an:02031086
Zbl 1040.35017
Diagana, Tocka
Some remarks about the Schr??dinger operator with a particular singular complex potential
FR
Bull. Belg. Math. Soc. - Simon Stevin 9, No. 2, 293-298 (2002).
00086638
2002
j
35J10 47B44 81Q05
Schr??dinger operator; complex potential; singular potential
In this paper Schr??dinger operators \(S=-\Delta+V\) in \({\mathbb R}^n\) are considered for which the sum in fact cannot be defined since \(D(\Delta) \cap D(V)=\{0\}\). It is assumed that \(V\in L^1({\mathbb R}^N)\), \(V\notin L^2_{\text{loc}}({\mathbb R}^N)\) and \(\Re (V) > 0\) (if \(N<4\) this implies that \(S\) is defined only for \(\{0\}\)). The author studies extensions of \(S\) by using the form sum of \(-\Delta\) and \(V\). It is shown that for sufficiently large \(\lambda\) the equation \((-\Delta \oplus V)u + \lambda u = v\) with given \(v\in L^2({\mathbb R}^N)\) has a unique solution in \(H^1({\mathbb R}^N)\). For this a fixed point argument is used.
Christiane Tretter (Bremen)