Estimates on Embedded Eigenvalues of Schr𝐨̈dinger Operators with Dilation Analytic Potentials

- Prospects for Theory of Fuzzy Schrödinger Operators -

Abstract

We report the results of research on spectral theory for 𝑆𝑐ℎ𝑟𝑜̈𝑑𝑖𝑛𝑔𝑒𝑟 operators 𝐻= −Δ+ 𝑉 with dilation analytic real/complex potentials 𝑉 rrecisely we first obtain a Lieb-Thirring type estimate on resonance eigenvalues of 𝐻 with complex potentials 𝑉 and we next derive an estimate on embedded eigenvalues of 𝐻 with special dilation analytic real potentials 𝑉 However it is generally so difficult to estimate embedded eigenvalues if 𝑉 are complex To realize that estimate we would like to propose introducing theory for `fuzzy 𝑆𝑐ℎ𝑟𝑜̈𝑑𝑖𝑛𝑔𝑒𝑟 operators’