The ground state of long-range Schrodinger equations and static $q\bar{q}$ potential

Motivated by the recent results in arXiv:1601.05679 about the quark-antiquark potential in $\mathcal N=4$ SYM, we reconsider the problem of computing the asymptotic weak-coupling expansion of the ground state energy of a certain class of 1d Schr\"odinger operators $-\frac{d^{2}}{dx^{2}}+\lambda\,V(x)$ with long-range potential $V(x)$. In particular, we consider even potentials obeying $\int_{\mathbb R}dx\, V(x)<0$ with large $x$ asymptotics $V\sim -a/x^{2}-b/x^{3}+\cdots$. The associated Schr\"odinger operator is known to admit a bound state for $\lambda\to 0^{+}$, but the binding energy is rigorously non-analytic at $\lambda=0$. Its asymptotic expansion starts at order $\mathcal O(\lambda)$, but contains higher corrections $\lambda^{n}\,\log^{m}\lambda$ with all $0\le m\le n-1$ and standard Rayleigh-Schr\"odinger perturbation theory fails order by order in $\lambda$. We discuss various analytical tools to tame this problem and provide the general expansion of the binding energy at $\mathcal O(\lambda^{3})$ in terms of quadratures. The method is tested on a soluble potential that is fully under control, and on various non-soluble cases as well. A supersymmetric case, arising in the study of the quark-antiquark potential in $\mathcal N=6$ ABJ(M) theory, is also exploited to provide a further non-trivial consistency check. Our analytical results confirm at third order a remarkable exponentiation of the leading infrared logarithms, first noticed in $\mathcal N=4$ SYM where it may be proved by Renormalization Group arguments. We prove this interesting feature at all orders at the level of the Schr\"odinger equation for general potentials in the considered class.


Introduction
Let us consider the Schrödinger operator in one dimension x ∈ R, λ > 0. (1.1) A variational argument shows that if V (x) ≤ 0, with V (x) < 0 on an open set, and V (x) → 0 at infinity, then H has a bound state for all λ > 0. The lowest eigenvalue E 0 (λ) is real analytic for λ > 0 under mild conditions on V [1]. Interesting questions are analyticity at λ = 0 and the existence of bound states for small λ when V is somewhere positive. 1 The analysis of [5] showed that when R dx V (x) ≤ 0 and R dx (1 + x 2 ) |V (x)| < ∞, there is a unique bound state for small λ. The ground state (binding) energy may be estimated in this case by the formula If for some a > 0 we have R dx e a |x| V (x) < ∞, then the quantity −E 0 (λ) is analytic at λ = 0. 2 If the potential is such that R dx (1 + x 2 ) |V (x)| = ∞, and V (x) ∼ −a |x| −β at infinity, the results of [11] imply the following. For 2 < β < 3, the estimate (1.2) is still valid. For β = 2, there is a unique bound state for λ → 0 + provided that R dx V (x) ≤ 0. However, (1.2) is violated because the r.h.s. develops a term λ 2 log λ. If 1 < β < 2, there are infinitely many bound states for any λ > 0 and (1.2) is valid when restricted to the first term only, i.e.
Here, we shall be concerned with the β = 2 case. Further discussion of other long-range cases may be found in [12,13] and in the review [14]. Thus, in this paper, we shall address the problem (1.1) with a potential in the class defined by the conditions and, for simplicity, we shall also assume it to be even V (x) = V (−x). The modified form of (1.2) under the conditions in (1.4) has been found in [11] and reads −E 0 (λ) = − 1 2 λ + a λ 2 log λ R dx V (x) + O(λ 2 ). (1.5) A non-analytic logarithmic factor enhances the second order term. Standard Rayleigh-Schrödinger perturbation theory completely fails and this term is infinite. 3 In general, it is very difficult to recover results like (1.5) from some kind of regularized perturbation theory. A very illustrative example are crude infrared cutoffs like |x| ≤ L. 4 Finite size effects may be suppressed at large L but by factors like e −λ g or similar. Expanding first in λ shows that the cutoff dependence is increasingly bad at higher orders. A modern application of results like (1.5) is to the study of the quark-antiquark static potential in flat space for N = 4 super Yang-Mills theory with gauge group SU (N c ). 5 The static potential is extracted from a pair of anti-parallel Wilson lines separated by the distance r [18,19]. In the planar limit N c → ∞, with fixed 't Hooft coupling λ = g 2 YM N c 6 , we can obtain the quark-antiquark potential − 1 r Ω( λ) from the expectation value of the associated Wilson loop. The function Ω( λ) is known at 3 loops at weak coupling [18,[20][21][22][23][24][25] and at one loop at strong coupling [26][27][28][29]. This construction can be extended by introducing an angular parameter ϑ that takes into account the relative flavours of the quark and antiquark. The generalized potential Ω( λ, ϑ) has been recently studied in [30] by means of the Quantum Spectral Curve methods developed in [31]. One of the remarkable results of [30] is the analytic weak coupling expansion of Ω( λ, ϑ) up to 7 loops, i.e. at order O(λ 7 ). A particularly interesting limit is the double scaling regime ϑ → i ∞ with fixed λ e −i ϑ . This limit resums the ladder diagrams in the gauge theory and the quark-antiquark potential is obtained as the ground state energy of a 1d Schrödinger equation with potential in the class (1.4). In the notation of (1.1), the problem addressed in [30] corresponds to the specific potential with Schrödinger coupling λ = λ e −i ϑ 16π 2 and identification E 0 = − 1 4 Ω 2 (λ). The first three terms of the expansion of √ −E 0 may be written [30] −E 0 (λ) = π 2 λ + π (log λ + 2 L − 1) λ 2 where L = log √ 2 e γ E π, and γ E denotes the Euler-Mascheroni constant. Inspection of the O(λ 7 ) additional terms in (1.7) shows that each power λ n comes together with all logarithms log m λ with 0 ≤ m ≤ n − 1. Also, the leading and next-to-leading logarithms seem to be captured by the resummation Ansatz whose leading part is a simple exponentiation. The expansion (1.7) and the resummation (1.8) pose several interesting questions. The first is whether the structure of (1.7) is general, 5 Another (older) interesting physical application of (1.1) with potential in the class (1.4) occurs in the study of wetting transition of 2d surfaces [17]. In that context, the interface free-energy density is given in the thermodynamic limit by the lowest eigenvalue E0 of (1.1) where x denotes the perpendicular distance of the interface from the substrate, and V (x) is the potential well that localizes the interface below the transition temperature, related to λ. 6 We introduce a slightly unconventional hat in the notation for the 't Hooft coupling to avoid confusion with the coupling λ in (1.1) that we prefer to keep in order to match the conventions in the mathematical physics literature.
i.e. if it is true that -at least at third order -for all potentials in the class (1.4) one has −E 0 (λ) = c 1,0 λ + (c 2,1 log λ + c 2,0 ) λ 2 + (c 3,2 log 2 λ + c 3,1 log λ + c 3,0 ) λ 3 + · · · . (1.9) The second question concerns the general validity of resummation formulas like (1.8). In the context of the static quark-antiquark potential, resummation of leading and next-to-leading logarithms has been performed by a Renormalization Group (RG) analysis in [20,23]. In particular, leading logarithms simply exponentiate and, since they are computed exactly by the ladder approximation, this explain the first term in (1.8). Nevertheless, it would be interesting to understand whether such exponentiation is expected for a general potential and can be determined at the level of the Schrödinger equation. This question is particularly intriguing because in gauge theories the RG exponentiation -at leading order -encodes the factorization scale dependence that remains after cancellation between real and virtual infrared divergences. In the Schrödinger equation, this deep machinery is somewhat absent and exponentiation of logarithms of λ is less clear and more elusive.
The plan and content of this paper is the following. In Sec.
(2), we discuss a fully solvable potential in the class (1.4) where we provide the expansion of the binding energy at high order showing that it admits the structure in (1.9) as well as an exponentiation formula similar to (1.8). In Sec. (3), we present a first simple extension of the result (1.5), by computing the subleading coefficient c 2,0 in (1.9). The derivation is rather straightforward, but cannot be easily extended at higher orders. To overcome these difficulties, we apply in Sec. (4) the general approach of matched asymptotic expansions to our problem. As a preparation, Sec. (4.1) is devoted to discuss a toy-model that captures the essence of the method. Then, in Sec. (4.2), we apply it to the determination of (1.9) and we compute the complete third order expansion for a general potential. Our main result, summarized in (4.39) and (4.40), is tested on various solvable and non-solvable examples in Sec. (5).
In particular, Sec. (5.5) presents several high precision numerical tests of the third order expansions to appreciate its accuracy at small coupling, despite being only asymptotic. As a final test, we discuss in Sec. (6) a consistency check of our expansion, when applied to a supersymmetric potential arising in the study of the quark-antiquark potential in N = 6 ABJ(M) theory. Finally, our third order expansion suggests exponentiation of the leading logarithms in the form (see the first term of (1.8) as a special case) In Sec. (7), we prove that this result holds at all orders. Finally, Sec. (8) summarises our conclusions and outlines possible further directions.

A solvable example
The potential (1.6) may be treated by the Quantum Spectral Curve as a special limit. To improve our knowledge, we also analyze the following solvable case discussed in [11,32] V (x) ≡ V II (x) = − 1 4 1 (1 + |x|) 2 . (2.1) Its ground state energy α ≡ −E 0 (λ) obeys the exact equation where K ν (α) are modified Bessel functions of the second kind. Expanding (2.2) at small λ, we get the asymptotic series where L = log λ−log 2+γ E + 1 2 . All the leading λ n+1 L n and subleading λ n+2 L n logarithms are captured by the following formula (see App. (A) for a proof) The first term in the square bracket is similar to the resummation (1.8  This is what we want to be able to compute for a general potential in the class (1.4). Of course, the (leading) second order rigorous result (1.5) is fully consistent with the entries in Tab. (1).

The complete second order expansion
A first simple extension of the result (1.5) obtained in [11] concerns the determination of the coefficient c 2,0 in (1.9). The analysis in [11] is based on the results of [5] that imply that the ground state energy E 0 = −α 2 is obtained from the condition where K α is the Birman-Schwinger operator [1] K α (x, y) = 1 2 α |V (x)| 1/2 e −α |x−y| |V (y)| 1/2 sign(V (y)).
We can expand (3.1) as follows The rigorous analysis of [11] leads to the result (1.5) for the λ and λ 2 log λ term. If we want to capture the λ 2 term (without logarithmic enhancement), we need to study in details the small α limit of the second order correction To this aim, we define the Fourier transform of V (x) with the convention and easily obtain the alternative form of F (α) For a specific potential, one can study this integral for α → 0 8 , but here we want to discuss (3.7) in general terms. One straightforward approach is the following.
can be split as follows An alternative expression for the second order term is . 8 For instance, if the Mellin transform of | V (ω)| 2 is known, then it is easy to obtain the small α asymptotic expansion of F (α).
The first integral is computed by adding and subtracting in the numerator ω U (0 + ) and splitting the elementary part that can be computed explicitly. In what remains, as well as in the second integral, we can safely send α → 0 if we are interested in the O(α log α) and O(α) terms of F (α). In conclusion, we obtain from (3.4) Denoting for brevity and inverting (3.11), we find In conclusion, we find the second order terms in the expansion (1.9) with the following coefficients (3.14) Notice that, to match (3.3), we need to show that U (0 + ) = a V . This is slighlty non trivial. For an even potential V (ω) is also even, and then U (0 + ) = 2 V (0) V (0 + ). Then, In the first integral we can safely send ω → 0 and then it vanishes because it is odd under x → −x. The remaining piece gives π ω e −|ω| |ω| = a π 2 . (3.16)

Check with the potential
In this case, Also, we can compute 9 (3.18) Replacing in (3.14), we obtain 19) in full agreement with the entries in Tab. (1).

Check with the potential
In this case, Also, we can compute the relevant non trivial integrals numerically at high precision finding Remarkably, we checked that up to 100 digits we have Replacing in (3.14), we obtain again in full agreement with Tab. (1). 9 Here, the exponential integral function is

Leading logarithms at third order
One may attempt to apply this approach to work out the third order contribution to √ −E 0 , but this is definitely a non-trivial task, although the leading order term ∼ λ 3 log 2 λ is heuristically derived in App. (B). In the next section, we shall introduce an alternative method that is powerful enough to fully compute the third order expansion and, in principle, also higher order terms.

Matched Asymptotic Expansion, the complete third order
A completely different approach is based on the method of matched asymptotic expansions for boundary layer problems [33,34]. Although the idea is the same as in the standard examples of singular perturbation of differential equations [35], the specific application to the present setup is rather different and somewhat unusual. 10 Earlier studies in the context of the quark-antiquark potential may be found in [37,38]. For the potential V I (x), this method has been successfully exploited in [21]. The method is fully general and we shall present its application for a generic potential in the class (1.4). A recent application to a different kind of Schrödinger problem may be found in [39]. 11 We now first present a simple example illustrating the procedure and then apply the method to our case.

A toy model
Let us consider the first order boundary problem Although apparently contrived, this problem captures the essential features of our problem. We are interested in the λ → 0 + limit and there are two boundary conditions, one in the UV region, at small x, and one in the IR region, at large x. The two boundary conditions determine α as a function of λ. Indeed, the exact solution of (4.1) before imposing the boundary condition at x = λ −1 is Imposing ψ(λ −1 ) = 1, we obtain the following asymptotic expansion 12 10 The application of matched asymptotic expansion methods to the Schrödinger equation in the semiclassical limit is of course a text-book topic, see for instance [36] for a clean pedagogical presentation. In that context, matching is applied around the inversion points. 11 The main new complication to be discussed here is the presence of non-analytic terms in a boundary problem eigenvalue. This is a feature that, for instance, is absent in the analysis of the Mathieu equation in [35]. 12 We remark that in this simple example, the non-analyticity due to logarithms is limited to the first term, while the remainder is actually a power series with unit convergence radius.
that is the result we want to derive by an alternative route. To this aim, we apply the ideas of matched asymptotic expansions and look for two approximate solutions of (4.1). The first is ψ IR (x) and is valid around x = ∞, with ψ IR (λ −1 ) = 1. The second is ψ UV (x) and must be used around x = 0, with ψ UV (0) = 1. The two solutions must be compared by imposing the symbolic matching For this procedure to make sense, the precise meaning of the relation (4.4) is that we need ψ UV (x) ψ IR (x) in an intermediate overlap region, as shown in Fig. (1). Besides, in our application the matching will be imposed perturbatively in λ. Following this approach, the first step is to consider the approximate equation that is obtained at large x and solve it with the IR boundary condition: (4.5) We have kept four terms in the large x expansion of the potential. This is instructive with the aim of keeping track of the role of the various corrections. In other words, the coefficients ξ 3 , ξ 4 are useful to identify terms that depend on including (ξ = 1) or excluding (ξ = 0) a certain subleading part of V . The solution of (4.5) is Now, we make the educated guess 13 and expand (4.6) at small λ, and fixed x. This is an expansion in powers of λ with possible powers of log λ. Explicitly, we find ψ IR (x) = 1 + λ ψ IR,1 (x) + λ 2 ψ IR,2 (x) + · · · , (4.8) 13 The appearance of a double expansion in λ and log λ is familiar in the context of matched asymptotic techniques where logarithmic terms are commonly named as switchback corrections [40].
In the UV region, we consider the unmodified initial problem, but impose only the boundary condition at x = 0 ψ UV (0) = 1. (4.10) To match (4.8), we solve (4.10) pertubatively in λ, taking into account (4.7), This gives immediately (4.12) Finally, we attempt to match the two solutions (4.8) and (4.11) by evaluating the ratio (4.13) The contribution from the UV region must be expanded at large x where the overlap region is located. Doing so, we obtain at first order (4.14) Thus, we see that matching (R n = 0) requires A = 1 in agreement with (4.3). This does not depend on the ξ-terms, i.e. the x −3 and x −4 corrections to V included in (4.5). The correct choice ξ 3 = ξ 4 = 1 cancels the various corrections in (4.14), but we repeat that this is irrelevant to fix A. With the choice A = 1, we can inspect R 2 (x, λ) and find (4.15) Again, the coefficient B is fixed at B = −1 in agreement with (4.3). The first two subleading corrections in (4.15) vanish if we include the x −3 correction in (4.5). Doing so, i.e. taking ξ 3 = 1, we get showing that the inclusion of the x −4 correction in (4.5), i.e. taking ξ 4 = 1, cancel two additional subleading terms.

Application to the third order correction
We now apply to our main problem the procedure that worked for the toy model (4.1).

Analysis of the infrared region
Let us consider an even potential admitting the expansion for x → +∞ The large x equation has the following solution decaying exponentially at infinity The second independent solution is Taking into account the next to leading correction to the potential at large x, the IR solution is (4.21) As we explained, we want to expand this solution in the limit α x → 0, and then λ → 0. It is convenient to set α = A(λ) λ and we find (c n are constant terms dependent of A, a, b, λ, but independent of x) where L A = log(2 A λ) + γ E . The higher orders λ n are complicated but have a similar structure

Analysis of the ultraviolet region
In the UV region, we first expand the exact equation This leads to , and so on. Given the solution, we want to expand it at large x and compare with (4.22), after the educated guess, including switchback logarithmic terms, To see how this works, let us begin with ψ 1 (x), i.e. 14 The large x expansion is obtained as follows (4.29) Computing the first integral and expanding inside the second one, we obtain with (4.31) Comparing the UV expansion (4.30) with the IR expansion (4.22), we see that we indeed recover the correct value of c 1,0 in agreement with the known result. At the next order, i.e. solving for the function ψ 2 , we find Doing the previous split in the integrals that cannot be computed by elementary means, we obtain (4.34) Finally, let us consider ψ 3 , we need the expansion of it up to a constant because this is the last stage of our computation.
(4.35) 15 An alternative more explicit form is Integrating, are new constants to be determined by quadrature.

Matching UV and IR, final result
The coefficients of the expansion (3.1) may be found by comparing the expansions in the IR and UV regions. After straighforward calculations, the general result reads It is convenient to write the asymptotic expansion (3.1) by isolating a relevant IR scale and making explicit the simplest coefficients (4.40) Notice that the exponentiation of the leading logs is recovered at this order. The quantities in (4.40) can be computed in terms of (3.12), (4.31), (4.34), and (4.37). These require quadratures that involves the functions ψ 1 (x) and ψ 2 (x) whose expression must be computed from, see (4.28), (4.41) and the quantity A may be replaced by − 1 2 V at this order.

Check with the potential
For the potential V I , we obtain the following explicit functions in (4.41) (ψ 2 (x) is real for x > 0 although this is not obvious from the following expression) − 12 x log(x 2 + 1) + 4 log 2i x + i + 2 arctan x + π 2 x (3x + 2 i) The various constants needed in (4.40) can be computed in closed form and read Replacing them in (4.38), we match the content of Tab. (1). In the form (4.39), it reads with log λ IR = 1 − 2 L.

(5.4)
The various constants are now Again, replacing them in (4.38), we match the content of Tab. (1). In the form (4.39), it reads
The various constants are Replacing them in (4.38), we obtain the new asymptotic expansion (5.10) with log λ IR = 2 log 2 − γ E − 13 12 . A similar calculation may be performed for the potential V IV . We omit the details and just write down the final expansion (5.11) with log λ IR = −γ E + 3 8 − log π.

Simple extensions
We remark that our expressions are sufficiently simple to deal with potentials that depend on parameters. One neat example is the potential For this potential we obtain the following generalization of the β = 1 case in (5.10), In general, it is clear that if the expansion may be worked out in details for V 1 and V 2 , then it may be computed for any linear combination of them. Also, the class of potentials may be enlarged. One interesting example is that of a potential with asymptotic form at infinity including a term ∼ log |x|/x 4 . Just to give a simple example, we studied the case where the relative weight of the first and second terms has been chosen to have V = −1.
We find in this case, with log λ IR = −γ E − 3313 3000 .

Numerical tests
We present a numerical test of the accuracy of the asymptotic expansions (5.3), (5.6), and the new (5.10,5.11). The convergence radius of these expansions is expected to be zero, but looking at sufficiently small λ, it is possible to appreciate the improvement associated with the higher order terms. The following tables will present the first, second, and third order expansion of −E 0 (λ) for the four considered potentials. The last column, labeled exact is obtained by finding the ground state in the interval x ∈ [0, L]. 16 For sufficiently large L, the number of stable digits in the result doubles as L is doubled showing exponential convergence at large L. Our results at λ = 10 −1 , 10 −2 , 10 −3 ( and 5 · 10 −4 for V IV ) are From the above tables, it is possible to check that the third order gives a sensible improvement at these small values of λ. The pattern is the same for all the considered potentials and, in particular, confirms the new expansion (5.10). Just as a final example, we also report the numerical results associated with the potential (5.14) Apart from general questions, our initial motivation came from the Quantum Spectral Curve results on the static quark-antiquark potential in N = 4 super Yang-Mills gauge theory. Quite recently, new interesting results appeared in the study of the cusp anomalous dimension in N = 6 ABJ(M) theory. A scaling limit has been identified where ladder diagrams dominate and the Bethe-Salpeter equation leads again to a Schrödinger problem [41]. This problem is studied for generic Wilson loop Euclidean cusp angle ϕ. The antiparallel limit ϕ → π is the one relevant for the calculation of the static potential, but requires some care as it is singular as in N = 4 SYM. On general grounds, it is expected that the larger amount of supersymmetry should be responsible for cancellation of the nonanalytic logarithms [42]. Technically, in the ladder approximation, this feature is due to the fact that the Schrödinger potential in the ladder approximation is supersymmetric, i.e. it has the the structure of Witten's supersymmetric quantum mechanics [43]. In particular, the N = 6 ABJ(M) version of the potential (1.6) -relevant for N = 4 SYM -reads 17 The exact bound state wave function is ψ(x) = N exp(−λ √ x 2 + 1), with binding energy E 0 = −λ 2 , without (logarithmic) corrections. We can put this problem in the form (1.1), by considering the potential where µ has to be replaced by λ at the end of the calculation. This is a well-defined procedure and the replacement µ → λ simply induces a rearrangement of the asymptotic expansion. Applying (4.39) to (6.2), we obtain the non-trivial expansion at generic fixed µ where log λ IR (µ) = π + 4 µ + 2 π µ 2 4 µ + 2 π µ 2 + 2 − π µ 2 + π µ log 2 − log(2 + π µ) − γ E . (6.4) Replacing µ = λ, the terms in (6.3) and (6.4) rearrange nicely to give −E 0 (λ; λ) = λ + O(λ 4 ), (6.5) in agreement with the exact binding energy E 0 = −λ 2 . This is a further non-trivial check of our expansion (4.39).

Universal resummation of leading infrared logarithms
The resummation results in (1.8) and (2.4) are consistent with (4.39), (4.40). It is tempting to conjecture that leading logarithms indeed exponentiate at all orders, i.e. that the expression captures all terms of the form λ n+1 log n λ. Exponentiation results like (7.1) appears in applications of matched asymptotic expansions often together with Renormalization Group techniques [44][45][46][47]. However, a general treatment is missing and we devote this section to the analysis of (7.1) in our specific setup. A physical argument for (7.1) is that the non-analytic logarithms of the coupling are an infrared effect related to the long-distance behaviour of the potential. This somewhat explains why resummation involves essentially the coefficient a in (1.4). Nevertheless, it is amusing that some global information about the potential is needed in the form of the prefactor in (7.1) that depends on V . An interesting further check of (7.1) may be presented by considering a soluble cut-off version of the potential V II (x), as discussed in App. (C).

(7.2)
According to the Birman-Schwinger principle [1], the Schrödinger operator in (1.1) has the eigenvalue −α 2 if and only if P α + Q α has eigenvalue α/λ or, equivalently, if 1 − λ α −1 (P α + Q α ) is not invertible -compare with (3.1). Since P α is rank 1, this translates into the condition 3) The first term is The second term can be computed by adding and subtracting the leading large distance term −a/(1 + x) 2 to V (x) and we obtain Tr(P α ) = − 1 2 V + 2 a α log α + subleading terms. (7.5) Let us analyze the next terms The most singular contribution comes from those permutations of {x k } with a specific maximal ordering with all large terms |x i − x j |. 18 Evaluating the leading logarithmic term integrating first in x n ∈ [x n−1 , ∞] and so on, one obtains Tr(P α Q n−1 α ) = (−1) n n! [−(2 a α log α) n ], (7.8) and therefore (7.3) reads at leading logarithmic order (−1) n+1 2 n n! λ n α n−1 (a α log α) n . (7.9) The series sums to α−α 1−2 a λ . Solving for α and manipulating up to subleading logarithmic terms λ n log m λ with n > m + 1, we get that is (7.1).

Conclusions
In summary, we have reconsidered the one-dimensional Schrödinger problem (1.1) for longrange potentials in the class (1.4) with the aim of deriving an improved asymptotic expansion of the binding energy of the (unique) bound state that is present at weak coupling.
Non-analytic terms are present at all orders in the coupling and are of course of infrared origin. Our setup is the simplest possible where such infrared problems occur and it is not possible to apply the powerful machinery of relativistic quantum field theory. In particular, there are no simple cut-off procedures that capture them accurately fixing the correct long-distance scales. We computed the general third order asymptotic expansion by methods coming from boundary layer theory. In principle, similar techniques may be applied to more difficult problems involving different behaviours at infinity, like for instance V (x) ∼ log(x)/x 2 or even log(x)/x 3 , where the extension is non trivial. It would be quite interesting to explore similar problems in more than one dimension.
is essentially F (α) in (3.5). For this quantity we know that its small α → 0 + expansion has leading term α (# log α + #). 19 There are further correction O(α 2 ), but an analysis of (3.5) shows that they do not contain log 2 α enhancement factors. This means that the λ 3 log 2 λ term comes entirely from the very last term of (3.3), i.e.
The general expression of the trace is .
(B.7) 19 Here and in the following we shall denote by # numerical coefficients.

(C.4)
Of course, the expansion (C.4) reduces to (2.3) for x 0 = 0 and is consistent with (4.39) and (4.40) for generic x 0 -the prefactor in (C.4) is − 1 2 V . The leading logarithms are precisely the same as in (2.3). We pushed further the expansion (C.4) identifying the rational coefficients of the higher leading logarithms and they continue to agree with (2.3). This strongly supports the exponentiation (7.1), where the prefactor depends on the global shape of V , while the exponential is universal in the sense that it depends only on the coefficient of the leading term of V at infinity. Instead, the subleading logarithms depend on x 0 both through the scale entering the definition of L as well as in the remaining coefficient of the λ n+2 L n contributions.