Maximally resonant potentials subject to p-norm constraints (Q582469)

\textit{E. M. Harrell} II and the author [Trans. Am. Math. Soc. 293, 723-736 (1986; Zbl 0591.34026)] studied the following problem: to what degree sharp resonances are due to tunnelling? For this they considered the differential equation \[ -d^ 2\psi /dr^ 2+V\psi =K^ 2\psi,\quad K^ 2=E-i\epsilon \] on \([0,L]\supset \sup p V,\) V is presumed bounded, and the wave function \(\psi\) satisfies the Dirichlet boundary condition at 0 and the outgoing condition at L: \(\psi\) (0)\(=0\), \((\psi '/\psi)(L)=iK\). This paper extends the obtained results to the class of potentials \[ s_ p(\Omega)=\{V:\quad V\geq 0,\quad \sup p V\subset \Omega,\quad \| V\|_ p\leq M\} \] for \(p\geq 2\) if \(n=2\) or 3 and \(p>1\) if \(n=1\).