Generic Behavior of the Density of States in Random Matrix Theory and Equilibrium Problems in the Presence of Real Analytic External Fields

Transcription

1 Generic Behavior of the Density of States in Random Matrix Theory and Equilibrium Problems in the Presence of Real Analytic External Fields A. B. J. KUIJLAARS Katholieke Universiteit Leuven AND K. T-R MCLAUGHLIN University of Arizona Abstract The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example, in random matrix theory: The limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the Euler-Lagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a bulk, in which there is universal behavior involving the sine kernel, and edge effects, in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any one-parameter family of external fields V /c, the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials, and integrable systems. c 2000 John Wiley & Sons, Inc. Contents 1. Introduction and Statement of Results Applications Proof of Theorem Preliminaries from Potential Theory Proof of Theorem 1.3(ii) Proof of Theorem 1.3(i), Part I Proof of Theorem 1.3(i), Part II 756 Communications on Pure and Applied Mathematics, Vol. LIII, (2000) c 2000 John Wiley & Sons, Inc.

2 GENERIC DENSITY OF STATES Auxiliary Results I: Rate of Convergence of a j (c) to a j Auxiliary Results II: Two Lemmas on Balayage Proofs of Lemmas 7.3 and Proof of Lemma Proof of Theorem 1.3(iii) Proof of Theorem 1.3(iv) 782 Bibliography Introduction and Statement of Results In a number of areas in classical analysis, a crucial role is played by a measure characterized by an extremal problem for logarithmic potentials. These areas range from approximation theory, orthogonal polynomials, and random matrices to singular limits of integrable systems; see, e.g., [5, 9, 14, 16, 28]. The extremal problem involves a continuous function V : R R, called an external field, satisfying the growth condition (1.1) V (x) lim x log x =+. The weighted energy of a Borel probability measure µ is 1 (1.2) I V (µ) := log s t dµ(s)dµ(t)+2 V (t)dµ(t), and the extremal weighted energy is { } (1.3) E V := inf I V (µ) : µ 0, dµ = 1. The equilibrium measure µ V in the presence of the external field V is the unique Borel probability measure on R minimizing I V (µ) among all Borel probability measures on R. It follows from the growth condition (1.1) that the support S V of µ V is a compact set. This problem arises in the study of ensembles of random matrices having probability distributions given by (1.4) Π n (M)dM = Z n 1 e 2nTr(V (M)) dm, where ( n dm = dm kk ) d ( ) ( ) ReM kj d ImMkj k=1 k< j is the Lebesgue measure on the space of n n Hermitian matrices and Z n is the normalization factor. It is well-known that the induced probability distribution on the eigenvalues λ 1 λ n is related in a direct way to the sequence of

3 738 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN polynomials orthogonal with respect to the measure e 2nV (x) dx. For example, if N n (M; ) is the fraction of eigenvalues of the matrix M that lie in the interval = [a,b], then the expected fraction of eigenvalues in is given by E(N n ( ; )) = 1 n e n 1 2nV (x) j=0 p 2 j(x;n)dx (see, for example, [22]), where p j (x;n) are the polynomials orthonormal with respect to the measure e 2nV (x) dx. Many of the central questions about ensembles of random matrices with probability distributions of the form (1.4) involve asking what happens when n, the size of the matrices in question, tends to infinity. For the quantity E(N n ( ; )), one has the remarkable connection to the equilibrium measure for V (see [2, 14, 15, 22]) (1.5) lim E(N n( ; )) = µ V ( ). n If µ V hasadensityψ = ψ V, then this density is called the density of states in random matrix theory. The classical example is the Gaussian unitary ensemble for which V (x) =x 2. The associated density of states is given by the so-called semicircle law, (1.6) ψ(x)= 2 π 1 x 2, x [ 1,1]. For different V s, the density of states can be quite different. It may be supported on many disjoint intervals, and it may vanish at interior points of its support. However, one might expect that the behavior exhibited in (1.6) that ψ is positive on the interior of its support and vanishes like a square root at endpoints is the typical behavior for the density of states. The purpose of this paper is to show that this is indeed the case if we restrict our attention to real analytic external fields. While the density of states depends on V, many local statistical quantities concerning the eigenvalues are independent of the exact form of the function V. This is known (loosely speaking) as the universality conjecture of random matrix theory. One precise form of this conjecture is that, independent of the choice of V, the following asymptotic description of the reproducing kernel for the orthogonal polynomials holds, K n (x,y)=e ( a + n 1 n(v(x)+v (y)) j=0 p j (x;n)p j (y;n) ) η sinπ(ξ η) = K n (a,a) π(ξ η) 1 lim n K n (a,a) K ξ (1.7) n K n (a,a),a + for each a supp(ψ) such that ψ(a) > 0. This result was first proved by Pastur and Shcherbina [24] (see also [1] and [8]). Our results imply that generically (1.7) holds for all interior points of the support of ψ.

4 GENERIC DENSITY OF STATES 739 Similarly, it is known that if a is an endpoint of the support S V of ψ V (a left endpoint, for example) and ψ V (t)=c t a(1 + O( t a )), c > 0, then the limit (1.7) is replaced by (1.8) 1 lim n cn 2/3 K n ( a + ξ,a + η cn2/3 ) cn 2/3 = Ai(ξ)Ai (η) Ai (ξ)ai(η) ξ η where Ai is the Airy function (see, for example, [31] and [1]). Our results imply again that the behavior in (1.8) is generic for endpoints of the support of ψ. Other applications of our results to orthogonal polynomials and integrable systems will be presented in Section 2. In [6] and [8], the notion of a regular external field V was introduced in terms of properties of the measure µ V and the following variational conditions associated with the minimization problem (1.3). There is a real constant l = l V such that (1.9) Lµ V (x) V(x)=l V, x S V, (1.10) Lµ V (x) V(x) l V, x R, where the operator L is defined by (1.11) Lµ(z)= log z t dµ(t), z C. If µ has a density ψ, we will also use the notation Lψ instead of Lµ. The properties (1.9) (1.10) characterize the measure µ V. Following [6] and [8], we now state the following definition: DEFINITION 1.1 The real analytic external field V is called regular if the following three properties hold: (i) The inequality (1.10) is strict for every x R \ S V. (ii) The density ψ V is positive on the interior of S V. (iii) The density ψ V vanishes like a square root at each of the endpoints of S V. That is, if a is a left endpoint of S V, then for some c > 0, ψ V (t)=c t a(1 + O(t a)) as t a, and if b is a right endpoint, then for some c > 0, ψ V (t)=c b t (1 + O(b t)) as t b. Remark A. For real analytic external fields, Deift, Kriecherbauer, and McLaughlin [5] showed that the support S V consists of a finite union of closed intervals and that the measure µ V has a density ψ = ψ V of the form ψ(x)= 1 (1.12) qv (x), x R, π where q V is a real analytic function on R and qv denotes its negative part. The endpoints of the intervals in S V are given by the zeros of q V with odd multiplicity.

5 740 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN In view of (1.12), property (ii) is equivalent to the fact that q V does not vanish in the interior of S V, while (iii) says that it has a simple zero at each of the endpoints of S V. Property (i) is satisfied if the set (1.13) SV := {x R : Lψ(x) V(x)=l} is equal to S V. The function q V has a zero of even multiplicity at each point of x SV \ S V. Thus SV is equal to S V plus a finite set of isolated points. We also see that V is regular if and only if q V does not have any multiple zeros on SV. Remark B. If V is nonregular, then any point x SV where q V has a multiple zero will be called a singular point for the external field V.Sinceq V is real analytic on R and SV is compact, the set of singular points is finite. According to properties (i) through (iii) of Definition 1.1, the singular points can be divided into three types: The points in SV \S V are called the singular points of type I. These are exactly the isolated points of SV. Points in the interior of S V where ψ V vanishes are called singular points of type II. If x is such a point, then we have ψ V (t) (t x ) 2k, t x, for some integer k 1. Here and in the sequel we use f g to denote that the ratio f /g lies between two positive constants for the range of parameters indicated. Endpoints of S V where q V has a multiple zero are called singular points of type III. At such a point x, we have for some integer k 2, ψ V (t) t x k 1 2 for t in a left or right neighborhood of x. It follows from this that Lψ V is a C k function in a neighborhood of x, but not a C k+1 function. Thus an endpoint x is a singular point if and only if Lψ V is a C 2 function in a neighborhood of x. We note that the number k here can, in fact, be an odd number only; see [5]. In this paper, we study the regularity of real analytic external fields. The following is our first result: THEOREM 1.2 Suppose V and V n,n= 1,2,..., are real analytic external fields on R such that the following hold: (i) For j = 0,1,2,3, we have lim n V n ( j) = V ( j) uniformly on compact subsets of R. (ii) The growth condition lim x V n (x)/log x = of (1.1) holds uniformly in n. Then there exist A,B > 0 such that S Vn [ A,A] and SV n [ B,B] for all n. Furthermore, if V is regular, then there is an n 0 such that V n is regular for all n n 0.

6 GENERIC DENSITY OF STATES 741 A large part of the paper is devoted to a detailed study of the one-parameter family of external fields (V /c) c>0,wherev is a given real analytic external field. This family appears naturally in the study of orthogonal polynomials and a number of its properties were obtained in [3, 4, 28, 30] using methods from logarithmic potential theory. The family (V /c) c>0 also plays a role in the description of the continuum limit of the Toda lattice [9]. In the study of a family (V /c) with fixed V, it is useful to write for c > 0, (1.14) and µ c := cµ V /c, ψ c := cψ V/c, l c := cl V/c, (1.15) S c := S V /c, Sc := SV /c, where SV /c is defined as in (1.13). A special property of this one-parameter family is that the measures µ c and the sets S c and Sc are increasing with c; see [3, 4, 28, 30] and Section 4 below. We also get from (1.9) (1.10), (1.16) Lψ c (x) V(x)=l c, x S c, (1.17) Lψ c (x) V(x) l c, x R, and (1.18) dµ c = c, µ c 0. The measure µ c minimizes the weighted energy I V (µ) of (1.2) among all positive measures µ on R having total mass c. We call c > 0aregular value for V if V /c is a regular external field. Otherwise, it is a singular value. Based on the types of their singular points, we may also divide the singular values into three types. That is, we say that c is a singular value of type I, II, or III if V /c has a singular point of the corresponding type. Note that a singular value may have more than one type. The following is our main result: THEOREM 1.3 Let V be a real analytic external field. Then the following hold: (i) The external field V/c is regular for every c > 0 except for an at most countable set of values without a finite accumulation point. (ii) The sets S c are increasing with c, and their intersection is equal to the set of all points where V assumes its global minimum. There is ε>0 such that c is a regular value for V for all c (0,ε) and S c = N 0 [a j(c),b j (c)], where N 0 is the number of points where V assumes its global minimum, and each interval contains precisely one such point. (iii) Let c 0 be a regular value for V, and let S c0 = N [a j (c 0 ),b j (c 0 )]. Then c is a regular value for V for every c in an open neighborhood of c 0 and S c = N [a j (c),b j (c)] with the same number of intervals N. The endpoints a j (c) are real analytic decreasing functions of c, and the endpoints b j (c) are real analytic increasing functions of c. (iv) Let c 0 be a singular value for V, and let x be a corresponding singular point.

7 742 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN 1. If x is a singular point of type I, then there is an open neighborhood O x of x such that O x S c is empty for c < c 0, and it is an interval for c (c 0,c 0 + ε) for some ε>0. 2. If x is a singular point of type II, then for c < c 0 sufficiently close to c 0, there are two intervals in S c close to x,say[a j (c),b j (c)] and [a j+1 (c),b j+1 (c)] with b j (c) < x < a j+1 (c). As c c 0, the endpoints b j (c) and a j+1 (c) tend to x. For c > c 0, the point x belongs to S c and ψ c (x ) > If x is a singular point of type III, then for c sufficiently close to c 0,there is exactly one interval in S c close to x. We see from Theorem 1.3 that a change in the number of intervals in S c (a phase transition) occurs only at singular values c and therefore at most countably often. Let c 0 be a singular value for V,andx a corresponding singular point. The local behavior of S c around x for c close to c 0 depends on the type of the singular point. From part (iv), it follows that a singular point of type I is associated with the formation of a new interval in the support, while a singular point of type II is associated with the closing of a gap. If ψ c0 (t) (t x ) 2k as t x, then we will show in Lemma 8.1 that a j+1 (c) x (c 0 c) 2k 1 as c c 0, x b j (c) (c 0 c) 2k 1 as c c 0. Thus the speed at which the gap closes is related to the order of vanishing of the density ψ c0 at x. A singular point of type III is not associated with a change in the number of intervals in S c. Thus if S c0 = N [a j (c 0 ),b j (c 0 )], then S c = N [a j (c),b j (c)] for c close to c 0, just as in the regular case. Here we assume that there are no singular points of type I or II. If the left endpoint a j (c 0 ) is a singular point, and ψ c0 (t) (t a j (c 0 )) k 1/2 as t a j (c 0 ), then it follows from Lemma 8.1 that a j (c) a j (c 0 ) (c 0 c) 1 k as c c 0. Thus the rate of convergence of a j (c) to a j (c 0 ) depends on the order of vanishing of ψ c0 at a j (c 0 ). Similarly, if b j (c 0 ) is a singular point and ψ c0 (t) (t b j ) k 1/2 as t b j, then b j (c 0 ) b j (c) (c 0 c) 1 k as c c 0. It is possible to obtain these relations for c c 0 as well. Note that k is always an odd integer. For k 3, we then see that the endpoint a j (c) or b j (c) does not depend in a real analytic way on c. In fact, at c 0 the curve c a j (c) has a vertical slope. This does not lead to a phase transition if we are varying the parameter c. However, if there is an additional parameter in the external field, like the time parameter in the continuum limit of the Toda lattice [5], then such a vertical slope indicates the possible breaking of an interval in the support in case that other parameter changes.

8 GENERIC DENSITY OF STATES 743 Theorems 1.2 and 1.3 allow us to prove that regular external fields are generic, i.e., open and dense in the space V of real analytic functions V satisfying the growth condition (1.1). To that end, we need to provide V with the right topology. In view of Theorem 1.2, we want that a sequence {V n } n=1 converges to V in V if and only if the conditions (i) and (ii) of Theorem 1.2 are satisfied. This can be done if we define the mappings G k : V R, and consider V with the metric ρ(v,w)= 3 j=0 k=1 V (x) V inf x k log x, k 1, 2 k V ( j) W ( j) [ k,k] 1 + V ( j) W ( j) [ k,k] + k=1 2 k G k (V) G k (W) 1 + G k (V ) G k (W), V,W V, where the norm [ k,k] is the supnorm on [ k,k]. It is easy to see that convergence of a sequence {V n } to V in the metric space (V,ρ) is indeed equivalent to the convergence of V n ( j) V ( j), j = 0,1,2,3, uniformly on compact subsets of R, together with the uniform growth condition. THEOREM 1.4 The set of regular V s in V is generic; i.e., it is an open dense subset of the metric space (V,ρ). PROOF: The openness follows directly from Theorem 1.2. Theorem 1.3 handles the denseness: If V is not regular, then there is a sequence {c n } converging to 1 such that V n := V /c n is regular by Theorem 1.3(i). The sequence {V n } tends to V in the space V with metric ρ. COROLLARY 1.5 The collection of real analytic V s for which the limit (1.7) holds at every interior point a of the support of ψ V and the limit (1.8) holds at every endpoint contains a dense and open subset of (V,ρ). PROOF: As we have seen above, the limit (1.7) holds whenever ψ V (a) > 0, and (1.8) holds whenever ψ V vanishes like a square root at the endpoint a of S V, that is, whenever V has no singular points of types II or III; see Remark B following Definition 1.1. Thus these limits hold if V is a regular external field, which by Theorem 1.4 is a dense and open subset of V. Note that the collection of external fields for which (1.7) holds at every interior point and (1.8) at every endpoint need not be open in V, since condition (i) of Definition 1.1 does not play a role here. Indeed, any V having singular points, but only of type I, belongs to this collection, and it is possible to approximate such a V in the space (V,ρ) by external fields having singular points of types II or III. The rest of the paper is organized as follows: In the next section, we discuss applications of our results to orthogonal polynomials and to integrable systems. The proof of Theorem 1.2 is in Section 3. The remaining part of the paper is devoted

9 744 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN to the proof of Theorem 1.3. We use techniques from potential theory, and some notions are reviewed in Section 4. In Section 5, we prove Theorem 1.3(ii). It uses the convexity of the external field near the points where it assumes its minimum. A similar idea is used to show that for a given c 0 > 0, there is ε>0 such that c is a regular value for every c (c 0,c 0 + ε). This proves half of Theorem 1.3(i). The other half concerns the case c (c 0 ε,c 0 ). Here we use an iterative method, first considered in [4, 17], where one solves the integral equation Lv n (x) V(x)=l n, x Σ n, with v n (t) = c, supp(v n ) Σ n, on a decreasing sequence of sets Σ n, all containing S c. The functions v n need not be nonnegative and are connected by a process called balayage. The idea is to prove that v n tends to ψ c and Σ n tends to S c as n. In addition, one needs to control the sets Σ n in order to obtain information about ψ c and S c. The technical difficulties that arise are dealt with in Sections 7 through 11, thus proving the other half of Theorem 1.3(i). In Section 12 we prove Theorem 1.3(iii) using the implicit function theorem and ideas from [5]. Finally, in Section 13 we prove part (iv). 2 Applications As mentioned in the introduction, the extremal problem (1.3) for logarithmic potentials arises in many areas of classical analysis. We discussed the significance of our results in random matrix theory in Section 1. In this section we discuss applications in two more areas: orthogonal polynomials and integrable systems. 2.1 Polynomials Orthogonal with Respect to Varying Weights Let p (n) k (x)=p k (x;n)=p k (x) denote the n th orthonormal polynomial with respect to the measure e 2nV (x) dx, (2.1) p j (x)p k (x)e 2nV (x) dx = δ jk (here δ jk is the Kronecker delta function). Such orthogonal polynomials and their asymptotic description play a central role in random matrix theory (see above) and in approximation theory (see, for example, [28, 30]). One asks for an asymptotic (z) as n, which is uniformly valid for all z C (note that the degree of the polynomial in question coincides with n, the parameter in the measure of orthogonality). The crucial role played by the equilibrium measure in connection with the asymptotics of orthogonal polynomials is well documented (see, for example, [20, 28] and the many references contained therein). For example, if x 1 < x 2 < < x n denote the zeros of the n th orthogonal polynomial p n, and if µ n = 1 n n δ x j denotes the normalized counting measure for the zeros of p n (here δ x j is the Dirac mass concentrated at x j ), then one has (see [11, 23, 25, 28]) description of p (n) n (2.2) µ n µ V in the weak- sense of measures.

10 GENERIC DENSITY OF STATES 745 As (2.2) indicates, the equilibrium measure is central in more detailed asymptotic descriptions of the orthogonal polynomials [19, 20, 21, 26]. Indeed, if V is real analytic (with sufficient growth), there are six different types of asymptotic formulae that are used to describe the polynomials p n (x) for x R, catalogued explicitly by the equilibrium measure [6] [8]. If x S V is such that ψ V (x) > 0, then the asymptotics are given by an oscillatory-type formula built out of a theta function associated to a Riemann surface related to the support S V.Ifxis an endpoint of S V and ψ V vanishes like a square root there, then the asymptotics are given in terms of Airy functions. If x R\SV, then the polynomial exhibits no oscillations, but rather exponential behavior. The remaining three cases are the singular cases: The behavior of p n in a vicinity of each type of singular point is obtained via a new family of special functions, determined by a local Riemann-Hilbert problem. One finds the following: Singular points of type I are points outside the support where nonetheless one expects to find some zeros of p n (x) (accumulating at a rate linked to the order of vanishing of Lψ V V l at the singular point). Points of type II are points within the support where the zeros accumulate at a slower rate (here the rate is linked to the order of vanishing of ψ V ). Singular points of type III are endpoints of S V where, ultimately, the scaling for the zeros changes from n 2/3 (in the Airy case) to a different scaling, determined by the order of vanishing of ψ V. The significance of regular V s is now clear: If V is regular, then the behavior of the polynomial p n at all endpoints of the support is given by Airy functions, the behavior within the support is given by a simple oscillatory-type formula, and there are no isolated points outside the support where additional zeros can accumulate. This could be thought of as an ideal situation. In this context, our work proves that the ideal situation described above is generic in the sense of Theorem Integrable Systems Our results have applications in integrable systems as well, as extremal problems like (1.3) appear in the analysis of singular limits of integrable systems. In the small dispersion limit of the KdV equation as studied in the remarkable papers [18] and [32], the number of gaps in supp(ψ) is the genus of the Riemann surface used to construct the approximate solution of the KdV equation via modulation theory [10] (here, in fact, the variational problem differs from (1.3) in several ways: The Borel measures are odd, there is an upper constraint, and there is no prescribed normalization). A similar relationship holds for the semiclassical limit of the NLS equation [12, 13] and the continuum limit of the Toda lattice [9]. In these integrable cases, the initial data is encoded in the external field, and the time and spatial parameters x and t appear explicitly as parameters in the variational problem. The Toda case actually reduces directly to the variational problem (1.3) (with an additional upper constraint), and in this case the spatial parameter x appears in the variational problem as the normalization of the equilibrium measure [9]. So, in light of (1.16) (1.18), we see that the evolution of ψ c in the independent variable c is precisely the spatial evolution of the continuum limit of the Toda

11 746 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN lattice. Initial data for the continuum limit of the Toda lattice is given in terms of two functions, α(x) and β(x). Att = 0, it turns out that for each x, the endpoints of the support of the equilibrium measure are precisely α(x) and β(x). At later times, the support can be quite complicated, and the central problem in describing the continuum limit is to describe, for t > 0 fixed, the support for all values of x in the given spatial domain. Our results (in particular Theorem 1.3) can be applied to the continuum limit of the Toda lattice. We note that as in [9], if there is an interval (0,d) on which β is increasing and α is decreasing, then for t > 0and x in a slightly smaller interval, the variational problem is posed without an upper constraint. To apply the results in this paper, we must be in a spatial region for which the continuum limit is described by the variational problem (1.3) without an upper constraint. This is the case, for example, if the function α(x) is strictly decreasing with limit and β(x) is strictly increasing with limit +. THEOREM 2.1 If the initial data α(x) and β(x) are such that the continuum limit is described by the variational problem without upper constraint, and the corresponding external field V(x) is real analytic, then for any t > 0 and any x (0, ), the support is a finite union of intervals. As we vary x, the number of intervals can change (through the occurrence of singular points of type I or II) at most countably many times, and the (discrete) set of points of transition has no finite accumulation point. The evolution of the support through a point of transition is described by Theorem 1.3. Remark. As mentioned, the support of the equilibrium measure provides a leadingorder asymptotic description of the continuum limit of the Toda lattice: To obtain higher-order asymptotics, as in [32], one would construct an approximate solution out of theta functions associated to the Riemann surface obtained by gluing together two copies of the complex plane, each copy cut along the support of the equilibrium measure. Thus Theorem 2.1 above provides a global description of the spatial evolution of the Riemann surface used to construct the approximate solution of the Toda lattice. Remark. In the continuum limit of the Toda lattice as studied in [5], the variational problem appears with an additional upper constraint, ψ φ, whereφ is obtained from a WKB analysis of the initial data α(x) and β(x). For this more complicated situation, our methods should enable us to prove that Theorem 2.1 holds provided the upper constraint φ is real analytic. 3 Proof of Theorem 1.2 For short, we write µ := µ V and µ n := µ Vn, E n := I Vn (µ Vn ),andl n := l Vn for each n. From assumptions (i) and (ii) of Theorem 1.2, it follows that the external fields V n are uniformly bounded from below. Adding a suitable constant (which does not change µ n or the regularity of V n ), we may assume that (3.1) V 0 and V n 0

12 GENERIC DENSITY OF STATES 747 for all n. From (i) it follows that V n V uniformly on the support of µ. Therefore Vn dµ Vdµ, andsoi Vn (µ) E V as n. From E n I Vn (µ) it then follows that the energies E n are uniformly bounded from above, say E n R for all n. Put M = R log2. From assumption (ii), we find that there exists A > e such that V n (x) log x M for x R \ [ A,A] and n 1. This easily implies that (3.2) V n (x) log x M for x R \ [ A,A] and all n. Then we claim that for all n, (3.3) V n (x)+v n (y) log x y E n + 1 whenever x R \ [ A,A] and y R. To prove the claim, we let x R \ [ A,A] and y R. We may assume x y, since otherwise we interchange the roles of x and y. Then we have x y 2 x, and we get for every n, using (3.1) and (3.2), V n (x)+v n (y) log x y log x + M log x y ( ) 2 x = R log E n + 1, x y as claimed in (3.3). As in the proof of [28, theorem 1.3(b)] it follows from (3.3) that (3.4) S Vn = supp(µ n ) [ A,A] for every n. Because the measures µ n are supported on a fixed compact interval, it readily follows from the fact that V n V uniformly on [ A,A] that (3.5) lim µ n = µ n in the weak- topology of measures on [ A,A] and that (3.6) lim n E n = E V, lim l n = l V. n Choose L such that l n L for every n. Then by assumption (ii) there is a B > 2A such that V n (x) log x L + 1 for x > B and all n. Let x > B. Then for all y S Vn,wehave y A by (3.4), and since x > 2A,weget x / x y 2 3. Hence ( 2 V n (x) log x y L log 3 ) > l n, y S Vn.

13 748 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN Integrating this with respect to dµ n (y), wegetv n (x) Lµ n (x) > l n whenever x > B. Thus by (1.13), the definition of S Vn,wehave (3.7) SV n [ B,B] for all n. It was shown in [5] that the function q V appearing in (1.12) is given by V q V (x)=[v (x)] 2 (x) V (y) 2 ψ V (y)dy x y ( 1 ) (3.8) =[V (x)] 2 2 V (tx+(1 t)y) dµ V (y). 0 Then ( 1 ) (3.9) q V (x)=2v (x)v (x) 2 0 tv (3) (tx+(1 t)y) dµ V (y) with similar formulae for q Vn and q V n. From (3.8) and the fact that V n V uniformly on [ A,A],wefind lim n 1 0 V n (tx+(1 t)y) = 1 0 V (tx+(1 t)y) uniformly for x and y in [ A,A]. Sinceµ n µ in the weak- sense on [ A,A] by (3.5) and V n V uniformly on [ A,A], it then follows from (3.8) that (3.10) lim q V n n = q V uniformly on [ A,A]. Similarly, we obtain using (3.9) and also the fact that V (3) n tends to V (3) as well that (3.11) lim n q V n = q V uniformly on [ A,A]. Now assume that V is regular and suppose that V n is not regular for n large enough. Thus infinitely many V n have a singular point, which we call x n. Then x n SV n (see Remark A following Definition 1.1), and so x n [ B,B] by (3.7). We can extract a subsequence (also called V n ) such that the points x n converge to x [ B,B]. Wehave Lµ n (x n ) V n (x n )=l n, and taking the limit n, we easily get Lµ V (x ) V (x )=l V. Thus x SV. Furthermore, we have that q V n has a multiple zero at x n ; see Remark A. Then it follows from (3.10) that q V (x )=lim n q Vn (x n )=0, and similarly from (3.11) that q V (x )=0. Thus q V has a multiple zero at x, and therefore x is a singular point of V. This contradicts the regularity of V, and we conclude that V n is regular for n large enough. This completes the proof of Theorem 1.2.

14 GENERIC DENSITY OF STATES Preliminaries from Potential Theory In this section, we collect known properties of the measures µ c and the sets S c and Sc defined in (1.14) and (1.15). These properties were derived using logarithmic potential theory, and we first present some facts from that theory. General references are [27, 28]. A basic role is played by the equilibrium measure ω(σ) of a compact set Σ, which for our purposes is a finite union of nondegenerate intervals. By definition, ω(σ) minimizes the (unweighted) energy 1 I(µ) := log s t dµ(s)dµ(t) among all probability measures µ on Σ. Let E(Σ) := I(ω(Σ)). Then the equilibrium measure satisfies (4.1) Lω(Σ)(x)= E(Σ), x Σ, (4.2) Lω(Σ)(x) E(Σ), x C. The constant E(Σ) is known as the Robin constant of Σ, and exp( E(Σ)) is called the logarithmic capacity of Σ. If Σ = N [a j,b j ] is a finite union of intervals with a 1 < b 1 < a 2 < < a N < b N, then there exist numbers y j (b j,a j 1 ), j = 1,...,N 1, such that the equilibrium measure ω(σ) has a density given by dω(σ)(t) (4.3) see [29, lemma 4.4.1]. Here R(z)= N = i N 1 (t y j) πr 1/2, t Σ; + (t) (z a j )(z b j ), z C \ Σ, and R 1/2 + (t), t Σ, denotes the boundary value from above. We also need g Σ, the Green function of the set Σ with pole at infinity. Thus g Σ is the unique function h on C that is harmonic on C \ Σ and continuous on C, vanishes on Σ, and is such that h(z) log z is bounded as z. Note that g Σ (z)=lω(σ)(z)+e(σ), z C, which relates the Green function to the equilibrium measure. The following result is due to Buyarov and Rakhmanov: LEMMA 4.1 For c > 0, we have (4.4) m c = c 0 ω(s γ )dγ,

15 750 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN (4.5) (4.6) PROOF: See [3]. V (x)=(minv )+ g Sγ (x)dγ, l c = c 0 0 E(S γ )dγ. From the first Buyarov-Rakhmanov formula (4.4), it follows immediately that µ c is increasing with the parameter c. The behavior of the sets S c and S c was studied in [3, 4, 28, 30]. LEMMA 4.2 We have for c > 0, (4.7) S c Sc S c+ε, (4.8) (4.9) S c = ε>0s c+ε, S c = ε>0s c ε, (4.10) (4.11) {t : ψ c (t) > 0} = ε>0s c ε, S c = {x : V (x)=minv }. c>0 PROOF: Properties (4.7) (4.9) were first proved in [30] (see also [28]). Property (4.10) is from [4], where it was shown for real analytic external fields on [ 1, 1]. The same proof can be used for real analytic external fields on R. Property (4.11) is taken from [3]. As already mentioned, the proofs of Lemmas 4.1 and 4.2 are based on the theory of logarithmic potentials. In the sequel, we will need some more results from logarithmic potential theory, most notably the following powerful result of de la Vallée Poussin: LEMMA 4.3 Let µ 1 and µ 2 be two positive Borel measures on R with compact supports and finite logarithmic potentials satisfying dµ 1 dµ 2. Suppose that for some constant c, (4.12) Lµ 1 Lµ 2 + c on supp(µ 2 ), and for some set A, (4.13) Lµ 1 Lµ 2 + c ona. Then we have for every Borel set B A, (4.14) µ 1 (B) µ 2 (B). PROOF: See [28, theorem IV.4.5].

16 GENERIC DENSITY OF STATES 751 Remark C. It follows from (4.12) and dµ 1 dµ 2 that the inequality (4.12) holds throughout the complex plane; see [28, theorem II.3.2]. Thus strict equality cannot hold in (4.13). We will need Lemma 4.3 only for measures µ 1 and µ 2 that are absolutely continuous with continuous densities ψ 1 and ψ 2. In that case Lemma 4.3 gives that if Lψ 1 Lψ 2 + c on the support of ψ 2,andLψ 1 (x 0 ) Lψ 2 (x 0 )+c for some x 0, then ψ 1 (x 0 ) ψ 2 (x 0 ). A variation of Lemma 4.3 is the following result, which will be used a number of times: LEMMA 4.4 Let c > 0. LetΣ be a compact set with S c Σ. Ifv is an integrable function such that (4.15) supp(v) Σ, v(t) = c, and for some constant l (4.16) then (4.17) Lv = V + l onσ, S c supp(v + ) and ψ c v +. Here v = v + v is the decomposition of v into its positive and negative parts. PROOF: From (1.16), (1.17), and (4.16) it is clear that L(ψ c + v ) Lv + + l c l on Σ, with equality on S c. Also note that (ψ c + v )(t) = v + (t); therefore by Lemma 4.3 we have ψ c + v v + on S c. This implies (4.17). As a first application of the above lemmas, we prove the following result: LEMMA 4.5 Let c 2 > c 1 0. Assume that for a fixed N, we have N (4.18) S c = [a j (c),b j (c)], c (c 1,c 2 ), with a 1 (c) < b 1 (c) < a 2 (c) < < a N (c) < b N (c). Then (i) for each j = 1,...,N, the function c a j (c) is continuous and strictly decreasing on (c 1,c 2 ), (ii) for each j = 1,...,N, the function c b j (c) is continuous and strictly increasing on (c 1,c 2 ), (iii) there are no singular values of types I or II in the interval (c 1,c 2 ).

17 752 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN PROOF: First we prove that for every c (c 1,c 2 ), there exists ε 0 (0,min(c c 1,c 2 c)) such that for each j = 1,...,N and ε (0,ε 0 ), (4.19) [a j (c ε),b j (c ε)] [a j (c),b j (c)] [a j (c + ε),b j (c + ε)]. Let c (c 1,c 2 ). For each j = 1,...,N 1, the interval (b j (c)),a j+1 (c)) isagap in S c. Since Sc differs from S c by a finite number of points only, we can take a point x j from (b j (c),a j+1 (c))\sc for each j = 1,...,N 1. Since x j Sc,wehave by (4.8) that there is ε 1 (0,c 2 c) such that x j S c+ε for each ε (0,ε 1 ). Then S c+ε is the union of the nonempty and disjoint sets S c+ε (x j 1,x j ), j = 1,...,N, where we write x 0 = and x N =+. There is also an ε 2 (0,c c 1 ) such that S c ε (x j 1,x j ) is not empty for each j = 1,...,N and each ε (0,ε 2 ). This can be seen by choosing, for each j, a point y j (a j (c),b j (c)) with ψ c (y j ) > 0. Then by (4.10) there is ε 2 (0,c c 1 ) such that y j S c ε for each ε (0,ε 2 ). Then clearly the sets S c ε (x j 1,x j ) are not empty. It is also clear that they are disjoint and their union is S c ε. Put ε 0 = min(ε 1,ε 2 ). Then for each c (c ε 0,c+ε 0 ), the sets S c (x j 1,x j ), j = 1,...,N, are disjoint and nonempty and have union S c. From (4.18) it then follows that S c (x j 1,x j )=[a j (c ),b j (c )]. If we recall that S c (x j 1,x j )=[a j (c),b j (c)] and use (4.7), we see that (4.19) holds. Having (4.19) for all c (c 1,c 2 ) and all ε sufficiently small, it is easy to extend the inclusion [a j (c),b j (c)] [a j (c + ε),b j (c + ε)], j = 1,...,N, to all c and c + ε satisfying c 1 < c < c + ε<c 2. Then it is clear that a j (c) a j (c + ε) and b j (c) b j (c + ε). Moreover, since ψ c+ε vanishes at the endpoint a j (c + ε), wehavea j (c + ε) S c by (4.10). Thus a j (c) > a j (c+ε), and we see that a j (c) is strictly decreasing on (c 1,c 2 ). Similarly, b j (c) is strictly increasing. It follows that a j (c) and b j (c) have left and right limits at each c (c 1,c 2 ). We denote the limits from the left by a j (c ) and b j (c ), and the limits from the right by a j (c+) and b j (c+), respectively. Using (4.10) and the fact that a j (c) is strictly decreasing and b j (c) is strictly increasing, we find that N {t : ψ c (t) > 0} = (a j (c ),b j (c )), and thus by taking closures N S c = [a j (c ),b j (c )].

18 GENERIC DENSITY OF STATES 753 Comparing this with (4.18), we easily get that a j (c)=a j (c ) and b j (c)=b j (c ). Hence a j (c) and b j (c) are continuous from the left. It also follows that {t : ψ c (t) > 0} = N (a j (c),b j (c)). Hence ψ c does not vanish in the interior of S c, which means that there are no singular values of type II. Next, we have by (4.8) S c = N [a j (c+),b j (c+)]. Since a j (c+) a j (c) < b j (c) b j (c+), wefinhats c has no isolated points. Hence S c = S c, and it follows that there are no singular values of type I. From S c = S c, it also follows that a j (c+) = a j (c) and b j (c+) = b j (c), which means that a j (c) and b j (c) are continuous from the right. This completes the proof of Lemma Proof of Theorem 1.3(ii) Put S0 := {x : V (x)=minv }, which is a finite set (since V is real analytic and satisfies the growth condition (1.1)), say S 0 = {a 1,...,a N }. Since V assumes its minimum at a j and V is real analytic, V is convex in a neighborhood of a j.leta j and B j be such that A j < a j < B j and V is convex in [A j,b j ]. Take ε>0 such that S ε N (A j,b j ), which is possible because of (4.11). Let c (0,ε). Then S c S ε N (A j,b j ). For every j, wehavea j S c by (4.11), and S c [A j,b j ] is an interval because of the convexity of the external field on [A j,b j ]; see [28, theorem IV.1.10(b)]. Thus S c [A j,b j ]=[a j (c),b j (c)] for certain a j (c) and b j (c) with a j (c) < a j < b j (c). It follows that N S c = [a j (c),b j (c)]. Since the number of intervals in S c does not depend on c if c (0,ε), we find from Lemma 4.5 that the interval (0,ε) contains no singular values of types I or II. Now, let x be a boundary point of S c,thatis,x is one of the points a j (c), b j (c), j = 1,...,N, and assume that Lψ c is a C 2 function in a neighborhood of x.since V is convex at x and Lψ c = V + l c in a left or right neighborhood of x,weget (Lψ c ) (x )=V (x ) 0.

19 754 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN On the other hand, we have (Lψ c ) (x )= 1 (x y) 2 ψ c(y)dy < 0. This contradiction shows that Lψ c is not a C 2 function in a neighborhood of x. Thus c is not a type III singular value either; cf. Remark B after Definition 1.1. We conclude that every c (0,ε) is a regular value for V. The remaining statements in part (ii) of Theorem 1.3 are contained in Lemma Proof of Theorem 1.3(i), Part I Using very similar arguments as given in the previous section, we are able to prove the following proposition, which covers half of the statement of Theorem 1.3(i). PROPOSITION 6.1 Let c 0 > 0. Then there is ε>0 such that c is a regular value for V for all c (c 0,c 0 + ε). (6.1) PROOF: Put Q = V Lψ c0. Then Q = l c0 on S c 0 by the definition of S c 0 and Q > l c0 elsewhere. Let (6.2) S c 0 = N [a j,b j ] with a 1 b 1 < a 2 b 2 < < a N b N. It is possible that a j = b j in case a j is an isolated point of S c 0. We want to show that Q is convex in a neighborhood of every component [a j,b j ] of S c 0. This is clear if a j = b j, since then Q is real analytic in a neighborhood of that point, and it assumes its minimum there. Assume a j < b j, and let us consider Q in the neighborhood of the point b j. The density ψ c0 vanishes at b j. There is an odd k N such that ψ c0 (t) (b j t) k 1 2 for t in a left neighborhood of b j. Then Lψ c0 is a C k function near b j. Thus Q is a C k function near b j by (6.1), and it follows that (6.3) Q (b j )= = Q (k) (b j )=0, since Q is equal to the constant l c0 in a left neighborhood of b j.wealsohavefor x > b j, x S c0, Q (k+1) (x)=v (k+1) ψ c0 (t) (x)+k!. (t x) k+1

20 Since k is odd, this implies that GENERIC DENSITY OF STATES 755 (6.4) lim x b j Q (k+1) (x)=+. Then it is clear from (6.3) and (6.4) that Q is convex in a neighborhood of b j. Similarly, Q is convex in a neighborhood of a j, and all together Q is convex in a neighborhood of S c 0. Choose A j and B j for j = 1,...,N, such that A j < a j b j < B j and such that Q is convex in N [A j,b j ]. Now we basically follow the proof of Theorem 1.3(ii). We take ε>0 such that S c0 +ε N (A j,b j ). This is possible because of (4.8) and (6.2). Let c (c 0,c 0 + ε). Then S c N (A j,b j ) by (4.7). Note from (4.8) that for each j = 1,...,N, S c (A j,b j ) is not empty. Define Then v c 0 and by (4.4) v c = ψ c ψ c0. (6.5) c dω(s γ )(t) v c (t)= c 0 dγ. From (6.5) it readily follows that (6.6) supp(v c )=S c. Using (1.16), (1.17), and the definitions of Q and v c, we then see that (6.7) (6.8) Lv c Q = l c l c0 on S c = supp(v c ), Lv c Q l c l c0 on R. Thus v c is the density of the equilibrium measure in the presence of the external field Q and normalization v c (t) = c c 0. Since Q is convex on [A j,b j ] and S c (A j,b j ) is not empty for j = 1,...,N, we have by [28, theorem IV.1.10(b)] that supp(v c ) [A j,b j ] is an interval for every j, say supp(v c ) [A j,b j ]=[a j (c),b j (c)]. Since supp(v c )=S c N (A j,b j ), we then find (6.9) N S c = [a j (c),b j (c)], c (c 0,c 0 + ε). By Lemma 4.5 it then follows that no c in (c 0,c 0 + ε) is a singular value of type I or type II. That c is not singular of type III either follows exactly as in the proof of Theorem 1.3(ii). Here one uses the convexity of Q on S c again.

21 756 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN 7 Proof of Theorem 1.3(i), Part II In order to complete the proof of Theorem 1.3(i), we now need to consider values c that are less than a given c 0 and prove the following analogue of Proposition 6.1: PROPOSITION 7.1 Let c 0 > 0. Then there is an ε>0 such that c is regular for V for every c (c 0 ε,c 0 ). Note that Theorem 1.3(i) follows from Propositions 6.1 and 7.1, together with Theorem 1.3(ii), which was already proved in Section 5. In this section, we introduce some notation that will be used in Sections 7 through 11. We choose c 0 > 0 and write (7.1) with {x : ψ c0 (x) > 0} = N (a j,b j ) (7.2) a 1 < b 1 a 2 < a N < b N. Strict inequality b j 1 < a j holds if a j is a left endpoint of S c0 and b j 1 a right endpoint. If b j 1 = a j, then a j is an interior point of S c0 where the density vanishes. We define the midpoints m j = a j + b j (7.3), j = 1,...,N. 2 By (4.10) we then have that every m j belongs to S c whenever c is sufficiently close to c 0. In what follows we will always assume that this is the case. For such c < c 0, we define (7.4) a j (c)=min{a m j : [a,m j ] S c }, j = 1,...,N, and (7.5) b j (c)=max{b m j : [m j,b] S c }, j = 1,...,N. Thus [a j (c),b j (c)] is the component of S c containing m j.sincea j and b j are not in S c by virtue of (4.10) and (7.1), we have a j < a j (c) < b j (c) < b j. So the intervals [a j (c),b j (c)], j = 1,...,N, are disjoint and N [a j (c),b j (c)] S c. We emphasize that equality need not hold. It may happen that S c consists of more intervals besides the intervals [a j (c),b j (c)]. These extra intervals would then be contained in either (a j,a j (c)) or (b j (c),b j ) for some j. Proposition 7.1 follows from the following lemma:

22 GENERIC DENSITY OF STATES 757 LEMMA 7.2 For every j = 1,...,N, there is ε>0 such that for all c (c 0 ε,c 0 ), (7.6) [a j,a j (c)) S c = and (7.7) similarly, ψ c (t) (t a j (c)) 1/2 as t a j (c); (7.8) (b j (c),b j ] S c = and (7.9) ψ c (t) (b j (c) t) 1/2 as t b j (c). Indeed, if Lemma 7.2 holds, then there exists ε>0 such that (7.6) (7.9) hold for all j and every c (c 0 ε,c 0 ). Then for such c,wehave N S c = [a j (c),b j (c)], which means that the number of intervals in S c is constant for c (c 0 ε,c 0 ). Thus by Lemma 4.5 there are no singular values of types I and II in (c 0 ε,c 0 ). From (7.7) and (7.9) it follows that there are no singular values of type III either. Thus every c (c 0 ε,c 0 ) is regular, and we see that Proposition 7.1 follows from Lemma 7.2. To establish Lemma 7.2, it is enough to prove the assertions (7.6) and (7.7), since (7.8) and (7.9) will then follow by similar arguments. We will distinguish the case that a j is a left endpoint of S c0 and the case that a j is an interior point of S c0. The proof of (7.6) and (7.7) uses an iterative procedure to be carried out in Section 11. In the case that a j is a left endpoint, the iterative procedure starts with the explicit construction of an auxiliary function v c, and Lemma 7.3 describes its important properties. In the case that a j is an interior point of S c0, the iterative procedure starts with a different auxiliary function, w c, whose important properties are described in Lemma 7.4. First assume that a j is a left endpoint of S c0.forc < c 0,wedefine (7.10) Σ c := S c [a j,a j (c)], which is a finite union of intervals, one of the intervals being [a j,b j (c)]. Letv c be an integrable function on Σ c satisfying (7.11) Lv c (x) V(x)=l, x Σ c, where l is a constant, and (7.12) supp(v c ) Σ c, v c (t) = c. Such a function exists and is unique; one may obtain it by minimizing the weighted energy I V (µ) of (1.2) among all signed Borel measures µ on Σ c having dµ = c. The function v c is the density of the minimizing measure. Note that v c is not

23 758 A. B. J. KUIJLAARS AND K. T-R MCLAUGHLIN necessarily nonnegative; in fact, as we will show, it is negative in a neighborhood of a j. We need the following properties of v c : LEMMA 7.3 Let a j be a left endpoint of S c0. Suppose Σ c and v c satisfy (7.10) (7.12). Then there is ε>0 such that for every c (c 0 ε,c 0 ), there is t 0 (c) (a j,a j (c)) such that (i) v c (t) < 0 for all t [a j,t 0 (c)) (ii) and there are constants K = K(c) > 0 and δ = δ(c) > 0 such that [ ] d (b j (c) t)(t a j )v c (t) > K for t [t 0 (c),a j (c)+δ]. As mentioned above, v c is used in the first step of the iterative procedure of Section 11 to establish (7.6) and (7.7) in the case that a j is a left endpoint. For example, from (i) it will follow that [a j,t 0 (c)) S c =, and using (ii) we will generate a sequence {t k (c)} k=0 that increases to a j(c) such that for all k 0, [a j,t k (c)) S c =, proving (7.6). Next consider the case that a j is an interior point. In Section 11, we will use arguments similar to those outlined above to prove (7.6) and (7.7) for this case, using a different auxiliary function, w c, defined as follows: For c < c 0, we set (7.13) ˆΣ c = S c [b j 1 (c),a j (c)], and we let w c be the integrable function on ˆΣ c satisfying (7.14) Lw c (x) V(x)=l, x ˆΣ c, where l is a constant, and (7.15) supp(w c ) Σ c, w c (t) = c. For this function we prove the following analogue of Lemma 7.3: LEMMA 7.4 Let a j be an interior point of S c0. Suppose ˆΣ c and w c are defined by (7.13) (7.15) for c < c 0. Then there is ε>0 such that for every c (c 0 ε,c 0 ), there exist s 0 (c) (b j 1 (c),a j ) and t 0 (c) (a j,a j (c)) such that (i) w c (t) < 0 for all t (s 0 (c),t 0 (c)), and (ii) there are constants K = K(c) > 0 and δ = δ(c) > 0 such that [ ] d (b j (c) t)(t a j 1 (c))v c (t) > K for t [t 0 (c),a j (c)+δ] and d for t [b j 1 (c) δ,s 0 (c)]. [ (b j (c) t)(t a j 1 (c))v c (t) ] < K

24 GENERIC DENSITY OF STATES 759 In Section 11, we will use (i) to prove that (s 0 (c),t 0 (c)) S c =, andwe will use (ii) to generate a decreasing sequence {s k (c)} k=0 converging to b j 1(c) and an increasing sequence {t k (c)} k=0 converging to a j(c) such that for all k 0, (s k (c),t k (c)) S c =. The proofs of Lemmas 7.3 and 7.4 require some preliminary work in Sections 8 and 9. In Section 10 we prove Lemmas 7.3 and 7.4. Finally, in Section 11 we use this to establish Lemma 7.2, which, as we have seen, yields Proposition 7.1, and this will then complete the proof of Theorem 1.3(i). 8 Auxiliary Results I: Rate of Convergence of a j (c)toa j In this section we investigate how fast a j (c) and b j (c) approach a j and b j, respectively. Recall that a j (c) and b j (c) are defined by (7.4) and (7.5). The rate of convergence depends on the order of vanishing of ψ c0 at a j and b j. First, we observe that for every j, wehavethata j (c) decreases and b j (c) increases as c increases towards c 0,and (8.1) lim a j (c)=a j, c c 0 Indeed, for every j, the sets lim b j (c)=b j. c c0 O c = {x (a j,b j ) : ψ c (x) > 0}, c < c 0, form an increasing family of open sets whose union is (a j,b j ). By a compactness argument, there is for every ε>0avaluec < c 0 such that [a j + ε,b j ε] O c. Choosing ε such that a j + ε<m j < b j ε,wethengeta j (c) a j + ε and b j (c) b j ε by (7.4) and (7.5), since O c S c. Hence (8.1) holds. LEMMA 8.1 (i) If a j is an endpoint of S c0 and ψ c0 (t) (t a j ) k 1/2 as t a j, then a j (c 0 ε) a j ε 1 k as ε 0. (ii) If b j is an endpoint of S c0 and ψ c0 (t) (b j t) k 1/2 as t b j, then b j b j (c 0 ε) ε 1 k as ε 0. (iii) If a j is an interior point of S c0 (thatis,ifb j 1 = a j ) and ψ c0 (t) (t a j ) 2k as t a j, then and a j (c 0 ε) a j ε 1 2k as ε 0 b j 1 b j 1 (c 0 ε) ε 1 2k as ε 0. PROOF: (i) Let a j be an endpoint of S c0 such that (8.2) ψ c0 (t) (t a j ) k 1/2 as t a j with k an odd integer. For ε>0, the function v(t)=ψ c0 (t) ε dω(s c 0 )(t), t S c0,