## Eigenvalues and Eigenfunctions of a Class of Potential Operators in the Plane |

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analytic annulus associated non-negative normalized Borel measures boundary component bounded sets characteristic convergence compact operator compact set compact subsets conclude conformal mapping connected admissible domains constant continuous extension continuous with respect D X D defined definition denote dielectric Green's function differential equation disc of radius Doctor of Philosophy domain of given dominated convergence theorem double-layer potentials dz dz eigen eigenfunctions associated eigenvalue and associated extremal problem follows given transfinite diameter Green's identity applied harmonic Hence Hilbert space implies interior minimum kernel Lebesgue measure log r2 log|z Moreover neighborhood non-negative normalized eigenfunction obtain operators G pointwise and boundedly satisfies Schwarz inequality sequence simply connected slit plane solution spectrum subharmonic sufficiently small superharmonic Theorem 2.4 uniform convergence uniformly unique negative eigenvalue vanish identically vanishes at infinity variational vergence weak derivative zero