On the negative spectrum of an elliptic operator (Q5893950)

From the text: Consider the operator \(L\in (L_2(\mathbb{R}^n), L_2(\mathbb{R}^n))\) generated by the differential expression \[ {\mathcal L}(u)\equiv (-1)^m(\Delta^m u+V(x)u), \] where \(\Delta\) is the Laplacian, \(x= (x_1, x_2,\dots, x_n)\), \(n\geq 1\), \(m\geq 1\), and \(V(x)\leq 0\) is a bounded function. It is well known that, if \(m= 1\), \(n< 3\), \(V(x)\leq 0\) and \(\text{mes}\{x\mid V(x)< 0\}> 0\), then the negative spectrum of \(L\) has at least one point. However, this assertion is not true if \(n\geq 3\). The main result of this paper is a similar assertion derived for \({\mathcal L}\) if \(m> 1\) and for a wider class of operators.