{'fraction_non_alphanumeric': 0.06285971223021583, 'fraction_numerical': 0.01991906474820144, 'mean_word_length': 3.542121923823139, 'pattern_counts': {'":': 0, '<': 35, '<?xml version=': 0, '>': 75, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Quantum Systems at the Brink: Existence of Bound States, Critical Potentials and Dimensionality

One of the crucial properties of a quantum system is the existence of bound states. While the existence of eigenvalues below zero, i.e., below the essential spectrum, is well understood, the situation of zero energy bound states at the edge of the essential spectrum is far less understood. We present necessary and sufficient conditions for Schr\"odinger operators to have a zero energy bound state. Our sharp criteria show that the existence and non-existence of zero energy ground states depends strongly on the dimension and the asymptotic behavior of the potential. There is a spectral phase transition with dimension four being critical.


INTRODUCTION
The existence of bound states plays a crucial role for the properties of quantum systems. Of special importance is the ground state, i.e., the eigenfunction corresponding to the lowest eigenvalue of the Hamiltonian describing the system. In this paper we consider a Schrödinger operator of the form is a real-valued potential such that the operator H is a well-defined self-adjoint realization of the formal differential operator −∆ + V which is bounded from below. Moreover, we need that eigenfunctions of H are continuous. The precise conditions are given in Assumption 1.1 below.
We are particularly interested in the special case when the ground state energy of the Schrödinger operator H is at the threshold of the essential spectrum. By shifting the potential by a constant, one can assume that the essential spectrum of H starts at zero. One also often assumes that the potential V decays to zero at infinity such that the essential spectrum σ ess (H) = [0, ∞), see Remark 1. 10. Under these conditions the zero-energy level is at the edge of two regions with very distinct behavior: the point and the continuous spectrum. It is well-known that positive eigenvalues embedded in the continuum appear only due to a special combination of oscillations and slow decay of the potential. This goes back to [54], see also [13,17,24,49] and the references therein. Also, with the help of the min-max theorem, the existence and non-existence of eigenvalues below zero is well-understood, see, e.g., [47].
Whether zero is actually a threshold eigenvalue, i.e., an eigenvalue at the edge of the continuum is a very difficult problem, in general. Early results on existence or non-existence of zero-energy eigenvalues go back to [1,25,29,33,32,38,42,45,46,51]. In [25] the authors studied the behavior of resonances and eigenstates at the zero-energy threshold in d = 3. Furthermore, based on the remark of a referee they note that resonances cannot exist in dimensions d > 4 based on properties of Riesz potential. However their approach is not applicable for d = 4. For slowly decaying negative potentials which, amongst other conditions, obey V (x) ∼ −c|x| −γ for some c > 0 and 0 < γ < 2 in the limit |x| → ∞, the non-existence of zero energy eigenstates was shown in [14,16], while it was noted in [10] that a long range Coulomb part can create zero energy eigenstates, see also [41,55]. An analysis of eigenstates and resonances at the threshold for the case of certain nonlocal operators appeared in [27].
In [9] it has been shown that, for Schrödinger operators on L 2 (R 3 ) with spherical symmetric potentials V ∈ L p (R 3 ) with p > 3/2 whose positive part satisfies V + (x) ≤ 3/(4|x| 2 ) for |x| large enough, zero is not an eigenvalue corresponding to a positive square integrable ground state eigenfunction. This extends to potentials with V + ≤ |x| −2 3/4 + ln −1 (|x|) near infinity in R 3 , the constants 3/4 and 1 are optimal. For similar results see [19], which reproved a slightly weaker non-existence result compared to [9] and additionally showed that if V (x) ≥ C|x| −2 for some constant C > 3/4 and |x| large then zero is an eigenvalue for critical potential, see Definition 1.5. Thus a repulsive part can stabilize zero energy bound states of quantum systems.
Strictly speaking, the paper [9] deals with continuous potentials on R 3 but they note that the condition V ∈ L p (R 3 ) with p > 3/2 is enough to guarantee continuity of ground states, due to a Harnack inequality for positive eigenfunctions. We also note that compactly supported zero-energy eigenfunctions were constructed in [30,34] for potential V ∈ L p (R d ) with p < d/2 and compact support. For these potentials, a Harnack inequality for the ground state cannot hold.
In this paper we significantly extend all previous results, in particular the ones of [9] and [19], by proving a family of sharp criteria for the existence and non-existence of zero energy ground states at the edge of the essential spectrum for Schrödinger operators in arbitrary dimensions. In particular, our results apply to Schrödinger operators with a so-called virtual level at zero energy and they explain when such a virtual level is a true ground state or when it is a resonance.
Our results clearly explain why increasing the dimension makes it easier for a virtual level to be a true ground state. In particular, our work explains why dimension d = 4 is critical. Dimension four shares some similarity with the case of lower dimensions but higher order corrections from our criteria are needed to settle this case.
Our main assumption on the potential V are given by Assumption 1.1. The potential V is in the local Kato-class K d,loc (R d ) and the negative part V − = sup(−V, 0) is relatively form small w.r.t. −∆ + V + , i.e., there exist 0 ≤ a < 1 and b ≥ 0 such that is the domain of the multiplication operator √ V + on L 2 (R d ), also called the form domain of V + and often written as Q(V + ).
Note that what we call relatively form small is usually called relatively form-bounded with relative bound a < 1. We will call a potential W infinitesimally form bounded (w.r.t. −∆ + V + ) if for all a > 0 there exist b ≥ 0 such that the positive and negative parts W ± satisfy (1.2) (with V − replaced by W ± ).
, whose definition is recalled below, see (1.14), contains most, if not all physically relevant potentials. This assumption is only made to guarantee that all weak local eigenfunctions of H are continuous, see [3,48,52].
One could relax the assumption that if some other condition guaranteed that weak local eigenfunctions of H are continuous. In fact, it would be sufficient to have that eigenfunctions are locally bounded and that a ground state of H is bounded away from from zero on compact sets. As will become clear from the proofs, we can allow for severe local singularities. E.g., it is enough to assume that V is in the local Kato-class outside some compact set K ⊂ R d .
If V ± ∈ L 1 loc and (1.2) holds, the KLMN theorem shows that there exists a unique selfadjoint operator H, informally given by the differential operator −∆ + V , such that its quadratic form, which with a slight abuse of notation we write as ψ, Hψ := ∇ψ, ∇ψ . Moreover, it is closed and bounded from below on the quadratic form domain Q(H). See also the discussion at the beginning of the next section.
To formulate our main results we recall the definition of the iterated logarithms ln n defined, for natural numbers n ∈ N, by ln 1 (r) := ln(r) for r > 0 and inductively for r > e n by ln n+1 (r) := ln(ln n (r)). Here e 0 = 0 and e n+1 = e en . Our first main result can be summarized as follows Theorem 1.3 (Absence of a zero energy ground state). Assume that the potential V satisfies Assumption 1.1 and σ(H) = [0, ∞). If for some m ∈ N 0 and R > e m for all |x| ≥ R, then zero is not an eigenvalue of the Schrödinger operator H.
As usual the empty product is 1 and the empty sum equals 0.
Remark 1.4. In particular, if inf σ ess (H) = 0 then Theorem 1.3 shows that zero is not an eigenvalue at the edge of the essential spectrum. Theorem 1.7 below shows the sharpness of condition (1.4) on the potential V for the absence of an embedded ground state at the edge of the essential spectrum.
Our second main result shows that critical potentials create zero energy ground states if they are not too small at infinity. We call a potential W ≥ 0 nontrivial, if it is strictly positive on a set of positive Lebesgue measure. Definition 1.5 (Critical potential). The potential V is critical if the Schrödinger operator H has spectrum σ(H) = σ ess (H) = [0, ∞) and for all nontrivial compactly supported potentials W ≥ 0 which are infinitesimally form bounded with respect to −∆ + V + the family of operators H λ = H − λW has essential spectrum σ ess (H λ ) = [0, ∞) and a negative energy bound state for all λ > 0.
Remark 1.6. The potential V is called subcritical, if the Schrödinger operator H has spectrum σ(H) = σ ess (H) = [0, ∞) and there exist a nontrivial potential W ≥ 0, which is infinitesimally form bounded with respect to −∆ + V + , such that H − λW ≥ 0 for some λ > 0. Theorem 1.7 (Existence of a zero energy ground state for critical potentials). Assume that the potential V satisfies Assumption 1.1 and that it is critical. If for some m ∈ N 0 , ǫ > 0, for all |x| ≥ R, then zero is an eigenvalue of H.
Remark 1.8. Clearly, the right hand sides of (1.4) and (1.5) are, for each fixed n ∈ N complementary. Thus our criteria for existence and non-existence of zero energy ground states at the edge of the essential spectrum are sharp! Considering the simplest case m = 0 we have for the absence and for the existence with ǫ > 0 and all |x| large enough. For d = 3 this recovers the results proved in [19] for the special case of three dimensions. Using the higher order corrections from Eqs. (1.4) and (1.5) we obtain a sharp distinction between existence and non-existence in the case of a critical potential. For example, the cases m = 1, 2 show that if for all large enough |x|, then zero will not be a ground state eigenvalue. Conversely, for critical potentials the bound for all large enough |x| and some ǫ > 0 implies that zero is a ground state eigenvalue. Using d = 3 in (1.8) recovers the non-existence result of [9]. The d = 3 case in (1.10) provides a complementary existence result which was missing in [9]. More importantly, our results provide, to arbitrary order, a whole family of complementary sharp criteria which are not restricted to three dimensions and our proofs are considerably simpler than the approaches based on delicate estimates for Green's functions.
One often says that a Schrödinger operator with a critical potential has a virtual level (at zero energy), see, e.g. [5,6,7]. Theorem 1.3 shows that such a virtual level is not a bound state of H if V obeys the bound (1.4), that is, it is a so-called zero energy resonance. Conversely, Theorem 1.7 shows that a virtual level is an eigenvalue at the edge of the essential spectrum when the potential V satisfies the complementary bound (1.5). Remark 1.9. In Appendix A we construct a family of potentials V α,d on R d for α ∈ R and d ∈ N such that the Schrödinger operator H α,d = −∆ + V α,d has spectrum σ(H α,d ) = [0, ∞). Moreover, V α,d is subcritical for α < 0 and critical for α ≥ 0. The Schrödinger operator H α,d has a zero energy resonance for 0 ≤ α ≤ 1 and a zero energy bound state for α > 1 in any dimension.
Remark 1.10. The operator H λ is well-defined with quadratic form methods for all λ, see Remark 2.1. In order to guarantee that σ ess (H λ ) = [0, ∞) in Definition 1.5, some decay of the potential V is required. A well-known sufficient criteria for this is that V is relatively form compact with respect to the kinetic energy P 2 = −∆, see [53]. This also implies that V is infinitesimally form bounded, i.e., relatively form small with relative bound zero, w.r.t. P 2 = −∆, which excludes Hardy type potentials.
A much less restrictive criterium for σ ess (H) = [0, ∞) only assumes that V vanishes asymptotically with respect to the kinetic energy. More precisely, if | ϕ, V ϕ | ≤ a n ∇ϕ 2 + b n ϕ 2 (1.12) for all ϕ ∈ H 1 (R d ) with support supp(ϕ) ⊂ {|x| ≥ R n } for some sequences 0 ≤ a n , b n → 0, and R n → ∞ as n → ∞, then σ ess (H) = [0, ∞), see [4,26]. This criterion is clearly in line with the physical heuristic that only the asymptotic behavior of the potential near infinity determines the essential spectrum and it allows for strongly singular potentials which are not infinitesimally form bounded. It also shows that σ ess (H λ ) = σ ess (H) = [0, ∞) for all λ > 0 when W has compact support and is infinitesimally form bounded w.r.t. −∆ and V is form small w.r.t. −∆ and satisfies (1.12).
Remark 1.11. The bounds on the potential in Theorems 1.3 and 1.7 are similar in spirit to logarithmic corrections to the Hardy inequality. For ψ ∈ C ∞ 0 R d \ {|x| < e m } and one has ∇ψ, ∇ψ ≥ ψ, see [43], which also discusses conditions on the potential such that −∆ + V has infinitely many, respectively finitely many negative eigenvalues. Bounds on the number of negative eigenvalues are given in [39,40]. Certain logarithmic refinements of Hardy's inequality have been used to study the existence of resonances of Schrödinger operators and the Efimov effect in low dimensions, see [8]. Moreover, in dimension d = 4, the leading order term vanishes and the next leading order term with m = 1 becomes dominant. Thus the four dimensional case is critical. Nevertheless, since the new leading order term for d = 4 is also positive, the four dimensional case is similar to the case of lower dimensions. In particular, non-positive potentials V cannot support zero energy ground states in dimensions d ≤ 4 while in dimension d ≥ 5 non-positive critical potentials have zero energy ground states.
Hence non-positive critical potentials will always create resonances in dimension d ≤ 4, while in dimension d ≥ 5 they have zero energy ground states unless their negative part is so long range such that the bound (1.5) does not hold anymore. Nevertheless, in dimension d ≤ 4 a 'long-range' positive tail of the potential can create zero energy ground states. See also the discussion in Section 2 of [22].
In particular, assume that the potential V is infinitesimally form bounded w.r.t. −∆ and has compact support. Then σ ess (−∆ + βV ) = [0, ∞) for all β ≥ 0, see [4,53], and a simple application of the min-max principle shows that as soon as negative eigenvalues of −∆+ βV exist, they are decreasing in β > 0, see [47, Proposition after Theorem XIII.2, page 79]. Let β 0 > 0 be the value of the coupling constant when the ground energy of −∆ + βV hits zero. Theorem 1.3 shows that −∆ + β 0 V has a zero energy resonance when d ≤ 4 and Theorem 1.7 shows that it has a zero energy ground state in dimension d ≥ 5. The asymptotic of the eigenvalues of the perturbed operators −∆ + βV in β − β 0 was studied in [31] for all dimensions.
The structure of our paper is as follows: In Section 2 we present all the necessary technical tools to precisely formulate our main results. Theorem 1.3 is proven in Section 3. The proof is by contradiction, assuming that a zero energy ground state exists and then deriving a lower bound which shows that it cannot be square integrable. To construct such a lower bound one only needs to know that a ground state, if it exists, can be chosen to be positive and that it is locally bounded away from zero. It is well-known that ground states of a Schrödinger operator H in L 2 (R d ) are unique, up to global phase, and can be chosen to be strictly positive as soon as they exist, see [15,18] or [47,Section XIII.12]. Thus, if one knows that the ground state is bounded away from zero, one can relax the condition on V to V ∈ L 1 loc (R d ) The assumption that V is in the local Kato-class is only needed to guarantee that eigenfunctions of H are continuous, see [3,52] and also [48]. This continuity then guarantees that the positive ground state is bounded away from zero on compact sets. The proof of Theorem 1.7 is given in Section 4. The main tool is an upper bound for the spacial decay of ground states of the approximating Schrödinger operators H λ , see Definition 1.5, which is uniform in λ > 0.
Since it will be necessary to have a positive ground state for the non-existence proof, we cannot prove the absence of ground state under symmetry constraints which destroy the positivity of ground states, such as fermionic particle statistics. However, the existence proof still works under symmetry restrictions, see Remark 4.10.
In Appendix A we construct an explicit example of a family of potentials which exhibits all possible different scenarios.
Lastly, recall that the Kato-class K d is given by all real-valued potentials V such that in The Kato class in one dimension is given by K 1 := L 1 loc,unif (R), the space of uniformly locally integrable functions on R. We say that the potential V is in the local Kato-class Moreover, it is well-known that any potential V ∈ K d is infinitesimally form small with respect to −∆, see [11].
, and V − ∈ K d then all of the claims of Assumption 1.1 hold. This class of potentials is large enough to include most, if not all, physically relevant potentials, except maybe for some highly oscillatory potentials.

DEFINITIONS AND PREPARATIONS
Assume that V ± ∈ L 1 loc (R d ) and (1.2) holds for V − . The KLMN theorem [47,53] then shows that there exists a unique self-adjoint operator H corresponding to a quadratic form with the usual slight abuse of notation. Here ψ ∈ Q(H) is the quadratic form domain of the multiplication operator V + . Since with respect to the norm 3) see [11,36]. In addition, Friedrich's extension theorem, see for example [53,Theorem 2.13], implies that the operator H and its domain D(H) are given by . So for any 0 < λ 0 < (1 − a 1 )/a 2 , we can construct the family of Schrödinger operators H λ as the unique self-adjoint operator given by the quadratic forms One can relax the conditions on V to hold only on a connected, open set U ⊂ R d , which contains infinity. In this case one assumes In this case H is the Schrödinger operator (with Dirichlet boundary conditions) defined by the quadratic form r.t. the norm given in (2.3). The same holds for H δ and any δ > 0 small enough. Now assume that the real-valued potential V ∈ L 1 loc (R d ), that its negative part V − is form small w.r.t, −∆ + V + , i.e., (1.2) holds, and let H be the associated Schrödinger operator defined by quadratic form methods as above. For an open set U ⊂ R d we consider weak (local) eigenfunctions of H at energy E, i.e., (weak local) solutions of the Schrödinger equation We are mainly interested in the case that E = 0.
With a slight abuse of notation, we denote by ϕ, Hψ the sesquilinear form given by whenever the right hand sides makes sense. This is the case if ϕ, ψ ∈ Q( is the vector space of functions which are locally (in U) in the quadratic form domain of H. Moreover, is the set of functions in Q(H) with compact support inside U. If ϕ ∈ Q U c (H) and ψ ∈ Q U loc (H), then the integral on the right-hand-side of (2.8) can be restricted to the set U.
. Similarly, one can define the local domain of H, relative to some open set U ⊂ R d , by Note that the definitions of Q U loc (H) and D U loc (H) are consistent in the sense that for any χ ∈ C ∞ 0 (U) one has χψ ∈ Q(H) (even χϕ ∈ Q U c (H)) for any ψ ∈ Q(H) and χψ ∈ D(H) for any ψ ∈ D(H). This is clear when is wrong in general, since the construction of the Schrödinger operator H with the help of quadratic forms allows for rather singular potentials V.
Thus we define weak solutions, supersolutions and subsolutions of (2.7) in the following quadratic form sense.  (2.11). Proof. This is probably a standard argument for weak eigenfunctions, but not standard for weak local eigenfunctions. Let f ∈ L 2 (R d ) and ψ be a weak solution of the equation Hψ = f , i.e., ϕ, Hψ = ϕ, f (2.17) for all ϕ ∈ Q(H). Then for any λ ∈ R we have for all ϕ ∈ Q(H). Since H is bounded from below, all large enough λ will be in the resolvent set of H. So for all large enough λ we can choose This holds for all ξ ∈ L 2 (R d ), so Note that if ψ is a weak eigenfunction of H, at energy E, then we can use f = Eψ. Thus weak eigenfunctions are eigenfunctions in the domain of H.
Using this and the definition (2.1) of the quadratic form in (2.22) yields for all ϕ ∈ Q(H). Adding again ϕ, λχψ on both sides and choosing ϕ = (H + λ) −1 ξ with λ large that enough, one sees that for any χ ∈ C ∞ 0 (U). Thus ψ ∈ D U loc (H). Again, using f = Eψ shows that any weak local eigenfunctions of H at energy E is locally in the domain of H.
Finally let us note that the definition of critical potential and virtual levels are rather natural. It is easy to see that any potential which creates a zero energy ground state is critical.
and that V − is form small and 0 ≤ W is infinitesimally form small w.r.t. −∆ + V + . Furthermore let H and H λ , 0 < λ ≤ λ 0 , be the associated Schrödinger operators, see Remark 2.1. Assume also that σ(H) = σ ess (H λ ) = [0, ∞) and that H has a zero energy ground state. Then the potential V is critical.
Proof. This is probably well-known. We provide the short proof for the convenience of the reader. Let ψ be a zero energy normalized ground state of H. Then for any small enough since we can choose the ground state ψ > 0. Thus as soon as σ ess (H δ ) = [0, ∞), the minmax principle shows that H δ has eigenvalues below zero.
The converse to Lemma 2.6 does not hold. See the example from Appendix A.
Our proofs of Theorems 1.3 and 1.7 rely on the so-called subharmonic comparison lemma which has already seen wide use in the study of the asymptotic decay of eigenfunctions of Schrödinger operators, see, e.g., [12,21]. We use the version of [2, Theorem 2.7] since it allows for a quadratic form version which needs only minimal regularity assumptions.

Theorem 2.7 (Agmon's version of the comparison principle). Let w be a positive supersolution of the Schrödinger operator H at energy E in a neighborhood of infinity
for some α > 1. If for some δ > 0 and 0 ≤ C < ∞ one has v(x) ≤ Cw(x) on the annulus R < |x| ≤ R + δ , (2.28) Remark 2.8. We note that the condition (2.26) is trivially satisfied as soon as v ∈ L 2 (R d ), but it also allows for subsolutions v which are not square integrable at infinity. This is crucial for the proof of our non-existence result. A slight extension of Agmon's comparison principle, which allows to relax the continuity assumptions and works for domains U which are not necessarily neighborhoods of infinity, is derived in [22].
Remark 2.9. Agmon also assumes that the supersolution w and the subsolution v are continuous in U R = {|x| ≥ R} in [2]. However, the form of Theorem 2.7 is what is really proven in [2], see also [22]. The additional assumption that the supersolution w > 0 and the subsolution v are continuous in {|x| ≥ R} are only made in [2] to guarantee that (2.27) holds with constant C = c 2 /c 1 where c 1 := inf R≤|x|≤R+δ w(x) > 0 and c 2 := sup R≤|x|≤R+δ |v(x)| < ∞, by continuity, for arbitrary δ > 0.
Before we give the proofs of Theorems 1.3 and 1.7, we sketch the simple proof of the m = 0 version of Theorem 1.3 using the comparison theorem: ), integration by parts shows that ψ γ is a weak local eigenfunction of −∆ + W γ in the sense of Definition 2.3 in the open sets U R for any R > 0. Moreover, Remark 2.2 shows that ψ γ ∈ Q U R loc (H) for any R > 0 and any Schrödinger for some γ > 0 and all large enough |x| > R, then , ψ γ is a zero energy subsolution of H. One easily checks that The last part shows that one should choose γ as large as possible in order to guarantee that (2.31) holds. Now assume that H ≥ 0 has zero as an eigenvalue with corresponding unique ground state ψ which can be chosen to be positive [15,18]. Since V is locally in the Kato class, one also knows that ψ is continuous, [52]. If then the above discussion shows that u = ψ d/2 is a zero energy subsolution of H which is not square integrable at infinity but for which (2.26) holds. Using c 1 for all |x| > R, in particular, ψ = L 2 (R d ), hence ψ is not an eigenfunction. Thus zero is not an eigenvalue of the Schrödinger operator H, which proves the n = 0 version of Theorem 1.3.
Remark 2.10. Note that for γ > d/2 the function ψ γ is in L 2 (U R ) for any R > 0, so γ = d/2 is the largest possible choice in order to get the non-existence result. The higher order condition for non-existence will have to use a suitably modified choices of subsolutions at γ = d/2.
For the existence result we want to reverse the roles of the eigenfunction ψ and ψ γ . If for all |x| > R and some ε > 0 then ψ γ is a zero energy subsolution of H in U R where γ > d/2 is the unique solution of γ(γ + 2 − d) = (d(4 − d) + ε)/4. Arguing as above, one sees that any positive zero energy ground state ψ of H satisfies the upper bound for all |x| > R, hence it is square integrable at infinity since ψ γ ∈ L 2 (U R ) as soon as γ > d/2. Of course, this is a circular reasoning, since we need the existence of a square integrable bound state, or at least the existence of a local zero energy bound state which satisfies (2.26). The rigorous argument uses the fact that H ≥ 0 is assumed to have a virtual level at zero, so the operators H δ have a negative energy ground states with negative energy for small δ > 0 These ground states will converge to a zero energy ground state of H in the limit δ → 0, see Section 4.

PROOF OF THE NON-EXISTENCE RESULT
Recall the iterated logarithms ln n defined by ln 1 (r) := ln(r) for r > 0 and, for r > e j , inductively by ln j+1 (r) := ln(ln j (r)) when j ∈ N. Here e 1 = 1 and e j+1 = e e j .
A convenient sequence of functions at the edge of L 2 -integrability near infinity is given by As usual, the empty product is one, so ψ ℓ,0 (x) = |x| −d/2 = ψ d/2 (x). We still have ψ ℓ,m ∈ C ∞ ({|x| > e m }), in particular, ψ ℓ,m ∈ Q U R loc (H), for R ≥ e m and any Schrödinger operator constructed via quadratic form methods as in Section 2. In order to mimic the proof sketched at the end of Section 2, we need to know the potential W m for which (−∆ + W m )ψ ℓ,m = 0 in U em = {|x| > e m }. This is a bit more complicated than the previous calculation for ψ γ .
By a straightforward but slightly tedious calculation one sees that where we used ∂ r ln 1 (r) = 1 r and ∂ r ln j (r) = 1 ln j−1 (r) and we have (−∆ + W m )ψ ℓ,m = 0 in U R as long as R > 0 is large enough, so that the iterated logarithms are well-defined.
Remark 3.2. Alternatively, one can compute W m = ∆ψ ℓ,m ψ ℓ,m inductively. For radial functions f, g, i.e., with the usual abuse of notation f (x) = f (r) and g(x) = g(r) for r = |x| we have Using (3.4) and ψ ℓ,m+1 (x) = ψ ℓ,m (x) ln . for all 0 ≤ ϕ ∈ C ∞ 0 (U R ) as soon as V ≤ W m in U R . So ψ ℓ,m is a zero energy subsolution of H in U R as soon as V ≤ W m in U R . Since V is in the local Kato class we know from [3,48,52] that any eigenfunction of H is continuous. Moreover, it is well-known that the ground state eigenfunction can be chosen to be strictly positive [15,18,47].

A straightforward calculation yields
So if ψ > 0 is a zero energy ground state of H then, with c 1 R = inf R≤|x|≤R+1 ψ(x) > 0 and c 2 R = sup R≤|x|≤R+1 ψ ℓ,m (x) < ∞, we can set C := c 2 R /c 1 R to see that ψ ℓ,m (x) ≤ Cψ(x) for R ≤ |x| ≤ R + 1. Moreover, ψ being a zero energy solution is also a zero energy supersolution, so Theorem 2.7 shows that for all |x| > R. In particular, ψ cannot be square integrable as soon as V ≤ W m on U R for some large enough R > 0, since ψ ℓ,m is positive and not in L 2 (U R ).  for large enough R > 0 and some m ∈ N 0 , then the Schrödinger operator H cannot have any zero energy ground state.

PROOF OF THE EXISTENCE RESULT
For the existence result we need to modify our comparison functions ψ ℓ,m to make them barely square integrable near infinity. Given an arbitrary ε > 0 we set where ψ ℓ,m is defined in (3.1). For each m ∈ N 0 and ε > 0 we have ψ u,m,ǫ ∈ C ∞ ({|x| > e m }) and it is not hard to see that for any ε > 0 and any large enough R > 0 the function ψ u,m,ǫ is barely in L 2 (U R ). The potential Y m,ǫ for which (−∆ + Y m,ǫ )ψ u,m,ǫ = 0 in U R = {x ∈ R d : |x| > R} is given by where the potential Y m,ǫ is given by
Proof. As in the proof of Lemma 3.1 we have to calculate Y m,ǫ := ∆ψu,m,ǫ ψu,m,ǫ . We use (3.4) to see that which is (4.2).
We want to show that ground states of Schrödinger operators H with critical potentials V exist using suitable eigenfunctions of H λ . For this the following is convenient.
Assume that the potential V satisfies Assumption 1.1 and W is a positive potential which is infinitesimally form small w.r.t. −∆ + V + . Let (H λ ) λ≥0 be the family of Schrödinger operators constructed in Remark 2.1. Moreover, assume that there exists a sequence 0 < λ n → 0 as n → ∞ such that the operators H n = H λn have eigenvalues E n = E λn with corresponding normalized weak eigenfunctions ψ n = ψ λn . If a) the sequence of eigenvalues (E n ) n of H n is bounded from above and b) the sequence ψ n is a Cauchy sequence in L 2 , then the sequence ψ n is Cauchy w.r.t. the quadratic form norm · 1 given in (2.3), hence its limit ψ = lim n→∞ ψ n ∈ Q(H). Moreover E = lim n→∞ E n exists and ψ is a normalized weak eigenfunction of H with eigenvalue E. Remark 4.3. Lemma 2.5 shows even that ψ ∈ D(H). Moreover, we do not need that V is in the local Kato-class, only that V − is relatively form small w.r.t. −∆ + V + .
Remark 4.4. We apply Lemma 4.2 when λ n converges monotonically to zero and E n is a ground state of H n , in which case one can simplify the proof. For example, if E n are ground state energies of H n , then since as quadratic forms H λ ≤ H λ ′ for all 0 < λ ′ ≤ λ ≤ λ 0 , the limit lim n→∞ E n exists, by monotonicity. However the result of Lemma 4.2 is needed when one considers not only ground states, but also excited states which hit the bottom of the essential spectrum.
Remark 4.5. Clearly any eigenvalue of H n is bounded from below uniformly in n ∈ N since H n ≥ H λmax with λ max = max n λ n as quadratic forms for all n ∈ N. In particular, all the eigenvalues E n are bounded uniformly in n ∈ N once they are bounded from above.
Moreover, if the essential spectrum of H is not empty and E n is an eigenvalue of H n below the essential spectrum of H n , then E n is bounded from above, since as quadratic forms H n ≤ H and by Persson's theorem [37,44] infσ ess (H n ) = lim for all n ∈ N, which shows that sup n |E n | < ∞ as soon as σ ess (H) is not empty.
Proof of Lemma 4.2: Let ψ n be a normalized sequence of eigenfunctions of H n with eigenvalue E n which is also a Cauchy sequence in L 2 . In particular, ψ n , H n ψ n = E n ψ n , ψ n = E n . (4.4) Let 0 < a 1 < 1 and b 1 ≥ 0, respectively a 2 , b 2 ≥ 0, such that (1.2), respectively (1.2) with V − replaced by W , holds. Then , since ψ n is normalized. We also used the quadratic form norm · 1 given by (2.3). Using (4.4) this implies which shows that we have lim sup n→∞ ψ n 1 < ∞, since a 1 < 1, λ n → 0 for n → ∞, and E n is bounded from above uniformly in n ∈ N. Thus both the Sobolev norm ψ n 2 H 1 = ψ n 2 + ∇ψ n 2 and √ V + ψ n are bounded in n.

Now consider
ϕ, H(ψ n − ψ m ) = ϕ, H n ψ n + λ n ϕ, W ψ n − ϕ, H m ψ m − λ m ϕ, W ψ m = E n ϕ, ψ n + λ n ϕ, W ψ n − E m ϕ, ψ m − λ m ϕ, W ψ m (4.6) for ϕ ∈ Q(H). The choice ϕ = ψ n − ψ m and Cauchy-Schwarz yields On the other hand, and plugging this lower bound into (4.7) and using that ψ n is normalized we arrive at (4.8) By assumption and Remark 4.5, the sequence of eigenvalues E n is bounded and, because of (4.5), we also have that ∇ψ n is bounded uniformly in n ∈ N. Since λ n → 0 and ψ n − ψ m → 0 as n, m → ∞, (4.8) implies lim sup n,m→∞ That is, the sequence of normalized weak eigenfunctions ψ n of H n is Cauchy in Q(H) with respect to the form norm · 1 as soon as it is Cauchy in L 2 and the sequence of eigenvalues (E n ) n∈N is bounded. In particular, the limit ψ = lim n→∞ ψ n exists in Q(H). Thus ∇ψ = lim n→∞ ∇ψ n , √ V + ψ = lim n→∞ √ V + ψ n , and, since W ≥ 0 is form bounded w.r.t.
Finally, it is easy to see that the sequence of eigenvalues E n must converge. Assume that E n does not converge as n → ∞. Since E n is bounded in n ∈ N, there exist two different limit points E 1 = E 2 of E n corresponding to two subsequences E σ 1 (n) → E 1 and E σ 2 (n) → E 2 where σ 1 , σ 2 : N → N are strictly increasing functions.
Clearly ψ = lim n→∞ ψ σ 1 (n) = lim n→∞ ψ σ 2 (n) . So (4.9) shows that ψ is a weak eigenfunction of H corresponding to the two different eigenvalues E 1 and E 2 , which is impossible. Hence the eigenvalues E n converges. This finishes the proof of Lemma 4.2.
Lemma 4.6. Assume that the potentials V and W satisfy Assumption 1.1, except that the relative bound of W does not have to be less than one. Let (H λ ) 0≤λ≤λ 0 be the family of perturbed Schrödinger operators constructed in Remark 2.1 for some small enough 0 < λ 0 . Moreover, assume that for some sequence 0 < λ n ≤ λ 0 the operators H n = H λn have eigenvalues E n with corresponding weak eigenfunctions ψ n .
If ψ n = 1 for all n ∈ N and sup n E n < ∞, then the weak eigenfunctions ψ n are pointwise locally bounded uniformly in n ∈ N, i.e., for any bounded set S ⊂ R d .
Remark 4.7. Since eigenfunctions are continuous if the potential is locally in the Kato-class, ψ n (x) makes sense for all x ∈ R d and n ∈ N.
Proof. Note that ψ n is a zero energy weak eigenfunction of the Schrödinger operator H n with potential V n given by V n = V − λ n W − E n . If V and W are in the local Kato class, so is V n . Hence for any x ∈ R d the subsolution estimate |ψ n (x)| ≤ C x,n |x−y|<1 |ψ n (y)| dy (4.11) holds, see [52, Theorem C.1.2] and also [3,48]. Moreover, the constants C x,n depend only on with ( V n ) − being the negative part of V n and the Kato norm · K d given by with g d = g d when d ≥ 3, g 2 = 1 + g 2 , and g 1 = 1, where g d is defined in (1.15). Adding 1 to g 2 is necessary since g 2 (x) = 0 when |x| = 1.
For any set S ⊂ R d and any potential V we have where S 1 = {y ∈ R d : dist(y, S) < 1}.

Now let S ⊂ R d be bounded. Then S 1 is bounded and, since the Kato norm of a constant function is finite and
for any bounded set S, using that sup n E n < ∞ and sup n λ n < ∞, by assumption, and 1 S 1 V − K d < ∞ and 1 S 1 W + K d < ∞, since S 1 is bounded and V and W are locally in the Kato class.
Thus for any bounded set S ⊂ R d there exist a constant C < ∞ such that |ψ n (y)| dy (4.14) for all x ∈ S and n ∈ N. Using the normalization ψ n = 1 we have for all x ∈ S and n ∈ N. Hence (4.10) follows immediately from (4.14).
The last result which we need is

r.t. −∆ + V + and ψ is a real-valued weak eigenfunction of H at energy E. Then |ψ| is a subsolution of H at energy E.
Proof. If ψ is real-valued eigenfunction of H at energy E then it is also a subsolution, hence [2, Lemma 2.9] shows that its positive part ψ + = sup(ψ, 0) is a subsolution. The same argument applied to −ψ, which is also a weak solution, shows that its negative part ψ − = sup(−ψ, 0) is a subsolution. Hence |ψ| = ψ + + ψ − is a subsolution of H at energy E. Remark 4.9. It is well-known that for the type of Schrödinger operator H we consider here the eigenfunctions can be chosen to be real-valued. Since H is self-adjoint all eigenvalues are real. Moreover, H commutes with complex conjugation, so for any complexvalued eigenfunction ψ of H also the real and imaginary parts Re(ψ) = 1 2 (ψ + ψ) and Im(ψ) = 1 2i (ψ − ψ) are eigenfunction of H at energy E. This is not true anymore if one considers Schrödinger operators with magnetic fields, since they do not commute with complex conjugation, in general.

Now we are ready to give the
Proof of Theorem 1.7 : By assumption, the potential V is critical. Thus σ(H) = σ ess (H) = [0, ∞). Moreover, for any non-trivial potential W ≥ 0 which is infinitesimally form small w.r.t. −∆ + V + and has compact support the Schrödinger operators H λ = H − λW , constructed in Remark 2.1, have non-trivial discrete spectrum below zero. That is, σ ess (H λ ) = [0, ∞) and there exist eigenvalues E λ < 0 of H λ with associated normalized weak eigenfunctions ψ λ for all λ > 0. We take any sequence (λ n ) n∈N which is monotonically decreasing to zero and abbreviate H n = H λn , E n = E λn , and ψ n = ψ λn .
Recall that we also assume that the potential V satisfies the lower bound for |x| > R, some m ∈ N 0 , ǫ > 0, and all large enough R > 0. We replaced ǫ by 2ǫ in (1.5).
Increasing R, if necessary, it is easy to see that this implies where the family of comparison functions Y m,ǫ is defined in (4.2). Since W has compact support, we can also assume that R is so large that its support Here ψ u,m,ǫ > 0 is defined in (4.1) and we used that E n ≤ 0. So for fixed m ∈ N, large enough R > 0, and small enough ǫ > 0 the function ψ u,m,ǫ is a supersolution of H n at energy E n in U R for all n ∈ N. Moreover, since ψ n = 1 we have by Lemma 4.6. Since ψ u,m,ǫ > 0 is continuous away from zero, we also have and using C R = c 1 R /c 2 R one gets |ψ n (x)| ≤ C R ψ u,m,ǫ (x), hence also ψ n (x) := |Re(ψ n (x))| + |Im(ψ n (x))| ≤ √ 2|ψ n (x)| ≤ √ 2C R ψ u,m,ǫ (x) (4.18) for all R ≤ |x| ≤ R + 1 and all n ∈ N. Clearly, |ψ n | ≤ ψ n . Since ψ n is a nonnegative subsolution of H n at energy E n by Lemma 4.8 and Remark 4.9 we can use w = ψ u,m,ǫ and v = ψ n in Theorem 2.7 to see that uniformly in n ∈ N. Since ψ u,m,ǫ is square integrable at infinity for any fixed m ∈ N and ǫ > 0, the bound (4.19) yields tightness in x-space, i.e., which is tightness in momentum space. Here ψ n is the Fourier transform of ψ n . Moreover, since ψ n is bounded in H 1 (R d ), there exists a subsequence which converges weakly in H 1 and L 2 . By a slight abuse of notation, we also write ψ n for this subsequence.
Lemma 4.2 shows that E = lim n→∞ E n ≤ 0 exists and that ψ is a normalized weak eigenfunction of H with eigenvalue E. Clearly, E = 0 since σ(H) = [0, ∞). So zero is the ground state eigenvalue of H which is at the edge of the essential spectrum of H. This finishes the proof of Theorem 1.7.
Remark 4.10. Note that we could have simplified some parts of the proof by using that ground states can be chosen to be strictly positive. We intentionally avoided the use of strict positivity of ground state eigenfunctions. This allows to use Theorem 1.7 also for systems with symmetry restrictions, or for the existence of higher eigenstates with energies above the ground state energy, provided one suitably modifies the assumption of a virtual level for such systems. These modifications are straightforward.

APPENDIX A. AN EXAMPLE IN SEARCH OF A THEOREM
It is well-known that the zero potential is critical in dimensions one and two, see [50] and also [35,Problems 1 and 2 in Chapter 45]. This phenomenon can be explained by the non-integrability of η → |η| −2 near η = 0 in R d , see [20]. The Iorio-O'Carrol theorem [47,Theorem XII.27] shows that shallow potential wells cannot create ground states in dimension d ≥ 3 and that the corresponding Schrödinger operators are even unitarily equivalent to the free Laplacian.
Of course, in order to construct zero energy resonances or zero energy ground states, one can take any Schrödinger operator H which has essential spectrum [0, ∞) and finitely many negative eigenvalues. Adding a suitable local positive perturbation then moves the ground state energy to zero, creating a zero energy resonance, or zero energy ground state, depending, for example, on which a priori bound from Theorem 1.3 of Theorem 1.7 holds.
Specific examples of critical potentials in dimension one and two which are different from the zero potential seem to be rare. In the following we construct a family of potentials V α,d in any dimension which are critical for α ≥ 0, having a zero energy resonance when 0 ≤ α ≤ 1 and a zero energy ground state when α > 1, and which are not critical when α < 0. To the best of our knowledge, our example is new.
Remark A.1. There are different definitions for a zero energy resonance available in the literature. One often calls ψ a zero energy resonance if it is a local positive eigenfunction of a Schrödinger operator H which is not square integrable on R d but its gradient ∇ψ is square integrable. We will follow this convention, except that we also allow that the L 2 -norm of ∇ψ is logarithmically divergent at infinity.
For α ∈ R and d ∈ N define the potential V α,d on R d by Clearly, V α,d is bounded and goes to zero at infinity. Thus it is a Kato-class potential for all d ≥ 1 and all α ≥ 0. In particular, V α,d is both infinitesimally operator bounded, hence also infinitesimally form bounded, w.r.t. −∆. Therefore the Schrödinger operator H α,d = −∆ + V α,d is a well defined self-adjoint operator on the domain H 2 (R d ) with form domain The key to understanding why the potentials V α,d are critical for all α ≥ 0 and d ≥ 1, not critical for α < 0, and switch from having zero energy resonances to having zero energy ground states at α = 1 is for x ∈ R d and the measure on the Borel sets B in R d . Then the map U α,d : is unitary with where ϕ = U −1 α,d ψ. Remark A.3. Lemma A.2 shows that the Schrödinger operator H α,d is equivalent to the Dirichlet form q(ϕ) = ∇ϕ, ∇ϕ L 2 (R d ,dµ α,d ) on the weighted L 2 -space with measure dµ α,d = (1 + |x| 2 ) −(d−2)/2−α dx. Note that this measure is finite if and only if α > 1.
We give the proof of the lemma at the end of the appendix. b) For all α ≥ 0 the potential V α,d is critical, that is, the Schrödinger operator H α,d has a virtual level. c) Zero is not an eigenvalue of H α,d when 0 ≤ α ≤ 1. For α > 1 zero is an eigenvalue.
The zero energy resonance for 0 ≤ α ≤ 1, respectively ground state for α > 1, is given by (A.2). d) For α < 0, the potential V α,d is subcritical, and zero is neither an eigenvalue nor a resonance.
Remark A.5. Using the early result of Kato, [28], see also [1,49], the operator H α,d has no strictly positive embedded eigenvalues. Since the potential V α,d is short range, the spectrum of H α,d is even purely absolutely continuous inside (0, ∞), see [11,Theorem 5.10].
Given ϕ ∈ L 2 (R d , dµ α,d ) let ψ := U α,d ϕ. Taking ϕ = 1 gives ψ = ψ α,d > 0 which is in H 1 (R d ) if and only if α > 1. In this case (A.6) shows that ψ α,d is the ground state of H α,d corresponding to the eigenvalue zero. This proves the second claim in c). Lemma 2.6 also shows that the potential V α,d is critical when α > 1.
In addition, note that the right hand side of (A.6) is strictly positive unless ϕ is constant. Hence zero is not an eigenvalue of H α,d when 0 ≤ α ≤ 1 because ψ α,d is not square integrable in this case.
Remark A.7. Claims b) and c) above are consistent with what is known about weakly coupled bound states in low dimensions. It is known that if V ∈ L 1 (R d ) and R d V dx ≤ 0, where V is supposed to be non-trivial when V dx = 0, then the operator −∆ + λV always has a negative bound state, no matter how small the coupling parameter λ > 0 is, when d = 1, 2. See, for example, [50] where this is proved under some additional assumptions, or [20] for the full result. In particular, this implies that critical potentials in one and two dimensions have to change sign and, if they are integrable, then R d V dx > 0 unless V is trivial. In addition, claim d) is consistent with our non-existence Theorem 1.3. Non-positive potentials cannot have a zero energy ground state in dimensions d ≤ 4. They need to have a strong enough positive tail in order to be able to have zero energy bound states. Moreover, claim f) together with the fact that the potential V α,d supports zero energy ground states if and only if α > 1, see Theorem A.4, is consistent with our Theorems 1.3 and 1.7.
It is illuminating to plot V α,d (x) for |x| = r to explicitly see the behavior of V α,d for various values of the parameters α and d.
Since for large |x| the second term in the definition of V α,d is much smaller than the first, the last claim f) follows from a straightforward computation.
It remains to give the proof of the ground state representation.
Remarks A.8. The proof of Lemma A.2 is clearly inspired by the proof of Hardy's inequality on L 2 (R d ) for d ≥ 3, where one considers ψ(x) = |x| −γ/2 ϕ(x) for ϕ ∈ C ∞ 0 (R d \ {0}) and optimizes in γ > 0. One needs to restrict to ϕ ∈ C ∞ 0 (R d \ {0}) due to the singularity of |x| −γ/2 in zero. Since C ∞ 0 (R d \ {0}) is dense in L 2 (R d ) only when d ≥ 3, this leads to the well-known fact that Hardy's inequality only holds in dimensions d ≥ 3.