Published online by Cambridge University Press: 05 March 2013
The development of spectral theory is strongly related to quantum mechanics, and the main operators that immediately appear in the theory are the operators of multiplication by x (in, say, L2(ℝ)), the operator of differentiation d/dx, and the harmonic oscillator −d2/dx2 + x2. These operators are unbounded and, in fact, not defined for any element of L2(ℝ). Of course, we can start by simply restricting the operator by introducing a smaller domain in the Hilbert space L2(ℝ) of definition, but what is the right notion of continuity for the operator? How do we choose the “maximal” domain of definition? This is what we shall start to explain in this chapter.
We consider a Hilbert space ℋ. We assume that the reader has some basic knowledge of Hilbertian theory. The scalar product will be denoted by 〈u, v〉ℋ or, more simply, by 〈u, v〉 when no confusion is possible. We adopt the convention that the scalar product is antilinear with respect to the second argument.