E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York (1997).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....any assumption on the recurrence coe#cients. We can also allow V to be quite arbitrarily. We assume the following V : R R is real analytic, 1.5) lim x ## log(x 1) #, 1.6) #(0) 0, 1. 7) where # is the density of the equilibrium measure in the presence of the external field V , [11, 34]. Let us explain the condition (1.7) Denote the space of all probability measures on R by M 1 (R) and consider the following minimization problem inf #M 1 (R) Z Z s t d(s)d(t) V (t)d(t) 1.8) Under the assumptions (1.5) and (1.6) it is known that the infimum is achieved [9, ....

....34] Let us explain the condition (1.7) Denote the space of all probability measures on R by M 1 (R) and consider the following minimization problem inf #M 1 (R) Z Z s t d(s)d(t) V (t)d(t) 1.8) Under the assumptions (1.5) and (1. 6) it is known that the infimum is achieved [9, 34] uniquely at the equilibrium measure V M 1 (R) for V . The measure V has compact support, and since V is real analytic, it is supported on a finite union of intervals. In addition it is absolutely continuous with respect to the Lebesgue measure, i.e. d V (x) #(x)dx, and # is real analytic ....

E.B. Sa# and V. Totik, " Logarithmic Potentials with External Fields," Springer-Verlag, New-York, 1997.

....satisfying the upper and lower constraints d(x) x) dx (17) for all Borel sets B # [a, b] and the normalization condition d(x) 1 . 18) The existence of a unique minimizer under the conditions enumerated in 1.3.1 and 1.3. 2 follows from the Gauss Frostman Theorem; see [26] for details. We will often refer to the minimizer as the equilibrium measure. It has been shown [21] that the equilibrium measure is the weak limit of the normalized counting measure of the zeros of pN,k (z) in the limit N with c = k N fixed. That a variational problem plays a central role in ....

E. B. Sa# and V. Totik. Logarithmic Potentials with External Fields. Springer-Verlag, New York, 1997.

....44 on the arc 1 = D 1 if n 1. Let Z(Pn ) denote the (multi )set of all zeros of Pn . Then we have n=1 m n Z(Pm ) 1 : 4.60) Because of (4. 60) the sequence f n=1 of probability measures is compact with respect to the weak convergence of measures (Helly s Selection Theorem, cf. [17], Theorem 1.3) Let 1 denote a cluster point of the sequence f n=1 , and assume that N N is an in nite subsequence with From (4.60) it follows that supp( 1 ) 1 . We have log jz tj d Pn (t) 4.62) From (4.62) limit (4.61) and the rst limit in (2.3) of Theorem 2.2 it ....

....1 ) 1 . We have log jz tj d Pn (t) 4.62) From (4.62) limit (4.61) and the rst limit in (2.3) of Theorem 2.2 it follows that log jz tj d 1 (t) for z 2 D 1 : 4.63) Since supp( 1 ) D 1 , it is a standard conclusion from potential theory (Carleson s Unicity Theorem. cf. [17], Theorem 4.13) that identity (4.63) implies that 1 = 1 : 4.64) The measure 1 is independent of the subsequence N , and consequently limit (4.61) holds for the full sequence f n=1 , which together with (4.64) proves the rst limit in (2.2) The third limit in (2.2) follows in exactly ....

Sa, E.B. & Totik, V., Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, New York (1997).

....in H(#) Next we have to put extra conditions on q. For this we consider the equilibrium problem in the presence of the external field q for the logarithmic potential V (z) x z d(x) of positive probability measures ( #, #] supported on [ 1, 1] It is well known (see [9] [15]) that there exists a unique measure # (called equilibrium measure in external field) # on supp #, # on [ 1, 1] 21) In what follows we assume q is such that supp# = 1, 1] 22) Remark. A su#cient condition for (22) is convexity of q on [ 1, 1] It is not so di#cult to see (see, for ....

E. Saff and V. Totik, "Logarithmic Potentials with External Fields," Springer Verlag, 1997.

....an arbitrary domain D C and a measure we define the Green potential as (5.4) g( D; z) g D (z, x) d(x) An useful tool in potential theoretical investigations is the technique of balayage. A definition for logarithmic potentials can be found in [22] Appendix VII, 13] Chapter IV, or [18], Chapter II.4. In our investigation we use this technique for Green potentials. In order to avoid technical subtilities, we assume that all domains involved are regular (with respect to Dirichlet problems) cf. 22] Appendix II, or [18] Chapter I.5) Let D C be a regular domain with cap(C ....

....can be found in [22] Appendix VII, 13] Chapter IV, or [18] Chapter II.4. In our investigation we use this technique for Green potentials. In order to avoid technical subtilities, we assume that all domains involved are regular (with respect to Dirichlet problems) cf. 22] Appendix II, or [18], Chapter I.5) Let D C be a regular domain with cap(C D) 0, a positive measure carried by D, i.e. D) and G D a regular subdomain, then there exists a positive measure , called the balayage measure, such that (5.5) g(#, D; z) g( D; z) for all z G, G, and we ....

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Sa#, E.B. and Totik, V..: Logarithmic Potentials with External Fields. Grundlehren der math. Wissensch., 316, Springer Verlag, 1997.

....(1.12) w(x) Qm i (x) # i where Qm i Zm i [x] and # i 0, i = 1, k. We develop the ideas of [21] and [7] and establish a connection with the weighted potential theory (or potential theory with external fields) that originated in the work of Gauss [14] and Frostman [13] see [25] for a modern account on this theory) An important part of the method is the analysis of the asymptotic behavior for the supremum norms of the weighted Vandermonde determinants (1.10) which is governed by the weighted capacity c w of [0, 1] corresponding to the weight w (cf. Section 2 below and ....

....for a modern account on this theory) An important part of the method is the analysis of the asymptotic behavior for the supremum norms of the weighted Vandermonde determinants (1. 10) which is governed by the weighted capacity c w of [0, 1] corresponding to the weight w (cf. Section 2 below and [25]) This method leads to the following lower bound for the integral of # function via c w . Theorem 1.1. Let w(x) be as in (1.12) and let # : # i m i . Then (1.13) #. We recover the results of Nair and Chudnovsky as a special case of Theorem 1.1. Corollary 1.2. If w(x) x ....

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E. B. Sa# and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997.

....assume that C S( is a regular domain with respect to the Dirichlet problem. In reference to this condition, for simplicity, we will say that S( is regular. The regularity of S( implies that cap S( 0. Let w be a positive continuous function on S( Set f(z) log w(z) It is well known (see [11], Sections I.1 and I.3) that among all probability measures # with support in S( there exists a unique probability measure w with support in S( called the extremal or equilibrium measure associated with w, minimizing the weighted energy I w (#) # # # t f(z) f(t) d#(z) d#(t) Let ....

.... behaviour of the polynomials p n,n may be expressed in terms of the equilibrium measure w in the presence of the external field P (#; Since the support of # is contained in L Co(S( it is well known that w is the balayage of # onto S( Therefore, Sw S( see, for instance [11], Chapter IV, Theorem 1.10) and P ( w ; z) P (#; z) Fw , z S( If S( is made up of several intervals, the measure w is absolutely continuous with respect to the Lebesgue measure dx and 1 #n (x; a) #n (x; a) d#(a) where g # ( a) is the Green s function of # with ....

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E. B. Saff and V. Totik, "Logarithmic Potentials with External Fields", A Series of Comprehensive Studies in Math. 316, Springer-Verlag, Berlin, 1997.

....and mathematics. Its motivations and applications can be found in several important areas: condensed matter physics, statistical mechanics and chaotic systems [14,26,27] multivariate statistics [8,10,13,15,19,28] the Riemann hypothesis [16,17] 2D potential theory and orthogonal polynomials [1,2,18], and numerical linear algebra [3 6,23] From the physics point of view, the dominance of spectral analysis for random matrices is mostly due to the significant physics meaning of eigenvalues, i.e. the correspondence between eigenvalues and nuclear energy levels, and between eigenvalues and ....

.... linear algebra [3 6,23] From the physics point of view, the dominance of spectral analysis for random matrices is mostly due to the significant physics meaning of eigenvalues, i.e. the correspondence between eigenvalues and nuclear energy levels, and between eigenvalues and Coulomb particles [14,18]. To numerical analysts, on the other hand, eigenvalue study is often restricted to the symmetric or Hermitian systems of linear algebraic equations (which can be further traced back to symmetric or Hermitian differential systems in the continuous world such as the Sturm Liouville problems, the ....

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E.B. Saff, V. Totik, Logarithmic Potentials with External Fields, Grundl. d. Math. Wiss., Vol. 316, Springer, Berlin, 1997.

....are the Chebyshev points # k,N = cos # (k 1 2)# N # , 1 # k # N. Then the measure # from (2.2) is d#(#) 1 # # 1 # 2 d#, # # [ 1, 1] 4. 1) The equilibrium measure # satisfies U # (#) log 2 if # # [ 1, 1] and U # (#) log 2 if # # C [ 1, 1] see, e.g. [19, 22]. It then follows from the relations (2.9) 2.10) that characterize the measure t and the constant F t that t = # and F t = log 2 for every t # (0, 1) Thus #(t; #) R [ 1, 1] 4.2) WHICH EIGENVALUES ARE FOUND BY THE LANCZOS METHOD 315 for every t # (0, 1) We see that eigenvalues ....

....measure on #, and rewriting the relations (2.9) 2.10) we can easily show that # t is the extremal measure in the external field Q t : 1 1 t U # , 5.5) with constants C t : t (1 t) F t . A comprehensive account about extremal measures in external fields can be found in [22]. See also [7] As far as the t dependence is concerned, it is important for us to know that the support of # t decreases as the parameter t increases; see [22, Theorem IV.1.6] Thus in view of (5.4) the support of # t t is decreasing. Also it is known that # : U # t (#) Q t (#) C t ....

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E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997.

....j ( d; and the inner product ( Delta; Delta) is the L 2 inner product on [A; B] f; g) Z B A f( g( d: The maximization problem (1.12) 1.13) is an extremal problem for logarithmic potentials. The function V ( Gamma t in the right hand side of (1. 12) is known as an external field, see [24]. The external field changes with time, and initially, at time t = 0, it is given by the spectral function V . The other spectral function OE is a constant of the motion and appears in (1.13) as an upper constraint for the maximizer. The spatial coordinate x appears as a normalization in (1.13) ....

....L ( Gamma V ( t ; if ( OE( 2.3) where is a constant, which may depend on x and t. The maximizer is the only function on [A; B] that satisfies (1.13) and the relations (2.1) 2.3) for some constant . We need some notions from logarithmic potential theory. Good references are [23, 24]. The Green function with pole at infinity of an unbounded domain Omega in the complex plane, is the unique continuous function of , that is harmonic in Omega Gamma vanishes on C n Omega and behaves like log jj as jj 1. We use g [ff;fi] to denote the Green function with pole at infinity of ....

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E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, New York, 1997.

....by defining V (z) V (jzj) for z 2 C . The logarithmic energy of a measure on C in the external field V is I V ( ZZ C 2 log 1 jw Gamma zj d(w)d(z) Z C V (z)d(z) 2 and the extremal energy is E V = inf 2M 1 (C ) I V ( By standard arguments from potential theory, see [12], there is a unique Borel probability measure V on C such that E V = I V ( V ) Because of the growth condition (1.1) the measure V has compact support. As V is circular symmetric on C , the measure V is circular symmetric. The circular projection of a circular symmetric is the ....

E.B. Saff and V. Totik, Logarithmic potentials with external fields, Springer-Verlag, Berlin, 1997.

....U oe of oe, which is the function U oe : C ( Gamma1; 1] 7 Z log 1 j Gamma 0 j doe( 0 ) 2. 1) For any finite Borel measure with compact support, the logarithmic potential U is superharmonic on C (in particular lower semi continuous) and harmonic on C n supp (oe) see [19]. Condition 2 U oe is a continuous, real valued function on C . 6 B. BECKERMANN AND A.B.J. KUIJLAARS Condition 2 is a regularity condition on oe. For example, it does not allow oe to have point masses. In applications, oe will typically have a density with respect to Lebesgue measure. ....

....measure (p) is defined by (p) P p( 0 ffi where each zero is counted according to its multiplicity. Then SHARP ERROR ESTIMATE FOR CG 9 ( 1 N (r m ) and ( 1 N (q n Gammam ) are two sequences of measures on [0; R] having total mass less than 1. By Helly s selection theorem, see e.g. [4, 19], we can extract from N 0 an infinite subsequence N 1 such that the following limits exist in the sense of weak convergence of measures on [0; R] lim N 1 N2N 1 1 N (q n Gammam ) lim N 1 N2N 1 1 N (r m ) ae: 3.6) From (3.4) we have that (p n ) q n Gammam ) r m ) so that by ....

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E.B. Saff and V. Totik, Logarithmic potentials with external fields, Springer-Verlag, Berlin, 1997.

....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] and [28]. The extremal problem involves a continuous function V : R R, called an external field, satisfying the growth condition lim jxj 1 V (x) log jxj = 1: 1.1) The weighted energy of a Borel probability measure is I V ( Z Z log 1 js Gamma tj d(s)d(t) 2 Z V (t) d(t) 1.2) and the ....

....n 0 . A large part of the paper is devoted to a detailed study of the one parameter family of external fields (V=c) c 0 , where V 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], and [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, c : c V=c ; c : c V=c ; c : c V=c ; ....

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E.B. Saff and V. Totik, Logarithmic potentials with external fields. Grundlehren der Mathematischen Wissenschaften, 316. Springer, Berlin, 1997. 60 A.B.J. KUIJLAARS AND K. T-R MCLAUGHLIN

....be the set of probability measures on . The equilibrium measure d V (z) for potential V (z) 2 (z z 1 ) on the unit circle is de ned by the following minimization problem, inf 2M ZZ log jz wj 1 d (z)d (w) Z V (z)d (z) 4. 6) The in mum is achieved uniquely (see, e.g. [ST]) at the equilibrium measure. Let J denote the support of d V . The equilibrium measure and its support are uniquely determined by the following Euler Lagrange variational conditions : there exits a real constant l such that; 2 Z log jz sjd V (s) V (z) l = 0 for z 2 J ; 2 Z log jz ....

E.B.Sa and V.Totik, Logarithmic Potentials with External Fields, Springer-Verlag, New York, 1997.

....is a convex function on [0; 1] which according to Theorem 2.17 in [5] implies that S ff;p is an interval. Case 1. ff p. In this case we focus first on the solution ff;p of the unconstrained weighted energy problem on the interval [0; 1] with weight function w ff;p = exp( GammaQ ff;p ) cf. [11]) One reason for considering the unconstrained problem is that there is a relationship between the support of ff;p and ff;p ; namely that S ff;p ae S ff;p (see [5, Theorem 2.6] The following lemma determines the support of ff;p . In Theorem 3(a) we determine the density of this measure ....

E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelgerg, 1997.

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E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York (1997).

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E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, 1997.

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E. B. SAFF AND V. TOTIK, Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, New York (1997).

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E. B. Sa# and V. Totik, Logarithmic Potentials with External Fields, Springer, Berlin-Heidelberg-New York, 1997.

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E.B. SAFF AND V. TOTIK, Logarithmic potentials with external fields, Springer, Berlin, 1997.

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Sa, E.B. and Totik, V..: Logarithmic Potentials with External Fields. Grundlehren der math. Wissensch., 316, Springer Verlag, 1997.

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Sa, E.B. & Totik, V., Logarithmic Potentials with External Fields, Springer-Verlag, Berlin, New York (1997).

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Sa#, E.B. and Totik, V., Logarithmic potentials with external fields, Springer Verlag, Berlin, 1997.

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E.B. Sa# and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag,

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E. B. Sa and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, Heidelberg, 1997.

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