Communications in Mathematical Physics Zero-Energy Bound State Decay for Non-local Schrödinger Operators

We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schrödinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay at infinity of both L2 and resonance solutions. We highlight the interplay of the kinetic term and the potential in these decay behaviours, and identify the decay mechanisms resulting from specific balances of global lifetimes with or without the potential.


Introduction
The study of spectral properties of Schrödinger operators H = − 1 2 ∆ + V on L 2 (R d ), featuring the Laplacian and a potential V , has a long history in mathematics. Potentials decaying to zero at infinity produce a fascinating variety of spectral behaviours and phenomenology, including the possibility of finite or countably infinite discrete spectra, (dense sets of) embedded eigenvalues or singular continuous spectrum, resonances, criticality, Efimov effect, enhanced binding, or scattering. In this, there is a split of qualitative behaviours according to the rates of decay of the potential (which led to various concepts of 'long range' and 'short range' potentials). The results indicate that the existence of embedded eigenvalues is a long range effect, and the appearance of positive point spectrum is a combination of slow decay and oscillations of the potential. For surveys we refer to [39,10,5,8] and the numerous references therein.
Zero-energy level, which coincides with the edge of the continuous spectrum, often marks a borderline between various regimes of spectral behaviour, in particular, between existence and nonexistence of bound states, thus also shedding light on the mechanisms of "birth" of such states. Whether zero is an eigenvalue is, in general, a difficult problem. Some early results on existence or non-existence of zero-energy eigenvalues go back to the papers [1,36,28,21,30,47,38,26]. For potentials which are negative at infinity, decaying at a rate V (x) ≍ −c|x| −γ , c > 0, as |x| → ∞, and satisfying some further conditions, it has been established in [12] that for γ ∈ (0, 2) zero is not an eigenvalue, see also [9,46]. For potentials that are positive at infinity the situation changes [49,35]. In [3] it has been shown that for Schrödinger operators on L 2 (R 3 ) with rotationally symmetric potentials V ∈ L p (R 3 ), p > 3 2 , whose positive part satisfies V + (x) ≤ C|x| −2 for x large enough, zero is not an eigenvalue corresponding to a positive eigenfunction if C = 3 4 , while a positive L 2eigenfunction does exist if C > 3 4 . (The result holds more generally for non-symmetric potentials and higher dimensions as well.) For some further results on the existence of compactly supported eigenfunctions at zero-eigenvalue for compactly supported V ∈ L p (R d ), p < d 2 , we refer to [27,29].
Apart from some cases using direct methods of analysis, the two most used techniques leading to these results are based on unique continuation or resolvent expansions.
Recently, a theory of non-local Schrödinger operators started to shape up, which has enriched the range of spectral phenomena and opened a new perspective to the understanding of classical Schrödinger operators as a specific case. Such operators arise by replacing the Laplacian with a suitable pseudo-differential operator such as the fractional Laplacian (−∆) α/2 , 0 < α < 2. In the present paper our primary aim is an analysis of eigenfunction decay at zero-eigenvalue or zeroresonance for a class of non-local Schrödinger operators H = −L + V on L 2 (R d ), with decaying potentials. One of our goals is to highlight the role of the potential in an interplay with the kinetic operator term −L in generating the decay behaviours.
The occurrence of zero or strictly positive eigenvalues for non-local Schrödinger operators just begins to be studied. In the recent paper [32], see Theorems 2.8 and 2.10, two sets of potentials generating zero-eigenvalues or zero-resonances for massless relativistic Schrödinger operators in dimension one have been constructed (as well as examples leading to a strictly positive eigenvalue). More recently, in [20] this has been generalized to fractional Laplacians of all order and arbitrary dimensions. Let κ > 0, α ∈ (0, 2), and P be a harmonic polynomial, homogeneous of degree l ≥ 0, i.e., satisfying P (cx) = c l P (x) for all c > 0, and ∆P = 0. Denote µ = d + 2l, and consider the potentials and functions (1.1) where 2 F 1 is Gauss' hypergeometric function. Then (−∆) α 2 ϕ κ + V κ,α ϕ κ = 0 holds in distributional sense with ϕ κ ∈ L 2 (R d ) if κ ≥ µ 4 , and . Furthermore, for large |x| we have that We note that the above examples by no means indicate that zero eigenvalues are common or easy to locate. By using methods of operator analysis, we have established further cases or conditions of existence as well as non-existence of embedded eigenvalues in [33]. For d = 3 it is known that provided |V |, |x · ∇V | and |x · ∇(x · ∇V )| are bounded by C(1 + x 2 ) −1/2 , with a small C > 0, jointly imply that (−∆) 1/2 + V has no non-negative eigenvalue [40]. Related work on unique continuation for fractional Schrödinger equations imply further non-existence results [11,44,45,41]. Some further recent work include non-positive potentials with compact support and L chosen to be the massive relativistic operator [34], and a class of generalized Schrödinger operators [6].
In our previous works [23,25] we have investigated eigenfunction decay for a large class of nonlocal Schrödinger operators. Using a probabilistic approach, L was assumed to be the infinitesimal generator of a 'jump-paring' Lévy process, i.e., having the property that |x−y|>1, |y|>1 ν(|x − y|)ν(|y|)dy ≤ Cν(|x|), |x| > 1, where ν is the Lévy jump density entering the symbol of the kinetic part −L of the operator (for details see the next section), and C > 0 is a constant. This condition means that double (and by iteration, any multiple) large jumps are stochastically dominated by single large jumps. As we have shown, there is a large family of such operators and related random processes of interest, including the fractional Laplacian and isotropic stable processes, relativistic Laplace operators and relativistic stable processes, and many others.
In the first paper quoted above we considered confining potentials V , i.e., increasing to infinity as |x| → ∞, and found that, for instance, if the potentials grow at a sufficiently regular rate in the sense that sup B V ≤ c inf B V for all unit balls B far enough from the origin, and where c > 0 is a constant, then the ground state (eigenfunction corresponding to the bottom of the spectrum) ϕ 0 behaves like ϕ 0 ≍ ν V . In this case the contributions of the kinetic and potential terms in H separate neatly. In the second paper we considered decaying potentials, i.e., decreasing to zero as |x| → ∞, such that λ 0 = inf Spec H < 0. In this case both V and ν decrease to zero, and now the interplay between the two terms determining the decay is much more intricate. We have shown that the decay behaviour depends on how some quantities related to intrinsic jump preferences of the Lévy processes compare with the width of the gap between the ground state eigenvalue and the continuum edge. Specifically, if λ 0 is sufficiently low-lying, then ϕ 0 ≍ ν. Moreover, when the gap is small and ν is chosen to have an increasingly light tail proceeding from a polynomial or sub-exponential, through an exponential, to a super-exponential decay rate, a sharp regime change in the decay of ϕ 0 can be observed and the rate of decay of the ground state suddenly becomes slower than the rate of decay of ν, see also [24].
The framework we have developed allows also to gain insight into the mechanisms generating these decay behaviours. When L is chosen such that the process (X t ) t≥0 it generates has the jump-paring property and V is an increasing potential, we obtain where τ B(x,1) = inf{t > 0 : X t ∈ B(x, 1)} denotes the first exit time of the process from a unit ball centered in x. This means that the fall-off rate depends on how soon the process perturbed by the potential on average leaves unit balls far out. Dependent on how negative λ 0 is and how light the tails of ν are, this relationship is preserved for decaying potentials up to a point when too light tails cause a drop below a critical level in the domination of single large jumps, which the ground state energy cannot compensate. Then the sojourn times due to multiple re-entries in unit balls of the process become comparable with the exit times piling up 'backlogs' in the fall-off events, and this slows the decay of ϕ 0 down. Indeed, when ν has so light tails that the jump-pairing condition no longer holds, ϕ 0 decays necessarily (and possibly much) slower than ν.
Our concern in the present paper is how ϕ 0 decays in the conditions when λ 0 = inf Spec H = 0. We note that in classical results on ground state decay of usual (local) Schrödinger operators by Agmon, Carmona, and other authors (see a discussion, e.g., in [31,Ch. 3]), a gap between the lowest eigenvalue and the edge of the continuous spectrum is an essential ingredient, and the results break down when this gap is brought to zero. In [2] it has been shown that for d ≥ 2 a zeroenergy eigenfunction ϕ for a potential V satisfying |x| 2−d/p V L p (R d ) < ∞, for some p ≥ 1 possibly being infinite, implies a power-law lower bound on the decay given by |x| −a ϕ ∈ L 2 (R d ), with some a ∈ (0, ∞). The authors also showed that for a potential slower than |x| −2 , exponential decay of a zero-energy eigenfunction is possible. For some upper bounds on the decay rates see [15]. A more encompassing study of exponential decay has been made in [14], in which also decays faster than exponential have been ruled out.
Our framework for non-local Schrödinger operators with λ 0 < 0 discussed above does not extend to the zero eigenvalue case, however, we will develop here a new framework by using a restricted set of operators L, which require a doubling property of the jump kernels instead of the more general jump-paring property. Since now λ 0 = 0, there is a complete lack of an energetic advantage from negative eigenvalues, however slight, leading to the behaviours discussed above, and now the influence of the potential appears through vestigial effects resulting from its sign at infinity. This will also reflect in the fact that in this case the contributions of the exits and the re-entries in local neighbourhoods even out and add up to such an extent that (1.5) obviously no longer holds, and the decay events become now governed by global lifetimes such as From this expression it can be appreciated that in the λ 0 = 0 case there is a rather delicate difference between the behaviour of paths under the perturbing potential and free fluctuations of the process, and this slight difference is responsible even for the very existence of a ground state at zero eigenvalue (for further details see Section 6). Below we start from the assumption that, for a class of operators L and decaying V , the eigenvalue equation (−L + V )ϕ = 0 is satisfied by a function ϕ ∈ L p (R d ), for some p > 0, describing zeroenergy eigenfunctions (when p = 2) or zero-resonances (when p = 2). Then we will study the asymptotic behaviour of ϕ(x) as |x| → ∞ which, following from the choice of the input operators, has a pointwise decay to zero. Our main results for asymptotically positive potentials are Theorems 4.1-4.3, giving upper and lower bounds on ϕ when these functions are positive or when they may have zeroes and distinct nodal domains. For asymptotically negative potentials we have Theorem 5.1, giving upper bounds. We note that our results apply to both zero-energy eigenfunctions and zeroresonances. As it will be seen in applications to specific cases (Theorems 6.1-6.4), these estimates perform remarkably well, giving exact or close hits of the precise asymptotics in (1.2) above.
The remainder of this paper is organized as follows. In Section 2 we introduce the class of non-local Schrödinger operators considered, and briefly summarize some of the relevant properties of the jump processes used in their Feynman-Kac representations. We also give an expression of the solutions of the eigenvalue equation using a path integral representation, which will be a key formula used throughout below. In Section 3 we derive and prove some self-improving upper and lower estimates on the solutions of related harmonic functions, on which our main conclusions will rely. In Sections 4 and 5 we obtain the decay behaviours separately for potentials positive and negative at infinity, respectively. In the concluding Section 6 we illustrate these results on specific examples, and discuss some mechanisms lying behind these decays.

Non-local Schrödinger operators and related random processes
We start by a general remark on notation. A ball centered in x ∈ R d and of radius r > 0 will be denoted by B(x, r). We write a ∧ b = min{a, b} and a ∨ b = max{a, b}. The notation C(a, b, c, ...) will be used for a positive constant dependent on parameters a, b, c, ..., dependence on the operator L or, equivalently, on the related Lévy process (X t ) t≥0 will be indicated by C(L) and C(X), while dependence on the dimension d is assumed without being stated explicitly. Since constants appearing in definitions and statements play a role, they will be numbered C 1 , C 2 , ... so that they can be tracked. We will also use the notation f ≍ Cg meaning that C −1 g ≤ f ≤ Cg with a constant C ≥ 1, while f ≍ g means that there is a constant C ≥ 1 such that the latter holds. By f ≈ g we understand that lim |x|→∞ f (x)/g(x) = 1. In proofs c 1 , c 2 , ... will be used to denote auxiliary constants.
Consider the pseudo-differential operator L with symbol ψ, defined by , and where the hats denote Fourier transform. We assume the symbol to be of the form where A = (a ij ) 1≤i,j≤d is a symmetric non-negative definite matrix, and ν is a symmetric Radon In the present paper we assume throughout without further notice that the measure ν(dz) has infinite total mass and it is absolutely continuous with respect to Lebesgue measure, i.e., ν(R d \ {0}) = ∞ and ν(dz) = ν(z)dz, with ν(z) > 0, z ∈ R d . For simplicity, we denote the density also by ν. It is a standard fact [19] that −L is a positive, self-adjoint operator with core C ∞ 0 (R d ), and the expression holds. Also, Spec(−L) = Spec ess (−L) = [0, ∞). Below we will use the symmetrization holds with suitable constants C 1 ∈ (0, 1), C 2 > 1, independent of A and ν. Also, it follows directly that r → H(r) is non-increasing and the doubling property H(r) ≤ 4H(2r), r > 0, holds, in particular, Ψ(2r) ≤ 4(C 2 /C 1 )Ψ(r), for all r > 0.
Furthermore, let Then for f s (x) = f (x/s) with f ∈ B and s > 0, we have Let f s (x) = f (x/s) for s > 0, and denote From the above observations it follows that A special feature of the operators of the form (2.1) with (2.2) is that they can be treated by a path integral approach. For each choice of the matrix A and the measure ν(dz) = ν(z)dz, the operator L is the infinitesimal generator of an R d -valued rotationally symmetric Lévy process, d ≥ 1, on the space of càdlàg paths (i.e., functions [0, ∞) → R d which are continuous from the right, having left limits). We denote by (X t ) t≥0 the Lévy process generated by L, the probability measure of the process starting in x ∈ R d by P x , and expectation with respect to P x by E x . It is a general fact that (X t ) t≥0 is a strong Markov process with respect to its natural filtration, and its characteristic function is given by where ψ is the symbol of −L as defined in (2.2) which, from a probabilistic perspective, is the Lévy-Khintchin formula for the class of Lévy processes we consider. In this context, A is the diffusion matrix describing the Brownian component of the process (X t ) t≥0 , and ν(dz) is the jump measure (called Lévy measure) describing the jump component, thus the Lévy triplet of the process is (0, A, ν). When A ≡ 0, the random process (X t ) t≥0 is a purely jump process, otherwise it contains an independent Brownian component.
The above properties jointly imply that (X t ) t≥0 is a strong Feller process, or equivalently, its one-dimensional distributions are absolutely continuous with respect to Lebesgue measure, i.e., there exist measurable functions p(t, x, y) = p(t, 0, y − x) =: p(t, y − x), corresponding to transition probability densities, such that P 0 (X t ∈ E) = E p(t, x)dx, for every Borel set E ⊂ R d , see [42,Th. 27.7]. Let D ⊂ R d be an open bounded set and consider the first exit time The transition probability densities p D (t, x, y) of the process killed on exiting D are then given by the Dynkin-Hunt formula The Green function of the process We also recall that when D ⊂ R d is a bounded open domain, the following formula due to Ikeda and Watanabe holds [18, Th. 1]: for every η > 0 and every bounded or non-negative Borel function f on R d such that dist(supp f, D) > 0, we have Furthermore, by [43,Rem. 4.8] we have with a constant C 4 independent of the process. For more details on Lévy processes and their generators we refer to [42,19].
In the remainder of the paper we will use the following class of Lévy processes and related operators L. Definition 2.1. Let (X t ) t≥0 be a Lévy process with Lévy-Khintchin exponent ψ as in (2.2) and Lévy triplet (0, A, ν), satisfying the following conditions.
(A3) There exists a constant C 7 = C 7 (X) such that sup x,y: |x−y|≥s/8 Assumption (A1) is a statement on the profile of ν, including a doubling property. Assumption (A2) is equivalent with e −t b ψ ∈ L 1 (R d ), for some t b > 0, which by the Markov property of (X t ) t≥0 extends to every t > t b . Assumption (A3) is a technical condition on the Green function for particular balls. The class of processes satisfying Assumptions (A1)-(A3) includes isotropic and anisotropic stable processes (corresponding to fractional Schrödinger operators), and layered stable processes (corresponding to another class of Lévy-operators), which will be discussed in Section 6. Throughout this paper we will use X-Kato class potentials. We say that the Borel function V : R d → R, called potential, belongs to K X associated with the Lévy process Also, we say that V = V + − V − is in X-Kato class whenever its positive and negative parts satisfy and by stochastic continuity of (X t ) t≥0 also K X loc ⊂ L 1 loc (R d ). Note that condition (2.10) allows local singularities of V . With an operator L given by (2.1) and an X-Kato class potential V , viewed as a multiplication operator, we call the operator defined by form-sum a non-local Schrödinger operator. To study the spectral properties of this operator, we use a Feynman-Kac type representation.

Feynman-Kac semigroups and generalized eigenfunctions
Consider the one-parameter family of operators By standard arguments based on Khasminskii's Lemma, see [31,, for an X-Kato class potential V it follows that there exist constants This implies that the operators T t , t > 0, are well defined on every L p (R d ), 1 ≤ p ≤ ∞, and Moreover, the family {T t : t ≥ 0} is a strongly continuous semigroup of operators on each L p (R d ), 1 ≤ p ≤ ∞, which we call the Feynman-Kac semigroup associated with the process (X t ) t≥0 and potential V . We define for those functions f ∈ L p (R d ) for which the limit exists. We denote the set of all such functions by Dom L p H and call it the L p -domain on H. It is known that H is a closed unbounded operator such that Dom L p H is dense in L p (R d ). For p = 2 the operator H can be identified with a self-adjoint operator as given by (2.11), defined in a quadratic form sense [7,Ch. 2]. A specific class of non-local Schrödinger operators given by H = Φ(−∆) + V , where Φ is a so called Bernstein function, has been defined and studied in [16,17]. Next we summarize some basic properties of the operators T t which will be useful below. Recall that for a function be a symmetric Lévy process with Lévy-Khintchin exponent satisfying (2.2) such that Assumption (A2) holds with some t b > 0, and let V be an X-Kato class potential. Then the following properties hold: (1) For all non-negative Borel measurable functions f, g we have (3) For all t ≥ 2t b , T t has a bounded measurable integral kernel u(t, x, y), symmetric in x and y, i.e., T t f (x) = R d u(t, x, y)f (y)dy, for all f ∈ L p (R d ) and 1 ≤ p ≤ ∞.
(4) For all t > 0 and f ∈ L ∞ (R d ), T t f is a bounded continuous function, i.e., {T t : t ≥ 0} is a strongly Feller semigroup.
(5) For all t > 0 the operators T t are positivity improving, i.e., The proof of these general properties is left to the reader, which can be obtained as an extension to the present set-up of the facts in [31, Sects. 3.2-3.3]. Note that we do not assume that p(t, x) is bounded for all t > 0, and thus in general the operators T t : Related to the Feynman-Kac semigroup, we define the potential operator by For further information on potential theory we refer to [4].
Let V be a decaying X-Kato class potential by which we mean V (x) → 0 as |x| → ∞. The main object of our investigations in this paper are the solutions ϕ ∈ L p (R d ), p ≥ 1, ϕ ≡ 0, of the equation (2.14) Hϕ = 0, or, equivalently, is the edge of the essential spectrum of H and (2.14) can be understood as the eigenvalue equation at Σ. Whenever the solution ϕ to (2.14) is such that ϕ ∈ Dom L 2 H, we call it a zero-energy eigenfunction (or zero-energy bound state) and then 0 is an eigenvalue. Otherwise, we call both (by a slight abuse of language) a zero-resonance. Throughout we will assume that every zero-energy eigenfunction ϕ is L 2 -normalized so that for every t > 2t b and p ≥ 1. Therefore, any solution ϕ to (2.14)-(2.15) is a bounded and continuous function, in particular, it makes sense to study their pointwise estimates.
Below we will make frequent use the following resolvent representation of solutions of (2.14). Choose θ > 0. Then by multiplying both sides of (2.15) and integrating with respect to time, we obtain Combining this with (2.13) applied to f = ϕ for an arbitrary open set D ⊂ R d and x ∈ D, and using the strong Markov property of the Lévy process (X t ) t≥0 , we readily obtain which will be a fundamental formula in what follows.

Self-improving estimates
In this section we show some key estimates which will serve to proving our main results below. Let R 0 ≥ 1 be a fixed number. Throughout this section we will consider non-increasing functions In what follows we will also use the notations and ω d will denote the volume of a unit ball.
Moreover, suppose that f is a bounded non-negative function on R d such that for a constant C 13 > 0, and let η := C 10 C 13 . If Proof. Observe that (3.7) follows directly from (3.6). We only need to prove (3.6). Let and R > 2R 0 be large enough such that By (3.4) and the part of (3.1) stated for u, for every |x| ≥ R we have From this we obtain the two independent estimates They give, respectively, Notice that if this holds, then by taking the limit p → ∞ it follows that which is the bound stated in the lemma. To prove (3.12) we make induction on p ∈ N. First observe that for p = 1 the estimate (3.12) follows from (3.11). Suppose now that (3.12) holds for p − 1 ∈ N. By using (3.10) and the induction hypothesis, we see for all |x| ≥ R that By the substitution h u,v (r) = u we obtain and thus which is the claimed bound.
It is direct to check that under the non-restrictive assumption that the function K u,v is almost non-increasing, i.e., there exists The next lemma deals with a lower bound on positive functions satisfying an integral inequality.
for a constant C 15 > 0. Then In particular, and by symmetrization of the second integral, for all |x| > 1. Next we prove that for every p ∈ N Clearly, if (3.17) is true for every p ∈ N, then estimate (3.14) also holds.
We use induction on p. For p = 1 the inequality (3.17) is an immediate consequence of (3.16). Suppose now that the induction hypothesis holds for some p ∈ N. By (3.16)-(3.17) and rotation symmetry we have we conclude that as required.
4. Decay of zero-energy eigenfunctions for potentials positive at infinity

Upper bound
Now we turn to discussing the spatial decay properties of eigenfunctions of non-local Schrödinger operators presented in Section 2. In this section we consider decaying potentials that are non-negative at infinity in the following sense: (A4) V is an X-Kato class potential such that V (x) → 0 as |x| → ∞, and there exists r 0 > 0 such that V (x) ≥ 0 for |x| ≥ r 0 .
It will be useful to introduce the notation Notice that V * (x) is a radial and non-increasing function such that V * (x) ≥ 0, |x| ≥ r 0 . We will need a uniform estimate of functions that are harmonic with respect to the operator H. Since our approach is via a Feynman-Kac type stochastic representation, we use throughout the following probabilistic definition. Let D be an open subset of R d and let V be a Kato-class potential such that V (x) ≥ 0 on D. We call a non-negative Borel function f on R d an (X, V )harmonic function in the domain D if for every open set U with its closure U contained in D, and a regular (X, V )-harmonic function in D if (4.1) holds for U = D (where τ U is the first exit time from U ). By the strong Markov property every regular (X, V )-harmonic function in D is (X, V )-harmonic in D. Whenever V ≡ 0 in D, we refer to f as a (regular) X-harmonic function.
An initial version of the type of bound we prove below has been first obtained in [25,Lem. 3.1] and it can be derived from the general results in [4]. Here we need a variant suitable for the purposes of the present paper. Note that the following estimate does not exclude the case V ≡ 0.
We can now make use of the above estimates and the technical results obtained in the previous section to derive upper bounds for the zero-energy solutions for potentials satisfying (A4).
Next we prove the remaining part (3). First note that ϕ ≥ 0 and similarly as in (4.6) we have Finally, by Hölder inequality with suitable p, q, which completes the proof.
As it will be seen below, Theorem 4.1 (1)-(2) gives sharp upper bounds, provided ϕ ≥ 0 (compare with the lower bounds in Theorem 4.3). We will now prove that if ϕ is antisymmetric with respect to a given (d − 1)-dimensional hyperplane π in R d with 0 ∈ π, and has a definite sign on both of the corresponding half-spaces, then the decay rate in (1) of |ϕ| at infinity far away from π improves, while the upper bound in (3) remains unchanged. By rotating the coordinate system if necessary, we may assume that π = {x ∈ R d : x 1 = 0}. We make the assumption and ϕ (x 1 , ..., x d ) ≥ 0 whenever x 1 > 0. (4.8) The next theorem deals with the case when ϕ has no definite sign, but does satisfy (4.8).
To establish (2), observe that similarly as above we have for |y − x| < |x| 4 and |x| ≥ 2r 0 that which yields for every |x| ≥ 2r 0 . By the first inequality in (4.13) and Corollary 4.1 (2), we have It suffices to estimate the latter integral. By the definition of the function f x and the second inequality in (4.13), we have Similarly as above, (2.8) implies that for all |z − x| < |x|/4, x 1 > 2r 0 , we have Inserting this estimate into (4.14), the claimed bound in (3) follows.
As it will be seen in specific cases in Section 6 below, by iterating the bounds in (1)-(2), we can often get the bound with 1/|x| instead of Ψ(1/|x|) V * (x) ∨ 1 |x| .

Lower bound
For a given potential V satisfying Assumption (A4) denote Clearly, V * (x) is a radial and non-increasing function. The auxiliary function Λ V has the suggestive meaning of lower envelope of the mean lifetime of the Lévy process under the potential V in a ball B(x, |x|/2), that is, The first lemma gives a lower estimate on Λ B(x,|x|/2) (x).
To conclude, it suffices to observe that when V * (x) ≥ Ψ 1 |x| , we have , |x| ≥ 2r 0 , With this lemma we also have the following estimate. Let (X t ) t≥0 be a Lévy process with Lévy-Khintchin exponent ψ as in (2.2) such that Assumptions (A1)-(A3) hold and let V be a potential satisfying (A4); specifically let (A4) hold with some r 0 > 0. Then for every positive solution ϕ of (2.14) there exists C > 0 such that In particular, Proof. By applying the resolvent formula (2.17) with any θ > 0 and D = B(x, |x|/2), and letting θ ↓ 0, we get Then by the Ikeda-Watanabe formula (2.8) and (A1), An application of Lemma 4.2 completes the proof of the first inequality. The second estimate is a direct consequence of the first.
We are now in the position to state the main theorem of this subsection. Theorem 4.3. Let (X t ) t≥0 be a Lévy process with Lévy-Khintchin exponent ψ as in (2.2) such that Assumptions (A1)-(A3) hold and let V be a potential satisfying (A4); specifically let (A4) hold with some r 0 > 0. Let ϕ be a positive solution of (2.14). Then the following hold.

Decay of zero-energy eigenfunctions for potentials negative at infinity
Now we turn to discussing the spatial decay properties of eigenfunctions of non-local Schrödinger operators with decaying potentials that are negative at infinity.
(A5) Let V ∈ L ∞ (R d ) be such that there exists r 0 > 0 and C > 0 such that Notice that under (A5) we have V (x) → 0 as |x| → ∞, i.e., V is indeed a decaying potential. It also covers potentials with compact support such as potential wells.
We will now prove a counterpart of Theorem 4.2 (3) in the case when the potential is negative at infinity and the negative nodal domain of ϕ is a subset of a given half-space. By rotating the coordinate system if necessary, without loss of generality we can assume that there exists l ∈ R such that supp ϕ − ⊂ y ∈ R d : y 1 < l . (5.1) Theorem 5.1. Let (X t ) t≥0 be a Lévy process with Lévy-Khintchin exponent ψ as in (2.2) such that Assumptions (A1)-(A3) and (A5) hold; specifically, let (A5) hold with some r 0 > 0. Moreover, suppose that for every ε ∈ (0, 1) there exists M ≥ 1 such that If ϕ is a solution of (2.14) such that ϕ ∈ L p (R d ), for some p > 1, and (5.1) holds, then for every ε ∈ (0, 1) there exist C > 0 and R > 3r 0 such that Moreover, if ϕ(l + x 1 , x 2 , ..., x d ) = −ϕ(l − x 1 , x 2 , ..., x d ), x ∈ R d , with l given by (5.1), then there exists R > 3r 0 ∨ |l| such that the same upper bound is true for ϕ(x) replaced by |ϕ(x)|, whenever |x 1 | ≥ R.
A further discussion of the potentials negative at infinity in some specific cases will be made at the end of Section 6.1 below. 6. Specific cases and decay mechanisms
First we consider potentials that are positive at infinity in the sense of (A4) and look at positive solutions of (2.15).
(2) If β ≥ α, then there exist γ ∈ (0, 1) and a constant C 3 > 0 such that On the other hand, if ϕ ∈ L p (R d ) for some p > 1, then there exists C 4 > 0 such that This has the following implication. Corollary 6.1. Under the assumptions of Theorem 6.1 we obtain that ϕ ∈ L 1 (R d ) if and only if α > β. Remark 6.1. Let V be a potential positive at infinity, and V (x) ≍ |x| −β as |x| → ∞, and consider the fractional Laplacian (−∆) α/2 , 0 < α < 2. Although the constants are hard to control in sufficient detail, a calculation using the above estimates shows that if β ≥ α and C 1 C 15 C 5 C 10 ≥ d, then zero is not an eigenvalue of (−∆) α/2 + V . We note that C 10 and C 15 play the more important role here, giving some best constants involving the jump doubling domination rate and another ratio related to jump activity. Also, from Theorem 6.1 (2) we see that whenever (0, 1) ∋ γ ≥ d 2 , which may occur when d = 1, the operator H has no zero eigenvalue.
It can already be seen from the above theorem that there is a transition in the localization properties of ϕ when α > β changes to α ≤ β. For a closer understanding of this transition around α ≈ β, we consider a more refined class of potentials.
Theorem 6.2. Let L (α) , 0 < α < 2, be a pseudo-differential operator determined by (6.1) and V be an X-Kato class potential for which there exists r 0 > 0 such that V (x) > 0 and V (x) ≍ |x| −α (log |x|) δ , for |x| ≥ r 0 , with some δ > 0. Suppose that there exists a positive function ϕ ∈ C b (R d ) which is a solution of (2.15). Then the following hold.
(3) If δ ∈ (0, 1), then there exist 0 < γ 1 ≤ 1 ≤ γ 2 and constants C 6 , C 7 > 0 such that In particular, ϕ ∈ L p (R d ), for every p > 1, but ϕ / ∈ L 1 (R d ). (4) If δ ≤ 0, then we have exactly the same bounds and L p -properties as in (2)  From the results above it is seen that the possible localization properties of the positive zeroenergy eigenfunctions or zero-resonances for decaying potentials positive at infinity splits naturally into disjoint regimes representing the following three different scenarios. (For the simplicity of the discussion here, we assume that V is a potential that is positive at infinity and regular enough so that V (x) ≍ V * (x) ≍ V * (x) far away from the origin). Let r 0 > 0 be large enough, and define h(r) = V (x) = 0 and the ratio ν(x) V (x) is integrable at infinity, then for large enough |x|. In particular, ϕ ∈ L 1 (R d ). Clearly, in this case the corresponding function h is bounded. V (x) at infinity breaks down, we have h(r) → ∞ as r → ∞. Hence (1) is no longer true and the function h contributes into the behaviour of ϕ at infinity like for large enough x, with some 0 < γ 1 ≤ 1 ≤ γ 2 and C 1 , C 2 > 0. Observe that Scenario (1) differs from (2) by the boundedness of h.
(2) If κ = d 2 , then by (1.2) we have V d/2 (x) ≍ |x| −α log |x|, and Theorem 6.2 gives that there exist 0 < γ 1 ≤ 1 ≤ γ 2 and constants C 1 , C 2 > 0 such that Hence we are now in scenario (2) above. In particular, this means that we recover the behaviour ϕ κ (x) ≍ 1/|x| d as in (1.1), with a near miss dependent on how close γ 1 ≤ 1 ≤ γ 2 are to each other. Clearly, this is a marginal and the most delicate case, and closing the gap would require a more refined analysis. Note that this special case coincides with the threshold κ for which ϕ κ / ∈ L 1 (R d ).
When the solution of the eigenvalue equation (2.15) is antisymmetric with respect to a given hyperplane and has a definite sign in each nodal domain, then at least far away from this nodal plane some of our upper estimates improve significantly. If the solution ϕ is no longer positive on R d but it satisfies (4.8), then the upper bounds in Theorems 6.1 (1)-(2) and Theorem 6.2 (1)-(3) also hold with ϕ replaced with |ϕ| (this is a consequence of Theorem 4.1 (1)-(2) and Theorem 4.2 (2)). However, due to Theorem 4.2 (1), in this case the upper bound for |ϕ| improves and the upper estimates in Theorems 6.1 (1) and Theorem 6.2 (1)-(2) upgrade as follows.
Corollary 6.2. The following situations occur: (1) Under the assumptions of Theorem 6.1, if β < α and ϕ ∈ C b (R d ) is a solution of (2.15) such that (4.8) holds, then there is C 1 > 0 such that (2) Under the assumptions of Theorem 6.2, if δ > 1 and ϕ ∈ C b (R d ) is a solution of (2.15) such that (4.8) holds, then there is C 2 > 0 such that Example 6.2. Comparing these upper bounds with the exact behaviours of ϕ κ for l = 1 and the potentials V κ (x) ≍ |x| −β in (1.1) with β = d + α + 2 − 2κ, κ ∈ ( d+2 2 , d+α+2 2 ), we see that in this case our result is not as sharp as in the cases above, however, it is still remarkably close to the exact rates. Indeed, we get here the upper bound |ϕ κ (x)| ≤ C 1 (1+|x|) 4(κ−1)−d , while the true behaviour is |ϕ κ (x)| ≍ 1 |x| 2κ−1 . We emphasize that the symmetry/antisymmetry properties of eigenfunctions as in (4.8) are of much interest in spectral theory and are known to have important consequences (see, e.g., [22]).
Roughly speaking, the above result says that for potentials that are negative in a neighbourhood of infinity and for antisymmetric solutions with nodal domains in the corresponding hyperplanes, L p -integrability, with p > 1, always gives a polynomial decay of order near to d/p, far from the antisymmetry axis. This result can be compared with the examples in (1.1). Specifically, for every i = 1, 2, ..., d and l = 1 we can take P i (x) = cx i and ϕ κ,i (x) = cx i (1 + |x| 2 ) κ , with κ ∈ 1, d−α 2 + 1 . As seen in (1.3), this leads to the case of potentials V κ,α negative in a neighbourhood of infinity. We clearly have |ϕ κ,i (x)| ≤ C(1 + |x|) −d/p and ϕ κ,i ∈ L p (R d ), for every p > d 2κ−1 .
Moreover, it follows from [48] that the probability transition densities p(t, x) exist and satisfy This, in particular, gives (A2) and, together with [25,Lem. 2.2], implies that Assumption (A3) holds as well. The operators L (α,γ) generate the class of layered α-stable processes.
We get the following result for potentials that are positive at infinity. Theorem 6.4. Let L (α,γ) , 0 < α < 2, γ > 2, be a pseudo-differential operator determined by (6.6) and V be an X-Kato class potential for which there exists r 0 > 0 such that V (x) > 0 and V (x) ≍ |x| −β , for |x| ≥ r 0 , with some β > 0. Suppose that there exists a positive function ϕ ∈ C b (R d ) which is a solution of (2.15). Then the following hold: (1) If β < 2, then there exist constants C 1 , C 2 > 0 such that In particular, ϕ ∈ L p (R d ), for every p ≥ 1.
(2) If β ≥ 2, then there exists a constant C 3 > 0 such that On the other hand, if ϕ ∈ L p (R d ) for some p > 1, then there exists C 4 > 0 such that For potentials negative at infinity a result similar to Theorem 6.3 holds as well.

Decay mechanisms
From the above it is seen that the decay of ground states at zero eigenvalue depends essentially on two factors. On the one hand, the sign of the potential at infinity makes a qualitative difference, and as seen in the case of classical Schrödinger operators, it has an impact even on the existence of ground states. From the decay results above one can appreciate that a positive tail of the potential has a (soft) bouncing effect tending to contain paths in compact regions, while a negative potential leaves more room for the paths to spread out to infinity. This difference makes the analysis of potentials negative at infinity much more difficult than of potentials positive at infinity.
On the other hand, the decay depends on some mean times spent in some regions by the paths. Using (2.9) and (4.15), we can give another interpretation of the results above, further highlighting the mechanisms. Assume, for simplicity, that V (x) ≍ V * (x) ≍ V * (x), for large enough |x|. Then we see that the conditions involving the ratios in Theorems 4.1-4.3 actually refer to a balance of the mean survival times of paths in a ball B(x, |x|/2) under the potential versus in the ball B(0, |x|) free of the potential. Due to the doubling property, these two times are comparable, and describe specific global lifetimes (note that B(x, |x|/2) can also be replaced by B(x, c|x|), 0 < c < 1, without qualitatively changing the results). This is in sharp contrast with the case of confining potentials or decaying potentials leading to a strictly negative and sufficiently low-lying ground state eigenvalue, where the decay is governed by local lifetimes as given in (1.5). When as in Scenarios (1)-(2) above, the potential has a relatively pronounced effect, making the paths favour (large) neighbourhoods of the origin than (large) neighbourhoods of far out points. This is reflected in the decay behaviours of ϕ by V entering explicitly in Scenario (1) discussed in Section 6.1. When, however, as in Scenario (3), the effect of the potential is weak also in relative terms, and the two lifetimes evolve on the same scale, being very near to (though clearly differing from) the situation of free fluctuations and absence of a ground state.