A new spectral analysis of stationary random Schrödinger operators

We consider random Schrödinger operators of the form

on $\operatorname{L}^{2}(\mathbb{R}^{d})$, where $V_{\omega}:\mathbb{R}^{d}\to\mathbb{R}$ is a realization of a “stationary” (that is, statistically translation-invariant) random potential $V$, constructed on a given probability space $(\Omega,\mathbb{P})$, and we study the properties of the corresponding Schrödinger flow on $\mathbb{R}^{d}$,

with initial data $u^{\circ}\in\operatorname{L}^{2}(\mathbb{R}^{d})$. This well-travelled equation models the motion of an electron in a disordered medium described by the potential $V_{\omega}$, where the coupling constant $\lambda>0$ stands for the strength of the disorder.

For comparison, let us first recall transport properties in the simpler case of periodic or quasiperiodic disorder. If $V$ is periodic, the energy transport is well-known to remain purely ballistic as for the free flow $u_{\lambda=0}$, cf. [3]. The proof relies on the absolute continuity of the spectrum of the periodic Schrödinger operator on $\operatorname{L}^{2}(\mathbb{R}^{d})$, and more precisely on the existence of so-called Bloch waves, which are extended states constructed by means of standard perturbation theory as deformations of Fourier modes, $x\mapsto e^{ik\cdot x}\varphi_{k,\lambda}(x)$ with $\varphi_{k,\lambda}$ periodic. In case of a quasiperiodic potential $V$, the problem is more involved and depends on the strength of the disorder: energy transport is expected to remain ballistic only at weak coupling $0<\lambda\ll 1$ or at high energies. This is rigorously established in dimension $d=1$ [54]. In higher dimensions $d>1$, for a specific class of quasiperiodic potentials, it was recently shown that there exist initial data at high energies that indeed display ballistic transport [25, 26] (the analysis of the discrete setting is more complete [6]). Inspired by the periodic case, the proof relies on the existence of corresponding Bloch waves as extended states of the form $x\mapsto e^{ik\cdot x}\varphi_{k,\lambda}(x)$ with $\varphi_{k,\lambda}$ quasiperiodic, but their construction is more intricate since standard perturbation theory no longer applies. In [13], we provide a simple method to construct “approximate” Bloch waves and deduce ballistic transport for all data at least up to “very long” timescales both at weak coupling and at high energies. These results in the periodic and quasiperiodic settings show how Bloch waves are crucial tools to infer transport properties of the Schrödinger flow.

The present work is concerned with the more general stationary random setting in the weak coupling regime $0<\lambda\ll 1$. In case of a random potential $V$ with short-range correlations, in stark contrast with the periodic and quasiperiodic cases, a celebrated conjecture by Anderson [2] states that in dimension $d>2$ every initial condition can be almost surely decomposed into two parts: a low-energy part that remains dynamically localized and a bulk-energy part that propagates diffusively. Despite the great recent achievements of rigorous perturbation theory in some asymptotic time regimes, e.g. [51, 18, 17, 16, 9], successfully describing the emergence of irreversible diffusion from the reversible Schrödinger dynamics, the full justification of this quantum diffusion phenomenon remains a major open problem in mathematical physics [49, 15]. More precisely, the ensemble-averaged Wigner transform of the quantum wave $u_{\lambda}$ is known to converge to the solution of a linear Boltzmann equation on the kinetic timescale $t\sim\lambda^{-2}$, and of a heat equation on longer times, but the justification is limited to a perturbative time regime $t\ll\lambda^{-2-\eta}$ for some small $\eta>0$. A simplified question concerns the behavior of time correlations in form of the averaged wavefunction $\mathbb{E}\left[{u_{\lambda}^{t}}\right]$, which is expected to display exponential time decay: more precisely, in Fourier space, on the kinetic timescale $t\sim\lambda^{-2}$, as $\lambda\downarrow 0$,

where the decay rate $\alpha_{k}>0$ would coincide with the total scattering cross section in the corresponding Boltzmann equation, and where corrections are added on longer times. A proof of this exponential decay on the kinetic timescale is given in [9] based on a perturbative expansion of a Feynman-Kac type formula. The perturbative analysis of [18, 17, 16] would further yield an improved result, but still restricted to limited timescales.

Motivated by these open questions, rather than trying to improve on perturbative expansions and renormalization techniques, we aim at developping an alternative spectral approach to describe the long-time behavior of the system beyond perturbative timescales. More precisely, we take inspiration from the periodic and quasiperiodic cases, although the behavior radically differs from the present random setting, and we investigate the role of a corresponding notion of Bloch waves. It appears that these Bloch waves are no longer extended states associated with absolutely continuous spectrum: they are expected to be only defined in a weak distributional sense in probability and to play the role of resonant modes associated with some kind of “continuous resonant spectrum”. Exploiting ideas from Malliavin calculus, we manage to appeal to perturbative Mourre’s theory, cf. Theorem 4, which leads to the construction of approximate dynamical resonances and constitutes the first spectral proof of (1.1), cf. Corollary 5. Non-perturbative refinements to reach longer times are postponed to future works, as well as the investigation of other possible dynamical consequences in closer connection with quantum diffusion.

We briefly describe the framework of our new approach to Schrödinger operators $H_{\lambda,\omega}=-\triangle+\lambda V_{\omega}$. First, we change the point of view and rather consider the operator $H_{\lambda}:=-\triangle+\lambda V$ as acting on the augmented Hilbert space $\operatorname{L}^{2}(\mathbb{R}^{d}\times\Omega)$, then studying the corresponding Schrödinger flow on $\operatorname{L}^{2}(\mathbb{R}^{d}\times\Omega)$,

with deterministic initial data $u^{\circ}\in\operatorname{L}^{2}(\mathbb{R}^{d})\subset\operatorname{L}^{2}(\mathbb{R}^{d}\times\Omega)$. This can be viewed as including stochastic averaging conveniently into the functional setup; see also [45, 24]. (Note that $H_{\lambda}$ on $\operatorname{L}^{2}(\mathbb{R}^{d}\times\Omega)$ has absolutely continuous spectrum as a consequence of Wegner estimates when $H_{\lambda,\omega}$ on $\operatorname{L}^{2}(\mathbb{R}^{d})$ has almost sure pure point spectrum.)

As we have shown and already used in [13], see Section 3 below for details (also [4, 24]), the operator $H_{\lambda}$ on $\operatorname{L}^{2}(\mathbb{R}^{d}\times\Omega)$ can be decomposed via a Fourier-type transformation as a direct integral of fibered operators $\{H_{k,\lambda}^{\operatorname{st}}\}_{k}$ acting on the elementary space $\operatorname{L}^{2}(\Omega)$, which is viewed as the space of stationary random fields,

The (centered) fibered operators take the form

where $\nabla^{\operatorname{st}}$ and $\triangle^{\operatorname{st}}$ denote the stationary gradient and Laplacian on $\operatorname{L}^{2}(\Omega)$; see Section 3.2 for proper definitions. In particular, the Schrödinger flow $u_{\lambda}$ is decomposed as

in terms of the fibered evolutions $\{e^{-itH^{\operatorname{st}}_{k,\lambda}}1\}_{k}$ on $\operatorname{L}^{2}(\Omega)$. This partial diagonalisation via Fourier is henceforth referred to as the stationary Floquet–Bloch fibration, in analogy with the well-known corresponding construction in the periodic setting, e.g. [32, 33].

At vanishing disorder $\lambda=0$, as the constant function $1\in\operatorname{L}^{2}(\Omega)$ is an eigenfunction with eigenvalue $0$ for the unperturbed fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$, the associated spectral measure coincides with the Dirac mass at $0$, and the decomposition (1.4) then reduces to the usual Fourier diagonalisation of the free flow,

When the disorder is turned on but small, $0<\lambda\ll 1$, the description of the Schrödinger flow $u_{\lambda}$ is reduced to a family of (hopefully simpler) fibered perturbation problems for the spectral measures. In case of a periodic potential $V$ (that is, $\Omega=\mathbb{T}^{d}$, cf. Remark 3.4(a)), the fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$ have compact resolvent in view of the Rellich theorem, and thus discrete spectrum. The eigenvalue at $0$ is then typically simple and isolated, which allows to apply standard perturbation methods, e.g. [32, 3], showing that it is perturbed into isolated eigenvalues $\{z_{k,\lambda}=\lambda^{2}z_{\lambda}(k)\}_{k}$. In other words, Fourier modes $\{\mathfrak{e}_{k}\}_{k}$ are perturbed into so-called periodic Bloch waves that diagonalize the Schrödinger flow and are associated with perturbed generalized eigenvalues of the form $\{|k|^{2}+\lambda^{2}z_{\lambda}(k)\}_{k}$. This entails that the flow is approximately conjugated to the free flow (1.5) in the sense of

and in particular the energy transport remains ballistic forever. The same conclusion holds in fact for all $\lambda$, cf. [3]. In case of a quasiperiodic potential $V$ (cf. Remark 3.4(a)), the situation is expected to be similar at weak coupling, but the existence of corresponding Bloch waves is a more subtle question: quasiperiodic fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$ are degenerate elliptic operators, for which compactness fails, and the simple eigenvalue at $0$ is no longer isolated but embeds in dense pure point spectrum, so that no standard perturbation theory applies; see [25, 26, 13].

In case of a random potential $V$ with short-range correlations, the situation differs drastically in link with the expected diffusive behavior. We show that $0$ is typically the only eigenvalue of the fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$, is simple, and embeds in absolutely continuous spectrum, cf. Proposition 1 below. According to Fermi’s Golden Rule, whenever the disorder is turned on, this embedded eigenvalue is then expected to dissolve in the continuous spectrum, cf. Proposition 3, and to turn into a complex resonance at

in the lower complex half-plane. In particular, this provides a spectral explanation why approximate Bloch wave analysis leading to ballistic transport as in [13] breaks down on the kinetic timescale $t\sim\lambda^{-2}$. For the averaged wavefunction, this leads to expect

which would indeed agree with the exponential decay (1.1) on the kinetic timescale, with $\Im z_{\lambda}(k)\sim-\alpha_{k}<0$, and a finer resonance analysis would yield a more accurate expansion. From a spectral perspective, fibered resonances are transferred via the fibration (1.3) to kind of a “continuous resonant spectrum” for the full operator $H_{\lambda}$ on $\operatorname{L}^{2}(\mathbb{R}^{d}\times\Omega)$, cf. Remark 2.1 below.

General spectral tools are however dramatically missing to rigorously study these fibered perturbation problems on $\operatorname{L}^{2}(\Omega)$, in particular due to the lack of any relative compactness of the perturbation. We start by performing a detailed study of rudimentary spectral properties of fibered operators, cf. Propositions 1–3, emphasizing the strong dependence on the structure of the underlying probability space $(\Omega,\mathbb{P})$. Next, we appeal to Mourre’s theory [37, 1] as a rigorous approach to fibered perturbation problems. More precisely, we start by constructing Mourre conjugates for the unperturbed operators $\{H^{\operatorname{st}}_{k,0}\}_{k}$. The construction requires a surprisingly nontrivial work and relies on a deep use of Malliavin calculus, which constitutes the core of our contribution, cf. Section 5. This construction is however not compatible with the perturbation $V$ in the sense that $\lambda V$ cannot be considered as a small perturbation of $\{H_{k,0}^{\operatorname{st}}\}_{k}$ in the sense of Mourre’s theory, in link with the infinite dimensionality of the probability space. For this reason, we only manage to apply perturbative Mourre’s theory under a suitable (weak) truncation, cf. Theorem 4. As a direct consequence, the decay law (1.1) is recovered at least on the kinetic timescale, cf. Corollary 5. Finally, we give a relevant formulation of resonance conjectures for fibered operators, cf. Conjectures Conjecture (LRC) — Local resonance conjecture and Conjecture (GRC) — Global resonance conjecture, which are motivated by our partial results and are shown to imply the expected decay law (1.1) to finer accuracy on all timescales, cf. Corollary 6. These conjectures are further illustrated in Section 7, where we display a toy model that shares various properties of Schrödinger operators and allows for a rigorous resonance analysis. Although these conjectures are left open, the present work sheds a new light on the study of random Schrödinger operators, in particular providing the first spectral proof of (1.1); our results will be strengthened in future works and hopefully serve as a starting point for a new line of research in the field.

1. $\bullet$

   We denote by $C\geq 1$ any constant that only depends on the space dimension $d$ and on the law of the random potential $V$. We use the notation $\lesssim$ (resp. $\gtrsim$) for $\leq C\times$ (resp. $\geq\frac{1}{C}\times$) up to such a multiplicative constant $C$. We write $\simeq$ when both $\lesssim$ and $\gtrsim$ hold. We add subscripts to $C,\lesssim,\gtrsim,\simeq$ to indicate dependence on other parameters. We denote by $O(K)$ any quantity that is bounded by $CK$.

2. $\bullet$

   We denote by $\widehat{f}(k):=\mathcal{F}f(k):=\int_{\mathbb{R}^{d}}e^{-ik\cdot x}f(x)\,dx$ the usual Fourier transform of a smooth function $f$ on $\mathbb{R}^{d}$. The inverse Fourier transform is then given by $f(x)=\int_{\mathbb{R}^{d}}e^{ik\cdot x}\widehat{f}(k)\,\,{{\mathchar 22\relax\mkern-12.0mud}}k$ in terms of the rescaled Lebesgue measure $\,{{\mathchar 22\relax\mkern-12.0mud}}k:=(2\pi)^{-d}dk$.

3. $\bullet$

   The ball centered at $x$ and of radius $r$ in $\mathbb{R}^{d}$ is denoted by $B_{r}(x)$, and we write for abbreviation $B(x):=B_{1}(x)$, $B_{r}:=B_{r}(0)$, and $B:=B_{1}(0)$. Without ambiguity, we occasionally also denote by $B$ the unit ball at the origin in the complex plane $\mathbb{C}$.

4. $\bullet$

   For a set $E\subset\mathbb{R}^{d}$ we denote by ${\operatorname{conv}}(E)$ its convex envelope, by $\operatorname{int}(E)$ its interior, and by ${\operatorname{adh\,}}(E)$ its closure.

5. $\bullet$

   We denote by $\mathcal{B}(\mathbb{R}^{k})$ the set of Borel subsets of $\mathbb{R}^{k}$, and for $E\subset\mathbb{R}^{k}$ we let $\mathcal{P}(E)$ denote the set of Borel probability measures on $E$.

6. $\bullet$

   For a vector space $X$, we write $X^{\otimes p}$ for its $p$-fold tensor product, and $X^{\odot p}$ for its $p$-fold symmetric tensor product.

7. $\bullet$

   For $a,b\in\mathbb{R}$, we write $a\wedge b:=\min\{a,b\}$ and $a\vee b:=\max\{a,b\}$.

We refer to Section 3 for a suitable definition of stationarity as statistical invariance under spatial translations. Throughout, the stationary random potential $V$ is assumed real-valued and centered, $\mathbb{E}\left[{V}\right]=0$. As we show, fine spectral properties crucially depend on the structure of the underlying probability space. We therefore mainly focus on Gaussian or Poisson settings, where Malliavin calculus is available and provides a useful Fock space decomposition of $\operatorname{L}^{2}(\Omega)$. More precisely, we consider the following:

1.  $\bullet$

   Gaussian setting: $V=b(V_{0})$ for some Borel function $b:\mathbb{R}\to\mathbb{R}$ and some stationary centered Gaussian random field $V_{0}$ with bounded covariance function

   $$\qquad\mathcal{C}_{0}(x):=\mathbb{E}\left[{V_{0}(x+y)V_{0}(y)}\right].$$

   Equivalently, the field $V_{0}$ can be represented as

   $$\qquad V_{0}(x)=\int_{\mathbb{R}^{d}}\mathcal{C}_{0}^{\circ}(x+y)\,dZ(y),$$

   (2.1)

   where $dZ$ is a standard Gaussian white noise on $\mathbb{R}^{d}$ and where the kernel $\mathcal{C}_{0}^{\circ}$ is the convolution square root of the covariance function, $\mathcal{C}_{0}=\mathcal{C}_{0}^{\circ}\ast\mathcal{C}_{0}^{\circ}$.

2.  $\bullet$

   Poisson setting: $V=b(V_{0})$ for some Borel function $b:\mathbb{R}\to\mathbb{R}$ and some $V_{0}$ of the form

   $$\qquad V_{0}(x)=\sum_{y\in\mathcal{P}_{0}}\mathcal{C}_{0}^{\circ}(x+y),$$

   (2.2)

   where $\mathcal{P}_{0}$ is a standard Poisson point process on $\mathbb{R}^{d}$ and where $\mathcal{C}_{0}^{\circ}$ is the single-site potential.

We say that the random potential $V$ is short-range if it has integrable decay of correlations: in the above settings, this amounts to choosing $\mathcal{C}_{0}^{\circ}\in\operatorname{L}^{1}(\mathbb{R}^{d})$.

For shortness, in the sequel, we shall mainly restrict to the Gaussian setting, although the same results can be transferred mutatis mutandis in the Poisson case (using the corresponding version of Malliavin calculus, e.g. [44]). For simplicity, we occasionally further restrict to a random potential $V=V_{0}$ that is itself Gaussian: although unbounded, such potentials have a simpler action on the Fock space decomposition of $\operatorname{L}^{2}(\Omega)$.

We refer to Section 3 for the construction of the stationary Floquet–Bloch fibration (1.3). Next, we start with a detailed spectral analysis of the unperturbed operators

Although the stationary Laplacian $-\triangle^{\operatorname{st}}$ is a natural operator on $\operatorname{L}^{2}(\Omega)$ and has been introduced in various settings (e.g. in the context of stochastic homogenization [42, 23]), its spectral properties have never been elucidated before, and we close this gap here. Note that some preliminary remarks on its spectrum have been made in [7, Section 3.1], see also [34, Section 2.C], namely that it is discrete if $\Omega$ is a finite set, that there is in general no spectral gap above $0$ in contrast with the periodic setting, and that it coincides with $[0,\infty)$ in case of an i.i.d. structure. Interestingly, the spectrum depends crucially on the structure of the underlying probability space $(\Omega,\mathbb{P})$, as precisely formulated in Section 4.1 below in terms of a notion of “spectrum” of the probability space. In the model Gaussian setting, our result takes on the following simple guise.

We turn to the perturbed fibered operators $\{H^{\operatorname{st}}_{k,\lambda}\}_{k}$ and start with a characterization of their spectrum. While we focus here on the Gaussian setting, a more general statement is given in Section 4.2. In case of an unbounded potential $V$, the essential self-adjointness of the perturbed operators $\{H_{k,\lambda}^{\operatorname{st}}\}_{k}$ is already a delicate issue, for which an (almost optimal) criterion is included in Appendix A, requiring $V\in\operatorname{L}^{p}(\Omega)$ for some $p>\frac{d}{2}$, in line with the corresponding celebrated self-adjointness problem for Schrödinger operators on $\operatorname{L}^{2}(\mathbb{R}^{d})$ with singular potentials, cf. [27, 19, 28]; see also Proposition 5.16 for the simpler case when $V=V_{0}$ is itself Gaussian.

The nature of the spectrum of the perturbed operators $\{H_{k,\lambda}^{\operatorname{st}}\}_{k}$ is a more involved question and is a main concern in the sequel. In view of the fibration (1.4), the perturbation of the eigenvalue at $0$ for the fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$ is of particular interest. According to Fermi’s Golden Rule, e.g. [48, Section XII.6], this eigenvalue embedded in continuous spectrum is expected to dissolve when the perturbation is turned on. The simplest rigorous version of this key conjecture is as follows. It is based on observing that the formula for the second derivative of a hypothetic branch of eigenvalues at $\lambda=0$ would be a complex number, cf. Section 4.3.

This basic instability result is however quite weak: for $k\in\mathbb{R}^{d}\setminus\{0\}$, the operator $H_{k,\lambda}^{\operatorname{st}}$ is in fact expected to have purely absolutely continuous spectrum in a neighborhood of $0$ for $0<\lambda\ll 1$. In addition, in view of the resonance interpretation of Fermi’s Golden Rule, which originates in the work of Weisskopf and Wigner [53], the perturbed eigenvalue is expected to turn into a complex resonance. Relevant conjectures are formulated in Section 2.4 below.

The perturbation problem for an eigenvalue embedded in continuous spectrum, in link with Fermi’s Golden Rule and resonances, is an active topic of research in spectral theory. Various general approaches have been successfully developed, see e.g. [14] and references therein, but none seems to be available in our probabilistic setting: a key difficulty is that the random perturbation $V$ is never compact with respect to the unperturbed operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$ on $\operatorname{L}^{2}(\Omega)$ (unless it is degenerate). This calls for the development of robust techniques for the spectral analysis of stationary operators on the probability space. In the present contribution, we appeal to Mourre’s commutator theory [37, 1], cf. Section 5.1, which is reputedly flexible and requires no compactness. Although not allowing to deduce the existence of resonances in any strong form, Mourre’s theory would ensure similar dynamical consequences, e.g. [41, 21, 8].

More precisely, we start with the construction of a natural group of dilations $\{U_{t}^{\operatorname{st}}\}_{t\in\mathbb{R}}$ on $\operatorname{L}^{2}(\Omega)$ in the model Gaussian setting, cf. Section 5.3: heuristically, it amounts to dilating the underlying white noise in the representation (2.1), which constitutes a unitary group since dilations preserve the law of the white noise. The generator $A^{\operatorname{st}}$ of this group is then checked to be a conjugate operator for the stationary Laplacian $-\triangle^{\operatorname{st}}$ in the sense of Mourre’s theory, cf. Proposition 5.7(i). In Section 5.6, by means of suitable deformations, we further construct corresponding conjugates for the whole family of fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$. This appears to be surprisingly more involved than for $k=0$, in link with the infinite dimensionality of the probability space: our proof makes a deep use of the Fock space structure of $\operatorname{L}^{2}(\Omega)$ as provided by Malliavin calculus, thus emphasizing the interplay between spectral theory and the functional structure of the probabilistic setup. Next, we turn to perturbed operators $\{H_{k,\lambda}^{\operatorname{st}}\}_{k}$. It appears that the perturbation $\lambda V$ is not compatible in the sense of Mourre’s theory, cf. Proposition 5.7(iii), again in link with the infinite dimensionality of the probability space. Perturbative Mourre’s theory can therefore not be applied unless we introduce a suitable (weak) Wiener truncation.

Based on this perturbative Mourre result, an approximate dynamical resonance analysis can be developed for truncated fibered operators in the spirit of [41, 21, 8] and leads to the exponential time decay of the corresponding averaged wavefunction. Further noting that the truncation error is easily estimated on the kinetic timescale, we can get rid of the truncation and rigorously deduce the validity of the exponential decay law (1.1) as stated below; the proof is postponed to Section 5.7.

Although our truncation argument could be compared with the truncation of the Dyson series in the perturbative analysis of [51, 18, 17, 16], it only requires to estimate a truncation error, which is often a simpler matter, while the truncated evolution is intrinsically analyzed by means of Mourre’s theory, avoiding any Feynman diagram analysis or any renormalization to handle the truncated Dyson series. In addition, formal computations indicate that the truncation of the evolution at time $t$ on Wiener chaoses of order $\leq K$ should be accurate provided $K\gg\lambda^{2}t$. Since our truncation $Q_{\lambda}$ in Theorem 4 amounts to projecting onto chaoses of order $\leq(L_{0}\lambda)^{-2}$, which is a particularly high order compared to truncations of Dyson series in [51, 18, 17, 16], the accuracy in Corollary 5 should thus follow in fact up to times $t\ll\lambda^{-4}$. Non-perturbative approaches to fibered resonances and accuracy on even longer timescales are postponed to future works.

The above results provide partial indications that the eigenvalue at $0$ of the fibered operators $\{H_{k,0}^{\operatorname{st}}\}_{k}$ should dissolve in the continuous spectrum upon perturbation and turn into complex resonances. In particular, in agreement with Fermi’s Golden Rule, Corollary 5 is consistent with resonances at

As resonance theories have never been constructed for operators on the probability space, we formulate relevant conjectures that will be investigated rigorously in future works. To emphasize the relevance of our formulation, we further consider in Section 7 an illustrative toy model that shares many spectral features of Schrödinger operators and for which resonances are explicitly shown to exist in a similar sense, cf. Theorem 7.1(iii).

According to the usual definition, the operator $H_{k,\lambda}$ has a resonance at $z_{k,\lambda}$ in the lower complex half-plane if the resolvent $(H_{k,\lambda}^{\operatorname{st}}-z)^{-1}$ on the upper half-plane $\Im z>0$, when viewed in a suitably weakened topology, extends to a meromorphic family of operators indexed by all $z\in\mathbb{C}$ (or at least in a complex neighborhood), and if this family admits a simple pole at $z=z_{k,\lambda}$. In the usual case of operators on $\operatorname{L}^{2}(\mathbb{R}^{d})$, the suitable weakening of the topology typically consists of viewing the resolvent as a family of linear operators $C^{\infty}_{c}(\mathbb{R}^{d})\to\mathcal{D}^{\prime}(\mathbb{R}^{d})$ rather than $\operatorname{L}^{2}(\mathbb{R}^{d})\to\operatorname{L}^{2}(\mathbb{R}^{d})$. In the present setting on the probability space, the role of $C^{\infty}_{c}(\mathbb{R}^{d})\subset\operatorname{L}^{2}(\mathbb{R}^{d})$ can be played for instance by the dense linear subspace $\mathcal{P}(\Omega)\subset\operatorname{L}^{2}(\Omega)$ of $V$-polynomials,

and the dual $\mathcal{D}^{\prime}(\mathbb{R}^{d})$ is then replaced by the dual space $\mathcal{P}^{\prime}(\Omega)$ of continuous linear functionals on $\mathcal{P}(\Omega)$. In these terms, we formulate the following resonance conjecture. The linear functionals $\Psi_{k,\lambda}^{+}$ and $\Psi_{k,\lambda}^{-}$ below are referred to as the resonant and co-resonant states, respectively. Since the imaginary part of the expected branch of resonances $\Im z_{k,\lambda}=-\lambda^{2}\alpha_{k}+O_{k}(\lambda^{3})$ vanishes to leading order both as $k\to 0$ (in dimension $d>2$) and as $|k|\uparrow\infty$, cf. formula (2.3), we henceforth restrict to $k$ in a compact set away from $0$.

Next, we state a global version of this resonance conjecture in the case of an unbounded potential $V$ with $\sigma(H_{k,\lambda}^{\operatorname{st}})=\mathbb{R}$ (see e.g. Proposition 2). A direct computation shows that the spectral measure of $H_{k,0}^{\operatorname{st}}$ associated with $V$ is typically supported on the whole half-axis $[-|k|^{2},\infty)$ and is only $\frac{d-2}{2}$-times differentiable at $-|k|^{2}$ in dimension $d>2$, cf. proof of Lemma 4.2; this suggests that the band on which the meromorphic extension of the resolvent exists must shrink as $\lambda\downarrow 0$ close to $z=-|k|^{2}$.

We show that the above conjectures imply the expected exponential decay law (1.1) for the averaged wavefunction to finer accuracy, thus providing a strong improvement and a very first workaround for the available perturbative methods [51, 18, 17, 16, 9]. Under Conjecture Conjecture (LRC) — Local resonance conjecture an accurate description of the decay law is deduced only for times $t\ll\lambda^{-2}|\!\log\lambda|$, but accuracy is reached on all timescales under Conjecture Conjecture (GRC) — Global resonance conjecture. The result is expressed as a resonant-mode expansion of the Schrödinger flow in the weak sense of $\mathcal{P}^{\prime}(\Omega)$, and the description of the averaged wavefunction follows as a particular case; the proof is quite standard and is postponed to Section 6.1.

In order to make the above resonant-mode expansion (2.6) more striking, we note that resonances and resonant states can be computed explicitly in form of a perturbative Rayleigh–Schrödinger series. In particular, in agreement with Corollary 5, the resonance is checked to coincide to leading order with $\lambda^{2}(\beta_{k}-i\alpha_{k})$; the proof is included in Section 6.2.