Criteria for Discrete Spectrum of 1D Operators

For discrete spectrum of 1D second-order differential/difference operators (with or without potential (killing), with the maximal/minimal domain), a pair of unified dual criteria are presented in terms of two explicit measures and the harmonic function of the operators. Interestingly, these criteria can be read out from the ones for the exponential convergence of four types of stability studied earlier, simply replacing the 'finite supremum' by 'vanishing at infinity'. Except a dual technique, the main tool used here is a transform in terms of the harmonic function, to which two new practical algorithms are introduced in the discrete context and two successive approximation schemes are reviewed in the continuous context. All of them are illustrated by examples. The main body of the paper is devoted to the hard part of the story, the easier part but powerful one is delayed to the end of the paper.