On the essential spectrum of differential operators over geometrically finite orbifolds (Q6562502)

Let \(O=\Gamma\setminus X\) be an \(m\)-dimensional complete connected Riemannian orbifold such that \(\Gamma\) is a countable group acting properly discontinuously and isometrically on \(X\), a complete simply connected Riemannian manifold. Let \(E\) be a Riemannian vector bundle over \(O\) and \(A\) be an elliptic differential operator on \(E\) and symmetric on \(C_c^\infty(O,E)\), the space of smooth sections of \(E\) over \(O\) and with compact support. We denote by \(\mathrm{spec}_{\text{ess}}(A,O)\) and \(\lambda_{\mathrm{ess}}(A,O)\) the essential spectrum of the closure of \(A\) and its bottom, respectively. The authors state three results (Theorems B, C and D). The first one states that if \(A\) is symmetric and bounded from below on \(E\) over \(X\), then \(A\) remains bounded from below on \(E\) over \(O\) and \(A\) is essentially self-adjoint on \(E\) over \(X\) or over \(O\) and \(\lambda_{\mathrm{ess}}(A,X)\le\lambda_{\mathrm{ess}}(A,O)\) and the equality is reached when \(A\) satisfies meaningful conditions. The second result states that\N\[\N\mathrm{spec}_{\mathrm{ess}}(A,O)\subset \mathrm{spec}_{\mathrm{ess}}(A,X), \tag{1}\N\]\Nwhenever \(\sigma_A\), the principal symbol of \(A\), is bounded. On the third one, the authors consider \(A=\Delta+B\) such that \(\displaystyle B=\sum_i\sigma_B(X_i)\nabla_{X_i}+V\) where \((X_i)_i\) is a local orthonormal frame of \(X\) or \(O\) and \(V\) is a field of endomorphisms of \(E\). Then, they state that \((1)\) is true whenever \(\sigma_B\) is bounded, \(V\) is lower bounded, and \(\nabla R\) is bounded where \(\nabla\) and \(R\) stand for the curvature and the Levi-Civita connection of \(O\), respectively.