Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators
We propose a simple approach allowing reducing the eigenvalue concentration analysis of a class of random operator ensembles with singular probability distribution to the analysis of an auxiliary ensemble with bounded probability density. Our results apply to the Wegner- and Minami-type estimates for single- and multiparticle operators.
The role and importance of the eigenvalue concentration (EVC) estimates in the spectral theory of random operators hardly needs to be explained today. It is, however, interesting that the very first general estimate, established over thirty years ago by Wegner , is very accurate; in particular, it features the optimal dependence upon the volume of the finite-size configuration space in which the random operator at hand is considered. In the general terminology of the point random fields, the Wegner estimate is an upper bound on the first-order correlation function of the (finite) point random field on formed by the eigenvalues of the respective random operator. The next significant step in this area was made by Minami  who proved the regularity of, and obtained an efficient upper bound on, the second-order eigenvalue correlation function for . Although this result appeared in the remarkable paper by Minami  only as an intermediate (but crucial) technical estimate its importance has been immediately recognized, and what is called today the Minami estimate has been encapsulated and further extended to correlation functions of all orders , in independent works by Bellissard et al.  and by Graf and Vaghi  (the order of citation is merely alphabetical). Combes et al.  extended these results to a large class of marginal probability distributions. Minami focused in  on the local Poisson statistics of the eigenvalues; his analysis was further developed by Nakano .
The ideal situation for the methods of the above-mentioned works [1, 3, 4], considering discrete Schrödinger operators with IID random potentials, is where the common marginal distribution of the random potential has bounded density (the compactness of its support is often an additional convenient feature).
Unfortunately, one encounters some analytical problems well before the marginal probability distribution of the random potential becomes singular; it suffices to take a distribution with unbounded density, for example, the so-called arcsine law with the probability density and the gentle, integrable singularities at suffice to ruin the original proof of the Wegner estimate, although it can be enhanced so as to apply to random potentials with -summable probability density. One example of this kind is given in Section 7 (where we cite the work by Kotani and Simon ).
Stollmann  proposed a simple method which allows establishing EVC bounds for random operators with singular distributions, sufficient for the purposes of the localization theory. While the method itself proved to be flexible and has been adapted to various operator ensembles, including the multiparticle Schrödinger operators (both discrete and continuous), and even to a large class of deterministic (e.g., quasi-periodic) operators, its main drawback is that the EVC bounds it provides have a nonoptimal volume dependence.
Combes et al.  employed a powerful analytic approach and established a Wegner-type estimate for a class of Schrödinger operators in with optimal volume dependence, essentially reproducing the continuity modulus of the alloy-type external random potential.
In the framework of multiparticle random operators, the first EVC bound suitable for the purposes of the multiparticle adaptation of the multiscale analysis (MPMSA) was established by the author and Suhov (cf. ) under a very strong hypothesis of analyticity of the common PDF of the IID random potential. A particularity of the variable-energy MSA (cf. [11–13]) is that its adaptation to multiparticle operators requires two separate EVC bounds: one for an isolated cube and another one for a pair of cubes (see Section 8). In the single-particle localization theory, the latter stems easily from the former, but the situation for the multiparticle operators is much more complicated: the proof of the two-volume EVC bound requires additional, sophisticated analytical, and geometrical arguments.
Kirsch  proved a Wegner-type bound for the multiparticle lattice Schrödinger operators (LSO) under the classical hypothesis of existence and boundedness of the marginal probability density. Recently, it has become clear (cf. ) that the one-volume multiparticle EVC bound suffices for the fixed-energy variant of the MPMSA; see details in . However, the derivation of the spectral, let alone dynamical, localization for the multiparticle systems from the results of the fixed-energy MPMSA still requires the two-volume EVC bound. Such bounds were proven for the LSO in our earlier work , for random potentials with Hölder-continuous (but possibly singular) marginal probability distribution. The proof is based on an adaptation of the Stollmann lemma on monotone functions , which resulted in a nonoptimal volume dependence. Later, the method of  was extended to the multiparticle random Schrödinger operators in with the so-called alloy-type random potential, featuring Hölder continuity of the marginal PDF; see [17, 18]. Again, the volume dependence of the obtained EVC bounds was nonoptimal, due to the use of Stollmann’s lemma.
The proof of the one-volume Wegner-type estimate for multiparticle alloy-type continuous Schrödinger operators with possibly singular marginal distribution, which feature the optimal volume dependence, has been recently announced by Combes et al. .
Sabri  proposed a simple proof, building on the works by Stollmann  and by Boutet de Monvel et al. (cf. ) and making use of the so-called uncertainty principles for spectral projections. His method applies both to single- and multiparticle operators; in the latter case, both the one-volume and two-volume estimates were obtained, in a simple way, with the optimal volume dependence. We note that the approach of  applies to operators in discrete and continuous configuration spaces, including countable graphs, Euclidean spaces, and the metric (a.k.a. quantum) graphs.
A common feature of all the above-mentioned works is that, whenever the probability distribution of the ensemble of random operators becomes “singular” (starting with the case where the marginal probability distributions cease to have a bounded probability density), special efforts are made to treat this technical difficulty by functional-analytical methods.
In the present paper, we change the general attitude and replace the given ensemble of random variables, for example, an IID random field on a configuration space , relative to some probability space , with singular common marginal distribution (probability measure) , (i)first, by a suitable, stochastically equivalent IID random field on some probability space , that is, by an equivalent, constructive version of ; (ii)secondly, by a uniformly small perturbation , , of the random field , satisfying the following key properties: (1)for all , ; (2)the random variables have bounded probability density with , where is the continuity modulus of the original PDF .
This approach allows for each given reducing the task of assessing the probabilities of the form , , , where are the eigenvalues of a given random operator, to the eigenvalue concentration analysis of the regularized (-perturbed) ensemble . The latter is much easier, for it suffices to apply the classical Wegner bound or its analog for the -point joint distribution of eigenvalues, respectively. Naturally, the key point here is the well-known Lipschitz continuity (in operator norm) of the eigenvalues of self-adjoint operators, stemming from the min-max principle.
The main technical tool used in our approach is the elementary method of generating random variables with continuous PDF from a standard random variable uniformly distributed in : assuming that the inverse of the PDF is well defined, the random variable has the probability distribution function . (See the details in the next section.) It is not difficult to see that replacing by a suitable, piecewise affine, strictly monotone function can provide a random variable satisfying both of the above-mentioned conditions (1)-(2). In simple terms, the probability distribution of is replaced by a mixture of uniform laws with -small disjoint ranges.
In Section 3 we describe an alternative, and quite natural, construction which gives rise to a similar result, except that is not necessarily piecewise affine, and the respective probability measure is simply the convolution of with a kernel , having the support of diameter .
2. Basic Notations, Facts, and Assumptions
A real-valued random variable on a probability space is a measurable mapping , where is the Borel sigma-algebra on . One can associate with its probability distribution function (PDF),
and a probability measure on ,
which is uniquely determined by the PDF: The choice of the form of the inequality in (3), namely, or , is quite arbitrary and varies from one textbook to another. With the choice made in (3), the function is upper continuous. is monotone increasing and may have at most countable (possibly empty) family of intervals (plateaus); we assume that they are numbered in some way (not necessarily in increasing order, of course): , , so .
The measure (with the PDF ) has at most countable (possibly empty) number of atoms , , with weights .
We shall consider only random variables with continuous probability distribution or, equivalently, with continuous PDF, so that for all . A number of conventions, notations, and facts will be contingent upon this assumption.
In the context of the present paper, the simplest case is where is a homeomorphism, thus with no plateaus (strictly increasing) and no jumps (no -atoms). Naturally, this implies that the associated random variable is lower and upper unbounded. A classical example is the standard Gaussian law (indeed, any Gaussian law , , ). The inverse is then a well-defined function from to and can be used for generating a Gaussian random variable , where has the standard uniform distribution . For this reason, we will often refer to the inverse as the generator function (not to be confused with the notion of generating function appearing in a different probabilistic context) and denote it by .
In very general terms, the notion of generator function can be defined as follows: a generator function of a probability measure on is a measurable function such that is the image by of the Lebesgue measure on . However, this degree of generality does not quite suffice for our purposes, and we have to pay more attention to the details of the mapping .
For the sake of clarity, we will first illustrate the main idea of the stochastic regularization in this case, thus avoiding a number of boring technicalities. The extension to the case where is an arbitrary interval is not difficult, either.
Minor technical adaptations are required in the general case, where may have plateaus; that is, may vanish on some subintervals of ; then its generator function cannot be defined as the inverse function , for the latter, strictly speaking, may not exist.
Notice that is strictly increasing, whenever the respective measure has no atom.
2.1. Generating a Random Variable with Known PDF
Consider the open interval and suppose that the function is strictly increasing and continuous.
A well-known fact, often used in the probability theory and in numerical simulations of random variables, is that the random variable with the PDF is stochastically equivalent to (i.e., has the same probability distribution as) the random variable defined on the probability space by Here is the restriction of to , and mes is the Lebesgue measure on . For all intents and purposes, is a faithful and convenient representation of , as long as one considers alone. In a more general situation, where there is a family of IID (independent and identically distributed) random variables on , with the common marginal PDF , such a family is stochastically equivalent to the IID family of random variables on with where are replicas of mes.
2.2. Smearing the Probability Distributions
For clarity, we start again with the simpler situation where admits a continuous inverse , that is, if had no plateau, and later adapt the main idea to discontinuous .
2.2.1. The Simplest Case , Is Continuous
Here, has no plateau, and is both lower and upper unbounded. We denote and call the generator function of the random variable (or, equivalently, of the probability measure with PDF ). We denote by the continuity modulus of the function ( is often called the concentration function of the measure ).
Introduce the following partition of into disjoint intervals: Note that so , . By definition of the continuity modulus , we have Next, introduce the piecewise affine, continuous function taking the same values as at all points , . The explicit interpolation formula is as follows: It follows directly from the definition that so that while for (i.e., for mes-a.e. ), we have by (11) As a result, the random variable (which is an -perturbation of in the norm ) admits probability density satisfying (recall: is fixed here) Therefore, for any , one has In particular, for any interval ,
The extension to more general uniformly continuous (e.g., Hölder-continuous) functions is straightforward. One can also consider a uniformly continuous mapping into a metric space with distance .
An important particular case is where : indeed, the eigenvalues of a self-adjoint operator are Lipschitz continuous with , with respect to the norm of the perturbation. In this case,
In the probabilistic terminology, the distribution of is a mixture of uniform distributions with weights . This means that can be determined by a double sampling, namely, (i)first by selecting a sample value of the random index , distributed with the probabilities ,(ii)then by sampling the random variable . Each distribution has bounded density, and so does the mixture.
2.2.2. The General Result
In a more general situation, where the PDF of a probability measure may have plateaus, the above construction becomes technically more involved, but the final result still remains true.
Proposition 1. Let be the generator function of a continuous probability measure on , with the continuity modulus . For any there exists a generator function with the following properties: (A);(B)the measuregenerated by admits bounded probability density with .Moreover, can be constructed in such a way that either(A) or, if desired,(B).
See the proof in appendix. One can also preserve both extremities of ; this would require some modifications of the construction and a minor change of the constants in the final formulae.
3. Stochastic Regularization by Convolution
Fix some and consider a bounded measurable function satisfying ; we will refer to such functions as convolution kernels. Given a probability measure on , which we still assume continuous, let be its PDF and its continuity modulus. Further, introduce the measure given by the convolution so that its PDF is given by Consequently, In particular, taking we obtain Also, we can write In other words, is the expectation of , where the random variable is distributed with density ; thus and
The following statement assesses the amplitude of the perturbation of the generator function, , induced by the convolution with .
Proposition 2. Let be a convolution kernel with . Let be the generator function of a continuous probability measure on , and consider the generator function for the probability measure . Then for Lebesgue-a.e. , the following inequality holds true:
The proof is a bit technical, so we put it in appendix. The main technical difficulty comes from the possible discontinuity of the generator functions—in the case where the respective PDF has plateaus. The reader willing to consider only the case where the measure does not vanish on any interval inside will find that this assumption simplifies the proof considerably and makes it much more transparent. Note that the most popular models of disorder used by physicists—Gaussian, exponential and uniform distributions—all fit in this class, along with their numerous modifications.
Finally, note that if, for example, is lower-bounded and the lower edge of its essential support is to be preserved, one can combine the convolution with the shift by .
4. The Stochastic Regularization Principle
The results of the previous sections can be summarized now in the following.
Theorem 3. Let be a random variable on a probability space with the continuous probability distribution function ; let be the continuity modulus of . There exists a stochastically equivalent random variable on the probability space , where is the Lebesgue measure on the Borel sigma-algebra restricted to , with the following property.
For any , there exists a random variable on such that (1); (2) admits a bounded probability density with . Property (2) implies, naturally, that the continuity modulu of the probability distribution of admits a linear bound; for any
Theorem 4. Let be a random field on a nonempty set relative to a probability space , with independent values and continuous marginal probability distribution functions , . Let be the continuity moduli of , . Then there exists a stochastically equivalent random field on the probability space , where (i),(ii) is the sigma-algebra generated by the cylinder sets in ,(iii) and are replicas of the Lebesgue measure on ,(iv), where is the generator function of the probability distribution of ,with the following property.
For any , there exists a random field with independent values such that (1); (2)each has the probability density , with .
Proof. The assertion follows directly from Theorem 3, for it suffices to construct individually the r.v. and with the required properties.
5. Minami-Type Estimates for Singular Distributions
Recall the following result by Minami. Let be the lattice Schrödinger operator on a finite lattice subset with IID random potential , and assume that the marginal probability distribution of admits bounded probability density . Denote , and let be its matrix elements in the standard delta basis on .
Lemma 5 (cf. [3, Lemma 1]). For any , any cube , and for any with , one has
This was the crucial observation for assessing the expectation leading, ultimately, to what is now called the Minami estimate: where is the spectral projection of on the interval . In particular, this implies the a.s. simplicity of the eigenvalues of , under the strong hypothesis of existence and boundedness of the marginal probability density .
More generally, under the same hypothesis, for any , This result does not rely upon the specific structure of the kinetic energy operator and remains valid for any ensemble of random matrices of the form where is self-adjoint and IID random variables have bounded common probability density . Therefore, we proceed in this general context.
Combes et al.  proved an analog of the Minami bound for singular marginal distributions. However, I would like to point out that the stochastic regularization technique allows extending—effortlessly—the Minami-type estimates well beyond the original class of random operators for which the proofs in [2–4] were designed. Therefore, the results of  on Minami-type estimates actually follow easily from [2–4] by a simple application of our general result on uniform stochastic regularization, as shown in the following statement, recovering a part of results of .
Theorem 6 (cf. [2–4]). Fix an integer and consider a family of IID real-valued random variables on a probability space and an ensemble of random matrices
where (hence, ) is Hermitian.
Let be the common PDF of the r.v. , and assume that admits the continuity modulus . Then for all , any , and interval of length , one has
Proof. Without loss of generality, we can assume that the probability space is endowed with the Lebesgue measure, and , where is the generator function of the probability distribution with the PDF . (Otherwise, we replace the ensemble by a stochastically equivalent one, having the described form.)
Fix and a bounded interval . By Theorem 3, there is a generator function with such that the IID random variables satisfy the following conditions: (1);(2) admits the probability density with .
Next, consider the ensemble of Hermitian random matrices Set for brevity , .
For any , Let be the eigenvalues of (numbered in increasing order) and the eigenvalues of , respectively. By the min-max principle, for any Therefore, if there are at least eigenvalues , , then necessarily , ; thus By the previously mentioned property (2), the random variables admit the bounded density , so the random matrix ensemble satisfies the hypotheses of Theorem 6, and therefore, as asserted.
With , we recover the Wegner-type bound with optimal -dependence.
Proposition 7. The following inequality holds true:
Proof . The expectation of a random variable with values in is given by the well-known formula (stemming from the Abel transformation):
Consequently, by (42) we have
6. Simplicity of Spectra of Anderson Hamiltonians
6.1. The Klein-Molchanov Result for Potentials with Bounded MPD
Lemma 8 (cf. [22, Lemma 1]). Let be an eigenvalue for the lattice Schrödinger operator in with two linearly independent eigenfunctions such that for some and Then there exists such that, with and , one has for all sufficiently large .
Corollary 9. Let be an eigenvalue for with two linearly independent eigenfunctions such that Then for any there exists such that, for any , with and , one has for all sufficiently large .
Hypothesis (KM). Let be the event that for all subintervals with length . Then
The choice of the constants in (KM) is dictated by the Minami estimate: In fact, the following more general hypothesis is sufficient for the proof of the a.s. simplicity of p.p. spectrum in the Anderson localization regime.
Hypothesis (KM′). Let be the event that for all subintervals with length . Then
We will say that a function on the lattice is fast decaying if it decays at infinity faster than any power law.
Theorem 10 (cf. [22, Main Theorem]). Under the hypothesis (KM′): (A)with probability one, the Anderson Hamiltonian cannot have an eigenvalue with two linearly independent fast decaying eigenfunctions;(B)consequently, in any interval where has a.s. p.p. spectrum with fast decaying eigenfunctions, the spectrum of is a.s. simple.
Proof . We slightly modify the original proof given by Klein and Molchanov, so as to make use of the more general hypothesis (KM′) replacing (KM).
Let be a bounded open interval, and set for . By (KM′) combined with the Borel-Cantelli Lemma, that for any and some with such that for all , there exists such that the event occurs for all .
Now the argument becomes deterministic, and we fix an arbitrary .
Let be an eigenvalue of with two linearly independent associated eigenfunctions with -decay for some . Then Lemma 8 implies that for all large enough, , where is a subinterval of with , which contradicts the definition of the events , since for there exists such that .
6.2. Extension to Hölder-Continuous Marginal Distributions
In this section, we show how the results of  on the simplicity of spectra of Anderson Hamiltonians can be easily inferred from the original Klein-Molchanov theorem , without any additional analytical work, by applying the stochastic regularization principle. We thus recover the statement of [5, Theorem 3.1].
Assume that the marginal PDF of the random potential field on is Hölder continuous of order .
By (43) replacing the Minami estimate with and , we have An interval can be covered by intervals of length in such a way that any subinterval of length will be contained in one of the intervals . Therefore, For any and , so that , we obtain
7. Wegner Estimates for Unbounded Random Operators
A number of EVC estimates have been obtained over the last decades for various classes of unbounded random operators in , including Schrödinger operators with the alloy-type random potential where is a -periodic lattice, as well as operators
Assuming that the random scatterers amplitudes admit a bounded probability density , Martinelli and Holden  proved the following bound:
Following (and further developing) the original Wegner’s strategy , Kirsch  gave a short proof of an EVC bound with the quadratic volume dependence for the alloy model, under the assumption of existence, boundedness, and compact support of the common marginal probability density of the IID scatterers amplitude.
A different simple approach was developed by Stollmann  who used his lemma on monotone functions. The EVC bound obtained by Stollmann also features the quadratic volume dependence. This is an artefact of the Stollmann lemma unrelated to the possible singularity of the marginal distributions (e.g., of the scatterers’ amplitudes ). The same quadratic volume dependence is obtained even for a.c. marginal distributions with a bounded density, as long as the Stollmann lemma is used in the proof. On the other hand, this method proved to be very general and flexible. (See, e.g.,  where it was applied to multiparticle systems and  where it was used in the framework of the multiscale analysis of quasiperiodic operators.)
Combes et al. proved an optimal EVC bound for Schrödinger operators in (possibly with magnetic field), with linear volume dependence.
Note that the possible singularity of the probability distribution of the random potential is certainly not the only technical difficulty encountered in the study of the eigenvalue concentration. Even if the marginal distributions in question admit uniformly bounded densities, the monotonicity arguments (for nonnegative potentials) are sufficient for a relatively simple proof in the case where (the so-called complete covering case). If the “bump potentials” (otherwise called scatterers) are only positive on small balls not covering the entire Euclidean space, one needs complementary arguments (the unique continuation principle = UCP, or, e.g., analyticity arguments).
While the stochastic regularization technique does not solve the hard analytic problems related to the incomplete covering and the unboundedness of the random operators, it allows concentrating one’s efforts on the most simple—from the probabilistic perspective—case where the disorder is induced by a countable family of random variables with bounded densities.
To illustrate possible applications of the stochastic regularization principle to the unbounded random operators, recall a well-known result by Kotani and Simon  published more than twenty years ago; developing the ideas and techniques from [28, 29], they proved the following.
Theorem 11 (cf. [7, Theorem 6.1]). Let be the random continuous Anderson-type Hamiltonian on , with the alloy-type random potential of the form where are the indicator functions of the unit cubes centered at and the random variables are IID with distribution . Suppose that and that has compact support. Then, for each there exists , depending only upon , and , such that and a similar bound holds for the restriction of to an arbitrarily large cube with periodic boundary conditions.
Further, it was pointed out in  that the proof of Theorem 11 can be adapted to the case where the common marginal density is unbounded, but for some (and still is compactly supported; the case of the unbounded support can be treated by a limiting procedure). For the reader’s convenience, we encapsulate Remark 3 from  in the following.
Proposition 12. Consider the same model as in Theorem 11, but suppose that the marginal probability density , , and has compact support. Then the integrated density of states (IDS) is locally Hölder continuous of order .
The proof, sketched in , makes use of the Hölder inequality, involving the function, where is the probability distribution function of the random variables and is the Radon-Nikodym derivative of the marginal probability measure (with the distribution function ) with respect to the Lebesgue measure . Formally, the (sketched) proof does not apply to probability measures singular with respect to the Lebesgue measure, even if is Hölder continuous of some order .
A direct application of the stochastic regularization technique leads to the following simple generalization of the Kotani-Simon theorem, where the marginal distribution is simply continuous with some continuity modulus , not necessarily admitting a power-law upper bound.
Theorem 13. Consider the same model as in Theorem 11, with compactly supported common probability measure of the random variables , but suppose that the marginal probability distribution function is continuous with continuity modulus . Then the IDS admits locally continuity modulus ; namely,
8. Optimal EVC Bounds for Multiparticle Systems
In this last section, we briefly describe another interesting application of the stochastic regularization technique—the spectral theory of random multiparticle operators. We do not formulate any specific result, and our discussion remains rather informal, for the simple reason that formulating any result in this direction would have required several pages of preliminary notions; this task is most certainly beyond the scope of the present paper.
The first EVC bounds for Hamiltonians of interacting quantum particle (lattice) systems subject to a common external random potential were obtained in our earlier work , under a strong assumption of analyticity of the marginal PDF in a strip around the real line. The main examples here are given by a Gaussian IID random field on and, more generally, by the so-called -stable laws, indexed by a real parameter ; corresponds to the Gaussian distributions and to the Cauchy distributions.
The localization analysis for multiparticle (even two-particle) disordered systems requires two kinds of EVC bounds, which are essentially equivalent for single-particle systems: with some functions , , , and ,(1)one-volume bounds of the traditional form (2)two-volume bounds of the form for disjoint cubes (or balls in some appropriate metric) , .
In the single-particle theory, the second EVC bound follows immediately from the first one by conditioning on one of the sets , , if the random potential is IID. Minor modifications are required to obtain a satisfactory bound for strongly mixing random potentials and sufficiently distant sets , .
The situation is much more complicated for multiparticle Hamiltonians, particularly for ; see the discussion in . It turns out that it is easier to prove the two-volume bounds for special pairs of cubes, called separable in . This notion does not appear in the fixed-energy multiparticle MSA, which is substantially simpler than its variable-energy counterpart developed earlier in .
Kirsch , following and adapting the original Wegner’s approach, gave an elegant proof of the optimal one-volume bound for the -particle lattice Schrödinger operator, under the assumption of existence and boundedness of the compactly supported MPD of the IID random potential in , for arbitrary and . However, it was not clear until recently if the one-volume bound suffices to establish any kind of localization in multiparticle systems. The following is shown in our work . (i)The one-volume EVC bound alone implies the a.s. absence of a.c. spectrum of multiparticle Hamiltonians, provided that the common PDF is Hölder-continuous of some order or at least log-Hölder continuous; namely, with sufficiently large . To this end, it suffices to make use of the multiparticle adaptation of the fixed-energy MSA proposed in .(ii)The two-volume EVC bound for separable pairs of sets , (again, with Hölder or log-Hölder type continuity modulus) is sufficient for the derivation of multiparticle spectral and strong dynamical localization from the probabilistic bounds provided by the fixed-energy MPMSA.
The two-volume (as well as the one-volume) EVC bound for discrete multiparticle Hamiltonians was proven in our works [10, 16], under the weaker assumption of continuity of the common PDF of an IID random potential. For such bounds to be useful in the context of (the multiparticle adaptation of) the multiscale analysis (MPMSA), one needs to assume that the common PDF is at least log-Hölder continuous. Due to the use of the Stollmann lemma, adapted to the multiparticle context, the volume dependence was nonoptimal in [10, 16]; in particular, for the spectra of -particle Hamiltonians in cubes , of radius centered at and , in the -particle configuration space , one has
Recently, Sabri  improved the results of [14, 16] and obtained optimal one- and two-volume EVC bounds for the multiparticle systems. As usual, special efforts are required in his proof (building on ) to treat singular (continuous) marginal distributions.
The stochastic regularization technique, in a certain way, does justice to Kirsch’s method: following the same path as in the previous sections, it is not difficult to infer from the main result of , combined with Theorem 3, the one-volume multiparticle EVC with optimal volume dependence.
The first EVC bounds (both the one- and two-volume versions) for multiparticle Schrödinger operators in with the alloy-type random potential were proven in . For the multiparticle version of the Hamiltonians on quantum (or metric) graphs, the EVC bounds were recently proven by Sabri in . Both [17, 33] rely upon the Stollmann lemma on monotone functions, thus featuring a nonoptimal volume dependence. Naturally, only the continuity of the marginal PDF is required for the EVC bounds, but one needs the stronger assumption of log-Hölder continuity for these EVC bounds to be efficient in the course of the MPMSA.
As was already said, Sabri improved his EVC bounds in .
A. Proof of Proposition 1
If the measure vanishes on some subintervals of , then, strictly speaking, is no longer a function well defined on the range of (cf. Figure 1). There is a finite or countable set , , and disjoint intervals such that . One can still define the right (but not the left) inverse ; we set and denote the obtained function again by (the generator function of ). Note that it is lower continuous (cf. Figure 1): .
Recall that in Section 2.2.1 we constructed by linear interpolation of on the sequence of intervals partitioning first the real line into subintervals of length . Such a complete covering is both unnecessary and inconvenient for our purposes, when vanishes on some intervals , so we modify the general strategy and construct only a covering, by a sequence of intervals of length , of the support of the measure , skipping its “gaps.”
Construction of on Some Interval . If is not in any plateau of , so that is a continuity point for , we set , , and .
Otherwise, we have , for some , and in this case we set In any case, Observe that with this definition is always a right growth point of ; that is, Next, assume that we have already defined, for some , and that we have Now define , with , by requiring that if the RHS is finite. If it is infinite, this means that , so we stop the recursive construction on the half-line . In the case where the new interval is actually constructed, set It is worth noticing that one always has : this follows immediately from the first equality in (A.8). This means that although we do not necessarily partition the entire real line (only ), we do partition the entire interval into the subintervals .
Further, for each constructed interval , define on by linear interpolation of : for all , thus .
At the same time, by definition of the continuity modulus , thus
Construction of on . Now we start with . If the interval had been constructed (i.e., does not vanish entirely on ), we make use of the point . Otherwise, we set formally , only for the purposes of this second stage of our inductive construction (but do not attempt to construct the interval which is useless when ).
Now we use essentially the same recursion as above, but in the negative direction of the real line. Given the point , with , define , with , by requiring that If the above RHS is infinite, this means that , so we stop the construction, since is covered by the intervals .
Again, while attempting to make the next step of recursion, we skip the useless part of the real line where the random variable takes values with probability .
As before, set
It follows from our construction that the intervals , , cover the entire interval , for we continue the progression to the right (resp., to the left) as long as the support of is not completely covered.
Conclusion. We come to the following formula for : may be discontinuous, but it is at least piecewise affine on with associated intervals .
Owing to the bounds (A.13) holding on each interval , we obtain the global bound for the perturbation (the latter is just a different name for ): For (a.k.a. ) we have another global bound, for all for which the derivative exists; namely, for all , As a result, the random variable admits bounded probability density satisfying the upper bound Therefore, for any , one has In particular, for any interval ,
A.1. Bounded or Semibounded
First, let . Then we can proceed essentially as above, but setting .
It is worth noticing that this procedure preserves the lower edge of the values of the random variables , which may be convenient in the Lifshitz-type analysis of the band-edge localization.
Respectively, if is upper bounded, with , one can proceed as in the case where we treated the positive half axis, but setting . This preserves the upper edge of the values of .
If is bounded, preserving both edges of may require minor quantitative adaptations; we do not discuss this matter here.
B. Proof of Proposition 2
Recall that there exists at most countable (and possibly empty) set of values where is discontinuous: every such value is given by the (constant) value taken by on an constance interval , , and there may by at most a countable family of such plateaus. In this case ; recall also that we set .
Since the probability measure on considered in our models is the Lebesgue measure, the set of discontinuity points of has measure zero and is negligible for the purposes of probabilistic analysis. Taking a quick look at the structure of the proof below (Case I.1 through Case II.3), the reader can see that it becomes much shorter and more transparent if is strictly increasing on the interval .
Fix some and let so that does not belong to a plateau of ; that is, is strictly increasing at .
We know from (28) that
Case I. Consider the first inequality:
then is not a discontinuity point of (for is not, by assumption), so is well defined as function () at , and we have
that is, is not a discontinuity point of . Then we apply directly to both sides of the inequality the function , which acts at both values as :
(3)Now suppose that the inequality (B.7) holds, but is a discontinuity point of . Then , where .
The problem with this situation is that may be far away from , making a direct estimate inefficient. However, we may act differently and notice that since the inequality in (B.7) is strict, there exists a point such that . Indeed, is nonempty, for is countable, and is continuous by assumption, thus taking all intermediate values. Then we have, with , so it follows from the first inequality, by strict monotonicity of , that and since ,
We conclude that in all three cases (B.6), (B.9), and (B.12)
Case II. Now consider the second inequality: (1)Again, consider first the case where , so is a continuity point for which acts here as , and apply : thus (2)Next, suppose that but is a continuity point for ; then, applying , we obtain so that (3)Finally, suppose is a discontinuity point of : for some