Spectral functions in quantum mechanics I'm a math student and a totally newcomer to quantum mechanics and I'm trying to teach myself this subject by studying Faddeev and Yakubovski's Lectures on Quantum Mechanics for Mathematics Students. 
The following paragraph appears on pp. 49-50 of the book.

For any self-adjoint operator there is a spectral function
  $P_A(\lambda)$, that is, a family of projections with the following
  properties:
  
  
*
  
*$P_A(\lambda)\le P_A(\mu)$ for $\lambda<\mu$, that is, $P_A(\lambda)P_A(\mu)=P_A(\lambda)$;
  
*$P_A$ is right-continuous;
  
*$P_A(-\infty)=0$ and $P_A(\infty)=I$;
  
*$P_A(\lambda)B=BP_A(\lambda)$ if $B$ is any bounded operator commuting with $A$.
  
  
  A vector $\phi$ belongs to the domain of the operator $A$ if
  $$\int_{-\infty}^\infty \lambda^2\, d(P_A(\lambda)\phi,
 \phi)<\infty,$$ and then $$A\phi=\int_{-\infty}^\infty \lambda\,
 dP_A(\lambda)\phi.$$

I've seen the construction of $P_A(\lambda)$ for the finite-dimensional case as $\theta(\lambda-A)$ where $\theta$ is the Heaviside function. However, 


*

*I wonder whether the reason that the authors say "there is a spectral function $P_A(\lambda)$" without introducing it explicitly in the infinite-dimensional case is because of the complexity of the derivation/formula or the fact that the existence of $P_A$ with the above 4 properties is just an "axiom?"

*Are the integrals merely consequences of items 1--4? (The book does not explain the meaning of $d(P_A(\lambda)\phi,
 \phi)$ and  $dP_A(\lambda)\phi$ in integrations, and I'm not even sure if I get them correctly.)


Any help with clarifications is greatly appreciated!
 A: This has nothing to do with physics, but it's pure functional analysis. 
What you have here is nothing but a slightly convoluted version of the spectral theorem, which is formulated for all self-adjoint operators (bounded or unbounded). It's slightly convoluted, because it includes the necessary definitions of spectral measure in the theorem itself - I'd usually see them defined beforehand.
To understand how this theorem comes about is nontrivial, because the spectrum of a (possibly un)bounded operator in infinite dimensions contains parts that are not eigenvalues (continuous spectrum). The way you construct $P$ and define $\int dP$ at least for bounded operators is the same as you do it for arbitrary measures in measure theory and/or probability theory. For unbounded operators, you need to take heed with domain issues. 
Remember probability theory? If you have a discrete measure $p_i$, an expectation value of a random variable $X$ is nothing else but $\sum_i X_ip_i$. If you have a continuous measure $\mu$, the expectation value will be given by something like $\int Xd\mu$. It's the same here: Given a self-adjoint matrix (or a compact operator), the spectral theorem reads $Av=\sum_i \lambda_i \langle v_i,v\rangle v_i$ where $v_i$ are the eigenvectors and $\lambda_i$ the eigenvalues. Your last expression for $A\phi$ is the continuous counterpart. Another complication arising here is that your measures are not real or complex-valued, but operator-valued in contrast to the usual measures.
If you want, I can try to add a rough sketch of how to prove the theorem.
