The fractional logarithmic Schrödinger operator: properties and functional spaces. (English) Zbl 07914778
Summary: In this note, we deal with the fractional logarithmic Schrödinger operator \((I+(-\Delta)^s)^{\log}\) and the corresponding energy spaces for variational study. The fractional (relativistic) logarithmic Schrödinger operator is the pseudo-differential operator with logarithmic Fourier symbol, \(\log (1+|\xi|^{2s})\), \(s>0\). We first establish the integral representation corresponding to the operator and provide an asymptotics property of the related kernel. We introduce the functional analytic theory allowing to study the operator from a PDE point of view and the associated Dirichlet problems in an open set of \(\mathbb{R}^N\). We also establish some variational inequalities, provide the fundamental solution and the asymptotics of the corresponding Green function at zero and at infinity.
MSC:
| 45K05 | Integro-partial differential equations |
| 45P05 | Integral operators |
| 45C05 | Eigenvalue problems for integral equations |
| 45A05 | Linear integral equations |
| 35R11 | Fractional partial differential equations |
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
| 26A33 | Fractional derivatives and integrals |
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
logarithmic symbol; pseudo-differential operator; Schrödinger operator; Green function; fundamental solution; regularity; maximum principleSoftware:
DLMFReferences:
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