Semibounded Forms and Self-adjoint Operators

Abstract

The main theme of Chap. 10 is the interplay between lower semibounded closed forms and lower semibounded self-adjoint operators. We begin with definitions and characterizations of closed and closable lower semibounded forms. Then the form associated with a self-adjoint operator is studied, and the first form representation theorem is proved. It establishes a one-to-one correspondence between lower semibounded self-adjoint operators and densely defined lower semibounded closed forms. Abstract boundary-value problems and abstract variational problems based on forms are formulated and solved. Forms are used to define an order relation between self-adjoint operators. The Friedrichs extension of a densely defined lower semibounded symmetric operator is the largest self-adjoint extension of this operator. We derive it from the form representation theorem. Numerous examples of forms and associated operators (differential operators  $-\frac{d^{2}}{dx^{2}}$ on intervals with various boundary conditions, Sturm–Liouville operators, etc.) are examined in great detail. Next, the Dirichlet Laplacian and the Neumann Laplacian on an open subset Ω of ℝ^d are studied as self-adjoint operators on the Hilbert space L ²(Ω). For these operators, the abstract boundary-value problems and variational problems of forms become the weak Dirichlet problem and weak Neumann problem, respectively. The final section of this chapter is devoted to the perturbation of forms and to the form sum of self-adjoint operators. The KLMN theorem is proved, and applications to Schrödinger operators are discussed.