In the above formula 1E denotes the indicator function of E. These measures νξ are called the spectral measures of T. If denotes the mapping z → z on C, then: This defines the functional calculus for bounded functions applied to possibly unbounded self-adjoint operators. Using the bounded functional calculus, one can prove part of the Stone's theorem on one-parameter unitary groups: As an application, we consider the Schrödinger equation, or equivalently, the dynamics of a quantum mechanical system. In non-relativistic quantum mechanics, the Hamiltonian operator H models the total energy observable of a quantum mechanical system S. The unitary group generated by iH corresponds to the time evolution of S. We can also use the Borel functional calculus to abstractly solve some linear initial value problems such as the heat equation, or Maxwell's equations. Existence of a functional calculus The existence of a mapping with the properties of a functional calculus requires proof. For the case of a bounded self-adjoint operator T, the existence of a Borel functional calculus can be shown in an elementary way as follows: First pass from polynomial to continuous functional calculus by using the Stone–Weierstrass theorem. The crucial fact here is that, for a bounded self adjoint operator T and a polynomial p, Consequently, the mapping is an isometry and a densely defined homomorphism on the ring of polynomial functions. Extending by continuity defines f(T) for a continuous function f on the spectrum of T. The Riesz-Markov theorem then allows us to pass from integration on continuous functions to spectral measures, and this is the Borel functional calculus. Alternatively, the continuous calculus can be obtained via the Gelfand transform, in the context of commutative Banach algebras. Extending to measurable functions is achieved by applying Riesz-Markov, as above. In this formulation, T can be a normal operator. Given an operator T, the range of the continuous functional calculus h → h(T) is the (abelian) C*-algebra C(T) generated by T. The Borel functional calculus has a larger range, that is the closure of C(T) in the weak operator topology, a (still abelian) von Neumann algebra. The general functional calculus We can also define the functional calculus for not necessarily bounded Borel functions h; the result is an operator which in general fails to be bounded. Using the multiplication by a function f model of a self-adjoint operator given by the spectral theorem, this is multiplication by