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Essential spectrum, quasi-orbits and compactifications: application to the Heisenberg group. (English) Zbl 1449.47028

Let \(H\) be the Heisenberg group. Using the natural bijection between \(H\) and \(\mathbb{R}^{3}\), there is introduced a compactification \(\bar{H}\) of \(H\) induced by the spherical compactification of \(\mathbb{R}^{3}\). Let \(T=-\Delta + V\) be the Schrödinger-type operator on \(L^{2}(H)\), where \(V\) is a continuous function on \(\bar{H}\). The main result of the paper gives a representation of the essential spectrum of \(T\) as the union of spectra of some simpler operators. There are also obtained some similar results.

MSC:

47A53 (Semi-) Fredholm operators; index theories
47L65 Crossed product algebras (analytic crossed products)
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
35J10 Schrödinger operator, Schrödinger equation