Polynomial boundedness of eigensolutions and the spectrum of Schrödinger operators
The spectrum of a Schrödinger operator \(T=-(\nabla-ib)^ 2+V\) is characterized as the closure of the set of those real numbers \(\lambda\) for which a polynomially bounded eigensolution exists, i.e. an \(u\in L_{2,loc}\backslash\{0\}\) with \(Tu=\lambda u\) in the distributional sense. To arrive at this goal, \(T\) has to be defined as an essentially self- adjoint operator on \(C^ \infty_ 0(\mathbb{R}^ n)\), which is achieved under the weakest possible assumptions on the magnetic term \(b\) and the positive part \(V_ +\) of the potential, whereas \(V_ -\) is assumed to be locally of Kato-type and no worse than \(O(| x|^ 2)\) at infinity. The existence of a polynomially bounded eigensolution \(u\) then allows to construct a singular sequence by the observations that \(u\) has locally square integrable first derivatives and that the nonincreasing part of \(V_ -\) is relatively form bounded with respect to \(-(\nabla-ib)^ 2+V_ +\); the rest of \(V_ -\) has to behave like \(o(| x|^ 2)\), a natural condition as indicated by an example. On the other hand, a thorough analysis of resolvents shows that expansion in generalized eigenfunctions applies, with sufficient information on the regularity of the solutions to employ an appropriate version of Kato's inequality, so that polynomial boundedness follows from a mean value inequality.
scientific article; zbMATH DE number 86199
scientific article