Constructing quantum observables and self-adjoint extensions of symmetric operators. III. self-adjoint boundary conditions
Let \(\widehat f^{(0)}\) be the initial symmetric operator with a given self-adjoint (SA) differential expression \(\breve f\), associated with \(L^2(a,b)\), and \(\widehat f^*= [\widehat f^{(0)}]^+\) be its adjoint with the domain \(D_*\). Any \(\psi_*\in D_*\) is reepresented as \(\psi_*= \psi+\psi_++ \psi_-\); \(\psi\in D_f\), \(\psi_+\in D_+\), \(\psi_-\in D_-\). The quadratic asymmetry form \[ \Delta_*(\psi_*)= 2ik(\|\psi_+\|^2- \|\psi_-\|^2) \] is nontrivial only on the direct sum \(D_++ D_-\) of the deficient subspaces. The authors derive \[ \Delta_*(\psi_*)= 2ik[\Psi^+_{\tau(+-)}\Psi_{\tau(+-)}- \Psi^+_{\tau(-+)}\Psi_{\tau(-+)}], \] where \(k =\tau^{-n+1}/4\) and \[ \Psi_{\tau(+-)}= ^t(\Psi_{\tau+}(b),\Psi_{\tau-}(a)),\;\Psi_{\tau(-+)}= ^t(\Psi_{\tau-}(b), \Psi_{\tau+}(a)). \] The SA boundary conditions defining the SA operator \(\widehat f_U\) are given by \(\Psi_{\tau(-+)}= U\psi_{\tau(+-)}\), where \(U: n\times n\) unitary matrix. As an illustration, they consider the expression \(\breve H= -d^2/dx^2+ V(x)\) on \([0,L]\) with integrable \(V(x)\) on \([0,L]\). The SA boundary conditions specifying the possible SA Hamiltonians are given by \[ ^t(\psi(L)- i\tau\psi'(L), \psi(0)+ i\tau\psi'(0))= U[^t(\psi(L)+ i\tau\psi'(L), \psi(0)- i\tau\psi'(0))],\;U\in U(2). \] They also study the expression \(\breve H= -d^2/dx^2+ V(x)\) on \([0,\infty)\) with deficiency indices \((1,1)\), under the condition \[ [\psi_*,\psi_*](\infty)= \lim_{x\to\infty}(\overline{\psi_*'(x)}\psi_*(x)- \overline{\psi_*(x)}\psi_*'(x))= 0. \] (I) \(V(x)\in L^2(0,\infty)\Rightarrow |\psi_*'|< C_2(\psi_*)+ C^{1/2}_1(\psi_*)(x- a)^{1/2}+ |\psi_*'(a)|\), \(x> a> 0\), \(\forall\psi_*\in D_*\). (II) \(V(x)> -Kx^2\), \(K> 0\), for \(x> N> 0\Rightarrow|\psi_*|^2> C_7(\psi_*) x^3/3- C_8(\psi_*)\).
scientific article; zbMATH DE number 5533444
scientific article