Rencently, I am reading some articles about time decay estimates or Strichartz estimates for Schrodinger equations with potential. When considering Strichartz estimates for potential $V$ with decay $|x|^{-2-\varepsilon}$, Rodnianski & Schlag said in Time decay for solutions of Schrödinger equations with rough and time-dependent potentials that the Strichartz estimte (nonendpoint case) holds if we assume the initial data $u_0 \in P_{[0,\infty)} X$, where $X$ is the total space. Moreover, in view of $0$ will contribute negative effection on the esimates, we should assume that $0$ is neither eigenvalue nor the resonance of $H=-\Delta+V$.
I am wondering what's the precise definition of resonance? Because I have seen at least three definitions of it, but what definition is the most essential?
definition (version 1: Rodnianski & Schalg: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials) A resonance is a distribution solution to $H \psi=0$ so that $\psi \notin L^2$ but $(1+|x|^2)^{-\frac{\sigma}{2}} \psi(x) \in L^2$ for any $\sigma>\frac{1}{2}$.
definition (version 2: Artzi & Klainerman: Decay and regularity for the Schrödinger equation ) zero is said to be a resonance of $H$ if for every $\varepsilon >0$ there exists a $0 \not= \psi \in H^{2,-1-\varepsilon}$ such taht $H\psi=0$.
definition (version 3: Nakanish & Schalg: Invariant manifolds and Dipersive Hamiltonian Evolution Equations) in odd dimensions, the reslovent $(H-z^2)^{-1}$ can be expanded by $(H-z^2)^{-1}=B_{-2}z^{-2}+B_{-1}z^{-1}+B_0+B_1z +... B_k z^k +R_k(z)$, where the expansion is in $L^{2,-1/2-\varepsilon} \to L^{2,1/2+\varepsilon}$ sense, we say $0$ is neither eigenvalue nor the resonance of $H$ iff $B_{-2}=B_{-1}=0$.
and so on...
I just know the aim of ruling out the resonance is to ensure the "good" behavior of $(H-z)^{-1}$ near $z=0$.
However, are the three definitions of resonance above equivalent to each other? Especially for definition 2 and definition 1. I think the definition 2 is a little peculiar, but Rodnianski and Schalg： "time decay estimates..." cited the results in Artzi & Klainermann: "decay and regularity..." to support their result. But at least we note that $\text{definition 1} \subset \text{definition 2}$, thus $\text{(definition 2)}^c \subset \text{(definition 1)}^c$. So we can see that if $0$ is not resonance in def 1, the conclusion that $0$ is not resonance in the sense of def 2 will not hold necessarily.
I am struggled about it, if someone has known something about it, please tell me, thank you!
