}\tilde H_{\varepsilon}(m)=\tilde H_{\beta}(m)\quad\text{with } \beta =(v;(1+K^2)^{-1} u)_{L^2(\mathbb R)}. \] Here NR lim means the one in the norm-resolvent topology.
scientific article; zbMATH DE number 4132930