Essential spectra of tensor product Hilbert complexes, and the $\overline\partial$-Neumann problem on product manifolds
Essential spectra of tensor product Hilbert complexes, and the $\overline\partial$-Neumann problem on product manifolds
We investigate tensor products of Hilbert complexes, in particular the (essential) spectrum of their Laplacians. It is shown that the essential spectrum of the Laplacian associated to the tensor product complex is computable in terms of the spectra of the factors. Applications are given for the $\overline\partial$-Neumann problem on the product of two or more Hermitian manifolds, especially regarding (non-) compactness of the associated $\overline\partial$-Neumann operator.
22 pages
essential spectrum, Mathematics - Complex Variables, Several-variable operator theory (spectral, Fredholm, etc.), \(\overline{\partial}\)-Neumann problem, Spectral problems; spectral geometry; scattering theory on manifolds, product manifolds, \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs, Functional Analysis (math.FA), Mathematics - Spectral Theory, Mathematics - Functional Analysis, Primary 58J50, Secondary 35N15, 58J10, 47A13, Mathematics - Analysis of PDEs, elliptic complexes, Differential complexes, FOS: Mathematics, Hilbert complexes, Laplacian, Complex Variables (math.CV), Spectral Theory (math.SP), Analysis of PDEs (math.AP)
essential spectrum, Mathematics - Complex Variables, Several-variable operator theory (spectral, Fredholm, etc.), \(\overline{\partial}\)-Neumann problem, Spectral problems; spectral geometry; scattering theory on manifolds, product manifolds, \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs, Functional Analysis (math.FA), Mathematics - Spectral Theory, Mathematics - Functional Analysis, Primary 58J50, Secondary 35N15, 58J10, 47A13, Mathematics - Analysis of PDEs, elliptic complexes, Differential complexes, FOS: Mathematics, Hilbert complexes, Laplacian, Complex Variables (math.CV), Spectral Theory (math.SP), Analysis of PDEs (math.AP)
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