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Supersymmetric quantum mechanics with Levy disorder in one dimension

We consider the Schroedinger equation with a supersymmetric random potential, where the superpotential is a Levy noise. We focus on the problem of computing the so-called complex Lyapunov exponent, whose real and imaginary parts are, respectively, the Lyapunov exponent and the integrated density of states of the system. In the case where the Levy process is non-decreasing, we show that the calculation of the complex Lyapunov exponent reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Levy process. We review the known solvable cases, where the complex Lyapunov exponent can be expressed in terms of special functions, and discover a new one.


Introduction
The disordered model we consider is a particular case of the Schrödinger equation In this expression, ψ is the unknown (wave) function of the independent variable x, E is the energy parameter, and V is the random potential. If V has stationary increments, then the most accessible quantities describing the model's behaviour are the integrated density of states N (E), which counts the number of energy levels below E per unit length, and the Lyapunov exponent γ(E), whose reciprocal provides a measure of the localisation length at level E. These are conveniently expressed in terms of the limit lim x→∞ ln ψ(x, E) x where ψ(·, E) is the particular solution of Equation (1.1) satisfying ψ(0, E) = 0 and ψ ′ (0, E) = 1. This limit is a self-averaging (non-random) quantity whose almostsure value we denote by Ω(E). The relationship between the real numbers γ(E) and N (E) and the complex number Ω(E) is then (see [31,33,34]) (1.2) Ω(E) = γ(E) − iπN (E) .
We call Ω the complex Lyapunov exponent, or the characteristic function, of the disordered system (1.1).
Building on the pioneering work of Frisch & Lloyd [18], Kotani [26] made a rigorous and detailed study of the integrated density of states when the potential is a Lévy noise. In the present paper, we shall instead examine the case where (1.3) V (x) = w 2 (x) 4 − w ′ (x) 2 and it is the superpotential w-rather than V itself-that is the Lévy noise. Historically, the study of the random supersymmetric case was initiated by Ovchinnikov & Erikhman [37] in the context of one-dimensional disordered semiconductors. The same model, expressed later on in terms of Equations (1.15) and (1.16), may also be derived as the square of a Dirac operator with a random mass. As such, it is relevant in several contexts of condensed matter physics, including one-dimensional disordered metals, random spin chains and organic conductors; see [13,44] for recent reviews. From the point of view of Anderson localisation, the one-dimensional case is special, for localisation takes place as soon as there is any disorder [20,27], in contrast with the richer higher-dimensional case where a localisation transition can occur. Nevertheless, there are compelling reasons for undertaking this study: in the deterministic case, supersymmetry permits a detailed mathematical treatment of the spectral problem, leading in a few cases to exact results [15]. It is therefore natural to ask whether this favourable state of affairs persists in the presence of randomness. Recently, we showed how to extend the Frisch-Lloyd approach to a general class of random, point-like, scatterers, including one of supersymmetric type [14], and one of our aims is to develop that preliminary work so as to bring out the very elegant structure peculiar to the supersymmetric case. Another powerful motivation is the equivalence between the supersymmetric model and the mathematical description of diffusion in a random environment [9,12,29,30,41,43]. In the remainder of this introduction, we review the terminology and give an outline of our main results.
1.1. Lévy processes. Any real-valued process with right-continuous, left-limited paths started at the origin, and stationary independent increments, is called a Lévy process [1,4]. Such a process, say W , is completely determined by its Lévy exponent Λ, defined implicitly by (1.4) E e iθW (x) = e xΛ(θ) .
Conversely, given numbers µg and g, and a Lévy measure m, there is a Lévy process W with Lévy exponent (1.4); µg, g and m are called the Lévy characteristics of the process.
Roughly speaking, the paths of a Lévy process consist of intervals of drifted Brownian motion separated by jumps whose height and frequency are controlled by the Lévy measure. In the particular case where the Lévy measure m is finite, i.e. (1.7) ρ := R m(dy) < ∞ we call W an interlacing process; it may be expressed in the form where B is a standard Brownian motion, the drift 2a is given by where P is a Poisson process of intensity ρ, and the h j are random variables with probability measure m/ρ. The processes B and P , and the random variables h j , in these expressions are mutually independent. Figure 1 displays realisations of two important particular cases: (a) a standard Brownian motion, corresponding to the choice a = ρ = 0, and (b) a compound Poisson process with exponentiallydistributed jumps h j , corresponding to the choice a = g = 0 and ρ = 1.  Not every Lévy measure is finite, and so interlacing processes do not exhaust the class of Lévy processes; see Figure 2 for a particular realisation of a Lévy process that is not interlacing, and [2] for a description of other important examples.
Nevertheless, every Lévy process W may be approximated by an interlacing process W ε as follows: let ε > 0, and define from m a new Lévy measure m ε by The measure m ε is finite. Hence the process W ε with Lévy characteristics µg, g and m ε is an interlacing process, and Λ ε converges to Λ pointwise as ε → 0. The sense in which the paths of W ε approximate those of W can be made precise; see the proof of Theorem 1 in [4]. The intuitive picture suggested by this construction is that non-interlacing processes experience very small jumps with a very high frequency, as illustrated by Figure 2.  In what follows, particular attention will be paid to the subordinator case, where the Lévy process is non-decreasing. For such processes, g = 0 and the Lévy measure does not charge (−∞, 0]. A subordinator is therefore of bounded variation on finite intervals and there holds with a ≥ 0. We shall see that this case, as in Kotani's study, affords a number of simplifications.
1.2. The Frisch-Lloyd-Kotani study. Let us review the formal steps involved in the calculation of Ω when the potential is the distributional derivative of a Lévy process W . We refer the reader to the original papers [18,26] for detailed explanations, hypotheses and proofs, and to [14] for heuristics.
Written in terms of the Riccati variable Z is a Feller process, started at infinity, whose infinitesimal generator, say G , is given by (see for instance [1,39]) Z has a stationary distribution whose probability density, denoted f , is in the kernel of the adjoint G † . Hence where the integrated density of states appears as a constant of integration; N may be determined by finding a positive integrable solution and imposing the normalisation condition. The Lyapunov exponent γ is then given by the Cauchy Principal Value integral The density f contains more information than is strictly necessary to determine Ω, and it is sometimes more convenient to work with its Fourier transform f satisfies the differential equation It is then readily verified (see for instance [23], Appendix B) that where, with a slight abuse of notation,f now denotes any non-trivial solution of the differential equation (1.12) such that This representation in terms of the decaying solution of a homogeneous secondorder linear differential equation was used by Frisch & Lloyd as the basis of their numerical study of the density of states [18]. Explicit formulae were obtained by Halperin in the case of a Brownian motion [23] and by Nieuwenhuizen in the case of a compound Poisson process with a gamma distribution of the jumps [34]. Kotani performed a semiclassical analysis of the differential equation, in the limit of low energy, for the particular case where W is a subordinator [26]. By using Langer's transformation, he was able to obtain an approximation of the decaying solution valid uniformly in the variable s, from which the low-energy behaviour of N (E) could be deduced.
1.3. The supersymmetric case. The Schrödinger equation with the supersymmetric potential (1.3) can be recast as the Dirac system The meaning of these equations when w = W ′ is a Lévy noise becomes clear if we introduce an integrating factor: Let us outline the modifications of the Frisch-Lloyd-Kotani approach required in the supersymmetric case. The appropriate Riccati variable is Then (1.20) and the Lévy noise now appears as a multiplicative term; as we have written it, this stochastic equation should be interpreted in the sense of Stratonovich. In the supersymmetric case, the essential spectrum is contained in R + . It will be convenient to consider in the first instance the case E < 0, so that N (E) = 0 and Z has a stationary distribution supported in R + , whose density we denote again by f . To find an equation for f , we start from the simple observation that the logarithm of Z satisfies a stochastic equation in which the noise appears as an additive term, just as in the previous subsection. It is then straightforward to deduce In particular, if Z(0) = ∞, we may use the stationarity of the increments of W to deduce that 1 So the reciprocal of the zero-energy Riccati variable has the law of an exponential of the Lévy process W . The study of these exponentials has received a great deal of attention in the probability literature [5], and its close connection with our disordered system makes it a rich source of ideas. In particular, Bertoin & Yor [6] studied the moments of the exponentials and found for them a first-order recurrence relation which we proceed to extend to the case E = 0.
1.5. Main results and outline of the paper. In the supersymmetric case, because the noise appears in Equation (1.20) as a multiplicative term, the equation for the Fourier transform of f contains an awkward integral term. Following the example of Bertoin & Yor, it is in fact more expedient to work with the Mellin transform We derive in §2 the supersymmetric counterparts of the Frisch-Lloyd formulae (1.12) and (1.14): for E < 0, The coefficient c(s) appearing in the difference equation (1.23) was defined earlier by Equation (1.13) for s = 0, and this definition may be extended to s = 0 by taking the obvious limit. To findf (0) andf (1), one can either solve the integro-differential equation (1.21) for the distribution and then compute the relevant integrals or, alternatively, seek a positive solution of Equation (1.23) satisfying the normalisation conditionf (0) = 1. The problem of computing the density of states and the inverse localisation length in the "physical" region E > 0 is thus reduced to a problem of analytic continuation-in the energy variable-from R − to the cut plane C\R + . This formulation in terms of the Mellin transform sheds new light on the few solvable cases that have appeared in the literature; we review them in §3.
The remaining sections are devoted to the subordinator case, where we have an effective tool for performing the analytic continuation. We show in §4 that for every complex E outside the essential spectrum, and we remark that the calculation of the density of states is tantamount to solving a Stieltjes moment problem. In §5, we look for instances where the general solution of the difference equation (1.23) is known explicitly. For the subordinator with Lévy measure 1 − e −2y dy , p, q > 0, we find, with the help of Masson [32], that the complex Lyapunov exponent is expressible in terms of the parabolic cylinder functions. Then, in §6, we take up the problem of determining the low-energy behaviour of the integrated density of states and show how to compute the leading term in a semiclassical approximation; we also provide a list of the possible singular behaviours. Some of the implications of these results for diffusions in a random environment are discussed briefly. We end the paper in §7 with a few concluding remarks.

Characteristic function and Mellin transform
This section provides a derivation of Equations (1.23) and (1.24). For the particular case where W is a subordinator with finite means and a finite Lévy measure, this derivation may be made completely rigorous by adapting the arguments of Frisch & Lloyd [18] and Kotani [26]. In the general case, however, we shall be content to view the formulae as merely plausible.  1)) . In particular, c(s) exists for every s ≥ 0 if W is a subordinator with finite means.
Next, to derive the difference equation, multiply Equation (1.21) by z −s and integrate over z. We use where we have assumed that By making the obvious substitution in the integral, we obtain When we put these results together, we eventually find thatf (s) solves the difference equation (1.23).
Let us now look back on Condition (2.2) in the case s ≥ 0. The fact that the limit at infinity vanishes is a consequence of the Rice formula lim z→∞ z 2 f (z) = N = 0 .
That the limit at zero vanishes is obvious if W is a subordinator and If the Lévy process is not a pure drift then the Riccati process will behave ergodically, and so the alternative formula will hold for E = 0. Our aim is to use this formula to express the Lyapunov exponent in terms of the Mellin transform of the Riccati variable. We shall suppose that W is an interlacing process of the form (1.8-1.10), with a finite mean at x = 1. By Equation (2.1) this implies that the jumps h j themselves have a finite mean. The Dirac system (1.15-1.16) is then Here x 0 := 0 and x j+1 := x j + ℓ j where the ℓ j are independent and exponentially distributed with parameter ρ. For On the other hand, by making use of Equation (1.18), we find and it follows that Then, by summing over j, we obtain In the limit as x → ∞, this becomes This establishes the formula (1.24) for the complex Lyapunov exponent when W is an interlacing process with finite means. We expect this formula to hold more generally even in cases where the Lévy measure is not finite since, as mentioned in §1.1, every such measure may be approximated by a finite measure.

Some known solvable cases
There are very few known cases where the complex Lyapunov exponent may be expressed in terms of familiar functions. Whereas the examples we review here were discovered by solving the integro-differential equation (1.21) directly, our presentation will emphasise the alternative approach based on solving the difference equation satisfied by the Mellin transform.
Its general solution is The Mellin transform of the invariant density is positive for every s and equals unity for s = 0. Hencef and, by using Equation (2.4), we find The spectrum is the interval E ≥ a 2 ; there, we have Hence γ vanishes in the spectrum-reflecting the fact that, for the non random potential V (x) = a 2 , there is no localisation.
Example 2. The case where the Lévy process is a Brownian motion with drift has been studied independently by Ovchinnikov and Erikhman [37] and Bouchaud et al. [9]. In this case, Hencef (s) satisfies the difference equation We recognise the reccurence relation satisfied by the modified Bessel functions. The general solution is and we obtain a positive solution by taking c I = 0. Alternatively, we can solve the equation (1.21) for the invariant density. Since, in this case, the Lévy measure is identically zero, this equation reduces to a differential equation. The relevant solution is (before normalisation) and its Mellin transform agrees, as it should, with the expression found earlier by solving the difference equation. Hence, by Formula (1.22), The analytic continuation of Ω from R − to R + may be done "by hand" and consists of replacing k by −ik: Expressions for N and γ follow easily. In particular, for µ = 0 (the driftless case), we find So the density of states has a so-called Dyson-type singularity at the bottom edge of the spectrum, and the model exhibits a very interesting "delocalisation" phenomenon there [9,13,44].
Example 3. Let the Lévy exponent be This corresponds to a driftless compound Poisson process of intensity ρ where the jumps are exponentially distributed with parameter q. Then Masson [32] showed that the general solution of the difference equation (1.23) is given by and B is the Euler beta function. To obtain a positive solution, we set c − = 0. The formula eqrefmellinFormula for the complex Lyapunov exponent then agrees with the calculation of our previous article [14], where we have worked directly from the invariant density (before normalisation), namely After normalisation we obtain the Lyapunov exponent from Equation (2.4) [14]. For negative energies γ coincides with the characteristic function Ω. After analytic continuation we find the integrated density of states. The low-energy behaviour is (see Bienaimé [7]) Remark 3.1. The low-energy behaviour of the density of states in the supersymmetric case is quite different from the usual Lifshits behaviour [28], namely The physical reasons for this difference will be explained in a forthcoming paper.

Continued fractions
There is a well-known connection between linear second-order difference equations and continued fractions [19,32]; we exploit it to derive a continued fraction that may, at least in the subordinator case, be used to compute the complex Lyapunov exponent.

4.1.
Pincherle's Theorem. We say that a non-zero solution u n of the difference equation is recessive (or minimal or subdominant) if, for every other linearly independent solution v n , there holds u n = o (v n ) as n → ∞ . Pincherle's Theorem says that the continued fraction converges if and only if the difference equation (4.1) has a recessive solution u n with u 0 = 0. In case of convergence, the limit is −u 1 /u 0 where u n is any such recessive solution. The proof of this well-known result is elementary, but we include it here because it contains information that will be useful later on. Introduce two particular solutions p n and q n of the difference equation (4.1) satisfying p 0 = q 1 = 1 and q 0 = p 1 = 0 .
These solutions are obviously linearly independent, and every other solution may be expressed as a linear combination of them. Set To say that the continued fraction converges is to say that K n has a (finite) limit as n → ∞. First, suppose that the continued fraction converges; call its limit K ∞ and set It then follows from this definition of the sequence u n that u 0 = −1 and Furthermore, u n solves the difference equation. To show that u n is recessive, suppose that v n is any other linearly independent solution. We can express v n in the form v n = αu n + βq n for some number α and some non-zero number β. Then This shows that u n is recessive. Conversely, suppose that the difference equation has a recessive solution u n such that u 0 = 0. Since q 0 = 0, the particular solutions q n and u n are linearly independent. Therefore lim n→∞ u n q n = 0 .
On the other hand, we can express u n as a linear combination of p n and q n : It follows that To complete the proof, there only remains to observe that there cannot be two linearly independent solutions that are recessive.

4.2.
The subordinator case. Let us now elaborate the relevance of Pincherle's Theorem to our study. For convenience, we shall omit the drift, so that the Lévy exponent is given by Equation (1.11) with a = 0, and the coefficient c(n) by The fact thatf (s) solves the difference equation (1.23) for s ∈ N implies that u n solves The corresponding continued fraction is We now observe that the coefficients c(n) are obviously positive. This means that the continued fraction (4.5) is of the kind studied by Stieltjes [3,36,42,45]. Stieltjes showed that a necessary and sufficient condition for the convergence of the continued fraction in C\R + is that the series Hence the continued fraction converges for every E ∈ C\R + , and the limit K ∞ is a function of E analytic in C\R + . Next, let us show that E nf (n) is a recessive solution of the difference equation (4.4) when E < 0: since it is a solution, we may write E nf (n) = α [q n K ∞ (E) − p n ] + βq n for some constants α and β. Note in particular that the q n are positive, because the c(n) are positive. Suppose that β = 0. Then, since q n K ∞ − p n is recessive, we must have E nf (n) ∼ βq n as n → ∞ .
But this is absurd because the right-hand side is of the same sign for every n whereas, since E < 0 andf (n) is by construction positive, the left-hand side alternates in sign. Hence β = 0 and the recessiveness of E nf (n) follows from Remark 4.1.
We deduce from Pincherle's Theorem that for E < 0, and thence for every E ∈ C\R + by analytic continuation. Remark 4.3. Continued fractions have been used before in the study of other one-dimensional disordered systems-most notably by Nieuwenhuizen [34] and Nieuwenhuizen & Luck [35]. In these works, however, the continued fraction arose only in very special cases where the noise or randomness could be modelled in terms of the exponential and other closely related distributions.

4.3.
The density of states in the subordinator case. In particular, we have the formula On the other hand, since the c(n) are all positive, the limit of the continued fraction is the Stieltjes transform of a certain measure, say κ, on the half-line R + and we may write (see for instance [3,36]) There is a simple relationship between the Stieltjes measure κ and the integrated density of states N : by the Stieltjes-Perron inversion formula, for every open in- We then deduce from Equation (4.6) Now, as is well-known, the coefficients c(n) appearing in the continued fraction K may be expressed in terms of the positive integer moments of the measure κ; see [42] for detailed formulae. So the problem of finding the density of states is essentially equivalent to a Stieltjes moment problem.

A new solvable case
Having reviewed in §3 the few examples where the complex Lyapunov exponent is known explicitly, we now look for new cases.

Masson's difference equation.
Masson's catalog of the solutions of the difference equation expressed in terms of the hypergeometric function or its confluent limits, is particularly valuable in this respect [32].
This is Masson's difference equation (5.1) with D = 0 and z = 1/ √ −E provided that There are two possibilities: either The first case leads to solutions expressible in terms of the hypergeometric function; this is our earlier Example 3. The second case, developed in the next subsection, leads to solutions expressible in terms of parabolic cylinder functions; we follow Masson's terminology and call it the "Hermite case".

The Hermite case.
Let the Lévy process be the subordinator with Lévy measure The only dimensional parameter in this expression is p, which has the dimension of an inverse length, or equivalently of √ E. For convenience, we shall for the present set it to unity, and reintroduce it later whenever it becomes relevant. To work out the corresponding Lévy exponent, introduce the tail of the Lévy measure: This makes it clear that m is not a finite measure and that (5.5) m(dy) ∼ dy y 3/2 as y → 0 + .
We have Here, we have made use of Formula 1 in §8.380 of [21]. Hence We note for future reference the large θ behaviour Masson's Theorem 4 [32] then says that the difference equation (4.4) has two linearly independent solutions u ± n given by This fact is easily verified by using the recurrence relation satisfied by the parabolic cylinder functions (see [17], Vol. 2, Chapter VIII): Furthermore, u + n is recessive. By using the identity Γ(2z) = 1 √ π 2 2z−1 Γ(z) Γ z + 1 2 we deduce the following expression for the Mellin transform of the invariant density: Therefore, after re-introducing the parameter p, we obtain (5. 10) The constant term is given by An analytic continuation to positive energies shows that The Stieltjes measure of a ratio of parabolic cylinder functions is given explicitly in Ismail & Kelker [25], Theorem 1.4. That result, combined with Equation (4.8), leads to the following explicit formula for the integrated density of states: The low energy behaviour is Plots of N and γ for various values of the parameter q are shown in Figure 3.
Remark 5.1. Let E < 0 and set k = √ −E. By using Equation (7.727) in [21], we can invert the Mellin transformf to obtain Remark 5.2. The limit q → 0+ is singular, since On the other hand, if we set q = 0 in Equation (5.4), we obtain the Lévy measure of the process with exponent This process is well-behaved; indeed Unfortunately, Equation ( Remark 5.3. One reason for the importance of the subordinator subclass of the Lévy processes is that they arise naturally in the study of diffusion processes. More precisely, let X = X(t) be a diffusion process on R, started at the origin, and driven by the Langevin equationẊ where η is white noise and v is the deterministic drift function. Then the inverse local time of X is a process, say τ t , that measures how much time elapses before X has spent a total time t in the vicinity of its starting point. It is a well-known fact that τ is a subordinator. Given the drift v, one may in principle compute the Lévy exponent of the inverse local time by studying a certain Sturm-Liouville problem associated with the diffusion. In particular, for the drift which corresponds to the familiar Ornstein-Uhlenbeck diffusion process, one finds that the Lévy measure of τ is exactly the measure m in Equation (5.4) for q = 1; see for instance [24]. It is unclear whether this intriguing coincidence is the manifestation of a deeper connection between the Ornstein-Uhlenbeck process and the disordered model.

The low-energy behaviour of the density of states
Throughout this section, we suppose that the Lévy process W is a subordinator. Whereas Kotani's study rested on a semiclassical analysis of a differential equation, we must, in the supersymmetric case, work with a difference equation.
Finite-difference versions of the WKB method were expounded by Dingle & Morgan [16] and Braun [11], and we shall draw on their work in what follows. It will be convenient to recast the difference equation (4.4) in the form The basic formula (4.6) for the integrated density of states becomes where y n (E) is any recessive solution of Equation (6.1).
6.1. The remainders. The starting point of our analysis is the represention of the recessive solution in terms of the remainders These remainders are certainly well-defined numbers for E / ∈ R + because the continued fraction on the right-hand side converges. We note the obvious identity If we take the empty product to equal 1, it is easy to show by induction on n that E n 2 y + n = E nf (n) for every n ≥ 0 .
This shows that y + n is a recessive solution of the difference equation (6.1). We remark also that, since c(n) decays to zero, z n and z n+1 have a common limit, namely − √ −E, as n → ∞. Hence, replacing z n+1 by z n in Equation (6.4) yields We select this particular solution of the quadratic equation because it yields the correct behaviour of z n in the limit E → 0; see Equation (6.4). Henceforth, we suppose that E > 0 and set k = √ E. The expression y + n (E) should then be interpreted as y + n (E + i0+).

6.2.
An alternative formula for the density of states. The difference equation (6.1) has a turning point n k , defined to be the largest integer not exceeding the root of We proceed to express the integrated density of states in terms of the recessive solution at the turning point. Let 0 < E ≤ c(1). For n > n k , we have It then follows from Equation (6.6) that, for n > n k and E > 0, there holds Since c(n) is real, the complex conjugate y − n := y + n is also a solution of the difference equation (6.1). Furthermore, y + n and y − n are linearly independent. Introduce their Wronskian Then On the one hand, by making use of the difference equation (6.1), it is easy to see that the Wronskian of any two linearly independent solutions satisfies W n = W 1 for every n ∈ N. On the other hand, by construction, y + n has the asymptotic behaviour (6.7). Hence We deduce and so (6.8) N (E) = k π y + n k 2 .
6.3. Semiclassical approximation. There remains to find the small k behaviour of y + n k . From this point on, we shall not aim at a completely rigorous treatment, but rather at a simple informative estimate based on heuristic considerations.
Following Braun [11], we introduce a small parameter h > 0 such that The relationship between h and E is assumed to be such that there exist functions C and Z defined on R + satisfying (6.9) Detailed examples will be given in due course. Equation (6.4) for the remainder may then be expressed as x = nh . We look for a solution of Equation (6.10) of the form When we substitute into Equation (6.10), use the MacLaurin series of the exponential function, and equate like powers of h, we find, for the leading term, Then, by making use of the Euler-MacLaurin formula, we obtain (6.13) In order to have as k → 0 for n < n k , we must select the following solution of Equation (6.12): Then where x k is defined by (6.14) C(x k ) = 1 .
In view of Formula (6.8), this leads to the estimate as k → 0. The validity of this estimate is open to question, for we have assumed implicitly that the O(1) correction term in Equation (6.13) is bounded uniformly for h < x ≤ x k . We proceed to discuss some examples where this assumption seems justified. The undesirable dependence of the function C on h could be removed by shifting the index n. As it is, however, Formula (6.15) makes sense and may be used directly; shifting the index would eventually lead to the same result. We have arccosh y y + arctan 1 We deduce for some constant A independent of E. This agrees with Equation (3.3).
Remark 6.1. We remark that, for every finite Lévy measure, Therefore the asymptotic relation holds more generally for every subordinator with a finite Lévy measure.
Example 5. For the solvable model of §5.2, we have Strictly speaking, the corresponding function C is defined implicitly by Equation (6.9), but a more tractable expression may be obtained by using the identity So, for the purpose of our calculation, we may take h := 2k 2 and C(x) := 1 Formula (6.15) then yields This is in agreement with the exact result found in §5.2; see Equation (5.14).
Example 6. The so-called alpha-stable subordinator has for its Lévy measure The corresponding Lévy exponent is Some of the properties of alpha-stable subordinators are reviewed in the appendix. For p = 1, the continued fraction coefficients are c(n) = n α−1 and so we set h := (2k) where we have set We deduce from Formula (6.15) 6.4. The case of infinite Lévy measure. Let us now discuss the case R+ m(dy) = ∞ in greater generality. A first interesting observation is that the two instances encountered earlier, namely the Hermite case and the alpha-stable case with α = 1/2, have led to the same leading behaviour of the integrated density of states at low energy; compare Equations (5.14) and (6.18). The mathematical explanation is that the corresponding Lévy measures have the same singularity at y = 0; the fact that their tails are very different has little bearing on the low-energy behaviour of N . The physical interpretation is that, for an infinite Lévy measure m with support R + , the low energy eigenstates of the supersymmetric Hamiltonian are strongly affected by the frequent small jumps but depend weakly on the rare large jumps of the Lévy process. This contrasts with the case of a finite measure, where the exponent appearing in Formula (6.16) indicates that there is no such distinction.
Turning then to the analysis of the low-energy states of the supersymmetric Hamiltonian, consider the finite interval [0, ℓ]. In the absence of any potential, this interval supports an eigenstate of (kinetic) energy E kin ≈ 1/ℓ 2 . A low-energy state E ≈ 1/ℓ 2 is possible if, in the interval 0 ≤ x ≤ ℓ, the potential V = w 2 /4 − w ′ /2 remains sufficiently small to ensure that V ≪ E kin . Because W (x) is a subordinator, we expect that the potential's average value is roughly proportional to the square of W (ℓ)/ℓ. Hence, if W (ℓ) does not exceed some small value, say ε, then the potential brings in a contribution V ≈ (ε/ℓ) 2 to the energy. Equating ℓ and 1/ √ E, we deduce We argued earlier that, in this limit, only the frequent small jumps of the process matter. With this in mind, let us begin with the case of a Lévy noise whose measure exhibits the "alpha-stable singularity" m(dy) ∼ p dy y 1+α , 0 < α < 1 , as y → 0+.
In the appendix, we describe the implications of this singularity for the form of the probability density function, say f W (· ; x), of the random variable W (x); we find Equation (A.4) in the appendix, which describes the small u behaviour of this density, then leads to (6.20) ln Modulo a constant factor on the right-hand side, these heuristic arguments extend our earlier result (6.18). Finally, let us mention very briefly the limiting case α → 0, where the Lévy measure satisfies (6.21) m(dy) ∼ dy y as y → 0+ .
The corresponding process W shares this singularity with the so-called gamma subordinator; see the appendix. By the same heuristic arguments as before, we arrive once again at Equation (6.19), and the result is 6.5. Comparison with the Kotani case V = w. It is interesting to compare these results with those obtained by Kotani [26] for the random Schrödinger equation −ψ ′′ + w ψ = Eψ .
For the compound Poisson process of Example 4, Kotani showed rigorously that where ρ is defined as before by (1.7). This is the well-known Lifshits singularity [28]. For the alpha-stable subordinator of Example 6, he found Kotani also obtained exact expressions for the constant factor A; see [26], Theorem 4.7. In the supersymmetric case, the constant factor may in principle be obtained by including the next term in the WKB expansion of the remainder Z, but the calculations involved are tedious. The asymptotic behaviour (6.24) may be recovered by a heuristic argument similar to the one developed in the supersymmetric case. The main difference is that, whereas in the supersymmetric case V = w 2 /4−w ′ /2 the Lévy process is dimensionless, in the case V = w it has the dimension of the reciprocal of length. Consider as before an interval [0, ℓ], and suppose that the Lévy process does not increase beyond a certain threshold ε in that interval. The contribution of the potential energy now reads V ≈ ε/ℓ. Therefore a state of low energy The different exponent, compared to Equation (6.20), comes from the fact that the threshold ε is now dimensional and scales with energy as √ E. The case m(dy) ∼ dy/y for y → 0+ leads to the same behaviour as in the supersymmetric case; see Equation (6.22). 6.6. Classical diffusion. The formal equivalence between supersymmetric quantum mechanics and classical diffusion is well-known and has proved fruitful in the study of diffusion in a random environment [9,29,30,43]. The purpose of this subsection is to outline the most immediate implications of our results for this field of study.
Consider a diffusing particle placed initially at a position x 0 . The Fokker-Planck equation for the probability density P (· ; t) of its position at time t is where v and a are, respectively, the drift and the absorption (killing) rate of the diffusion. The transformation leads to the Schrödinger equation When v and a are random, the expected value of P (x 0 ; t) turns out to be independent of the starting point x 0 ; it is related to the density of states N ′ (E) associated with the disordered system via By standard Tauberian arguments, the long-time behaviour of this expected density is therefore completely determined by the low-energy behaviour of the density of states: The Kotani case V = w corresponds to the choice v = 0 and a = w. Here, the decay of the return probability density follows from the decay of the total probability due to absorption. For finite Lévy measures, the Lifshits singularity (6.23) leads to the well-known exponent ν = 1/3 [31]. On the other hand, when the Lévy measure is infinite, the behaviour (6.24) leads to the exponent The supersymmetric case V = w 2 /4 − w ′ /2 corresponds to the choice v = w and a = 0. In this case, there is no absorption, the total probability is conserved, and the decay of the return probability density is due to the expected drift of the Lévy process: For a pure deterministic drift, ν = 1 and the return density decays exponentially. For a non-zero Lévy measure, however, the fluctuations in the drift bring about a slower rate of decay. For example, in the case of a finite measure, we find, as in Kotani's case, ν = 1/3; for the alpha-stable case, we find For the singular measure (6.21), the spectral singularity (6.22) leads to the probability decay ln P (x 0 ; t) ∼ −t 1/3 ln 2/3 t . In this marginal case, the singular nature of the measure is only reflected in the logarithmic correction. This behaviour holds for both the random drift case and the random absorption case.

Conclusion
In this paper, we have developed the supersymmetric version of the Frisch-Lloyd methodology for computing the density of states of a system with Lévy disorder. The main novelty is that, in the supersymmetric case, the complex Lyapunov exponent may be expressed in terms of the positive solution of a difference equation. When the Lévy process is non-decreasing, the complex Lyapunov exponent has a continued fraction expansion whose coefficients have simple expressions in terms of the Lévy exponent, and the problem of computing the density of states reduces to a Stieltjes moment problem. One pleasing outcome was the discovery of a new solvable case. Our efforts to adapt to the supersymmetric case Kotani's semiclassical analysis of the low-energy regime have also borne some fruit, although our approach only gives the leading term. This analysis has shown in particular that, for infinite Lévy measures, the low-energy behaviour of the integrated density of states is controlled by the small jumps of the process. This result is in some sense the dual of that obtained in an earlier study by one of us, namely that it is the tail of the Lévy measure that determines the high-energy regime [7,8].
The paper raises a number of questions that deserve further study: (1) Much of the work presented here has been concerned with the subordinator case. From the point of view of localisation, however, the most interesting processes are those with zero means, such as Brownian motion. Hence we require a less restrictive theory. (2) As mentioned earlier, exponentials of Lévy processes may be viewed as the zero-energy case of the supersymmetric model. There has been a great deal of recent work on such exponentials, including their close connection with self-similar processes [38] and a generalisation to complex exponentials [22]. Our own findings encourage us to seek interpretations of some of these results in terms of disordered systems.
Appendix A. Some Lévy processes We recall in this appendix some basic facts concerning the distributional properties of two important classes of Lévy processes.
The distribution (A.1) may be analysed as follows: for u < 0, the analyticity of the integrand in the lower complex plane implies f (u) = 0. For u > 0 we can deform the contour of integration in order to follow the branch cut along the positive imaginary semi-axis; the result is (A.2) f W (u ; 1) = ∞ 0 dt π sin (t α sin πα) e −tu−t α cos πα and so we deduce an asymptotic power law tail similar to the Lévy measure The behaviour of the density function (A.1) for small u may be studied by using a stationary phase approximation. Let us write Since Re[ϕ(z)] < 0 for Im z < 0, we can deform the contour of integration in order to pass through the stationary point z 0 = −iα 1/(1−α) such that ϕ ′ (z 0 ) = 0; this leads to where A α = (1 − α)α α/(1−α) and B α = (1 − α)α −1/(1−α) . For example, we can verify that these behaviours are the correct ones in the case α = 1/2, since L 1/2,1 (u), the density of the so-called Lévy distribution, admits the simple expression f W (u ; 1) = 1 2 √ π u −3/2 e −1/(4u) 1 (0,∞) (u) .
A.2. The gamma process. The gamma process is characterised by the infinite Lévy measure [1] m(dy) = dy y e −by .
Using From the point of view of the statistical properties of the frequent small jumps, the gamma process can be understood as the α → 0 limit of the alpha-stable process. The distribution can be studied by considering the Fourier transform In particular, the moments are finite, given by