About This Course

This course is intended for fourth-year students at NHSM (the National Higher School of Mathematics) and aims to introduce them to the basics of the spectral theory of differential operators. This course details chapters for students on unbounded operators, closed operators, symmetric and self-adjoint operators, and the classifications of their spectra and application of the min-max principle, and finally, some aspectes of asymptotics analysis. These chapters will familiarize the student with mathematical formalism in order to develop clear, structured, and rigorous reasoning that can be applied in complex situations.

By the end of this course, students will be able to: Explain why unbounded operators are necessary in differential equations. Explain the difference between symmetric and self-adjoint operators. Explain the role of compact resolvent in obtaining a purely discrete spectrum. Compute the adjoint of simple differential and multiplication operators. Determine whether a given operator is closed, closable, symmetric, or self-adjoint. Construct Weyl sequences to prove that a number belongs to the spectrum. Apply the Friedrichs extension to obtain self-adjoint realizations of semibounded operators. Use the Kato-Rellich theorem to prove self-adjointness of Schrödinger operators. Compare the Dirichlet and Neumann Laplacians on a given domain. Distinguish between discrete and essential spectrum using Weyl criteria. Analyze the stability of the essential spectrum under relatively compact perturbations. Justify the choice of boundary conditions (Dirichlet, Neumann, Robin) for self-adjoint realizations. Evaluate the compactness of resolvents using the Riesz-Kolmogorov criterion. Argue whether a perturbation preserves essential self-adjointness. Formulate a min-max estimate for eigenvalues of a perturbed operator. Propose a domain where the Neumann Laplacian has non-compact resolvent. Develop a Weyl asymptotic argument for Schrödinger operators.