On the essential spectrum of Schrödinger operators with spherically symmetric potentials
This paper discusses how the essential spectrum, denoted by \(\sigma_ e\), of a higher-dimensional Schrödinger operator \(-\Delta +V(x)\) in \(L_ 2({\mathbb{R}}^{\nu})\), \(\nu\geq 2\), is connected with that of a one- dimensional Schrödinger operator \(-d^ 2/dr^ 2+q(r)\) in \(L_ 2({\mathbb{R}})\), when V behaves like q on a strip in rectangular coordinates or in the radial direction in spherical coordinates. Theorem 1: If V is a real-valued function in \(L_{2,loc}({\mathbb{R}}^{\nu})\) and q in \(L_{\infty,loc}({\mathbb{R}})\) such that \(-\Delta +V\) is essentially selfadjoint on \(C_ 0^{\infty}({\mathbb{R}}^{\nu})\) and \(-d^ 2/dr^ 2+q\) on \(C_ 0^{\infty}({\mathbb{R}})\) with closures T and t, respectively, and if V(x), \(x=(x_ 1,..,x_{\nu})\), behaves like \(q(x_ 1)\) for large \(| x_ 1|\) but does not change too much in \((x_ 2,...,x_{\nu})\), then \(\sigma_ e(T)\) includes the interval \([\lambda_ 0,\infty)\) whenever \(\lambda_ 0\in \sigma_ e(t).\) Theorem 2: If \(T=-\Delta +q(| \cdot |)\) and \(t=-d^ 2/dr^ 2+q\) are selfadjoint operators defined in the sense of quadratic forms, where q is symmetric, i.e. \(q(-r)=q(r)\), and \(q=q_+-q_-\) with \(q_{\pm}\) nonnegative, \(q_+\in L_{1,loc}({\mathbb{R}})\) and \(q_-\in L_{1,loc,unif}({\mathbb{R}})\), then \(\sigma_ e(T)=[\inf \sigma_ e(t),\infty).\) [See also the following review (Zbl 0629.35029).]
scientific article; zbMATH DE number 4023827
scientific article