Solvability in the sense of sequences for some non-Fredholm operators with the logarithmic Laplacian

Efendiev M., Vougalter V.

Monatshefte fur Mathematik, vol.202, no.4, pp.751-771, 2023 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 202 Issue: 4
  • Publication Date: 2023
  • Doi Number: 10.1007/s00605-023-01834-1
  • Journal Name: Monatshefte fur Mathematik
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.751-771
  • Keywords: Solvability conditions, Non-Fredholm operators, Logarithmic Laplacian, INTEGRODIFFERENTIAL EQUATIONS, PROPERNESS PROPERTIES, ELLIPTIC-OPERATORS, HOLDER THEORY, DIRICHLET, SYSTEMS
  • Marmara University Affiliated: No

Abstract

We establish the solvability of certain linear nonhomogeneous equations and demonstrate that under reasonable technical conditions the convergence in L2(Rd) of their right sides implies the existence and the convergence in L2(Rd) of the solutions. In the first part of the work the equation involves the logarithmic Laplacian. In the second part we generalize the results derived by incorporating a shallow, short-range scalar potential into the problem. The argument relies on the methods of the spectral and scattering theory for the non-Fredholm Schrödinger type operators. As distinct from the preceding articles on the subject, for the operators involved in the equations the essential spectra fill the whole real line.