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6. Spectrum of \overline{\partial}-Neumann Laplacian

    1. Problem 6.1.

      [Fu] Compute the spectrum of the \overline{\partial}-Neumann Laplacian on the Hartogs triangle and on the Diederich-Fornaess worm domain. In particular, determine whether or not the corresponding spectra are discrete.
        • Problem 6.2.

          [Fu] Relate the infimum of the essential spectrum of the \overline{\partial}-Neumann Laplacian on the worm domain to the winding of the domain.
            • Problem 6.3.

              [Fu] Is the spectrum of the \overline{\partial}-Neumann Laplacian on a bounded smooth pseudoconvex domain in \mathbb{C}^n always discrete? Is the infimum of the spectrum always a point eigenvalue (ground state energy)?
                • Problem 6.4.

                  [Haslinger] Let \phi:\mathbb{C}^n\to\mathbb{R} be a C^2 plurisubharmonic function. Determine the relation between the eigenvalues of the complex Hesssian of \phi and the compactness of the \overline{\partial}-Neumann operator on L^2_{(0,q)}(\mathbb{C}^n,e^{-\phi}). See [MR3029187] for n=1. Determine the relation between the spectrum of the complex Laplacian \Box_{\phi} and the weight function \phi. In particular, understand the bottom (infimum) of the essential spectrum.

                      Cite this as: AimPL: Cauchy-Riemann equations in several variables, available at http://aimpl.org/crscv.