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In the interaction picture,At infinity this is $-(i\lambda /\hslash ){\sum }_k{\alpha }_k{W}_k{F}_k$, whereso $b_1=1,b_2=1,b_3=-1$ and ${\omega }_{f{i}_k}=(E_f-E_{i_k})/\hslash$. The probability is $({\lambda }^2/{\hslash }^2)|{\sum }_k{\alpha }_k{W}_k{F}_k{|}^2$, displaying interference between initial components. For $N$ equal response factors, Cauchy-Schwarz gives $|{\sum }_k{\alpha }_k{|}^2\le N$, attained by equal phases and $|{\alpha }_k|=N^{-1/2}$; the largest probability is $N{\lambda }^2|WF{|}^2/{\hslash }^2$ within perturbation theory.

The flow of $W=\varphi u{\partial }_u+\sum {\alpha }_i{x}_i{\partial }_{x_i}$ is $(u,x_i)\mapsto (e^{\varphi s}u,e^{{\alpha }_{i}s}{x}_i)$; invariants are exactly solutions of $Wf=0$. Two scaling symmetries of the PDE areInvariants for $W_1$ include $y,xt,u{t}^2$; for $W_2$, $u,y/x,t/x$. Two common invariants are $ut/x$ and $y^2/(xt)$. Invariance under $u{\partial }_u+x{\partial }_x+\frac{1}{2}y{\partial }_y$ and ${\partial }_y$ forces $u=xF(t)$. Substitution gives $F^{\prime }=F^2$, hence $u=0$ or $u=-x/(t-C)$.

A subset is dense when every nonempty open set meets it, equivalently its closure is the whole space. Baire category theorem says a complete metric space has dense intersection for every countable family of open dense sets. Hereso Baire makes $S$ dense. It need not be open: enumerate the rationals $q_j$ and take $A_j=[0,1]\setminus \{q_1,\dots ,q_j\}$, giving the irrationals. Nor need ${\cap }_j{A}_j$ be nonempty: take $A_1=\varnothing$ and $A_j=[0,1]$ for $j\ge 2$.