WEYL-TITCHMARSH TYPE FORMULA FOR ... - UT Mathematics
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR WITH SMALL PERTURBATION SERGEY SIMONOV Abstract. Small perturbations of the Jacobi matrix with weights √ n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schr¨odinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for ”free” Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained.
1. Introduction ∞ Let {an }∞ n=1 be a sequence of positive numbers and {bn }n=1 be a sequence of real numbers, {en }∞ n=1 be the canonical basis in the space l2 (N) (i.e., each vector en has zero components except the n-th which is 1), let also lf in be the linear set of sequences with finite number of non-zero components. One can define an operator J in l2 , which acts in lf in by the rule
(J u)n = an−1 un−1 + bn un + an un+1 , n ≥ 2, (J u)1 = b1 u1 + a1 u2 . The operator is first defined on lf in and then the closure is considered. ∞ P 1 Then J is self-adjoint in l2 provided = ∞ [3] (Carleman conan dition), and it has the following the canonical basis: b1 a1 J = 0 .. .
n=0
matrix representation with respect to a1 0 b2 a 2 a2 b3 .. .. . .
··· ··· ··· . ...
1991 Mathematics Subject Classification. 47A10, 47B36. Key words and phrases. Jacobi matrices, Absolutely continuous spectrum, Subordinacy theory, Weyl-Titchmarsh theory. 1
2
SERGEY SIMONOV
Consider the spectral equation for J : (1)
an−1 un−1 + bn un + an un+1 = λun , n ≥ 2.
1 Solution Pn (λ) of (1) such that P1 (λ) ≡ 1, P2 (λ) = λ−b is a polynomial a1 in λ of degree n − 1 and is called the polynomial of the first kind. Correspondingly the solution Qn (λ) such that Q1 (λ) ≡ 0, Q2 (λ) ≡ a11 is a polynomial of degree n − 2 and is called the polynomial of the second kind. For two solutions of (1) un and vn , the expression
∞ W (u, v) := W ({un }∞ n=1 , {vn }n=1 ) := an (un vn+1 − un+1 vn )
is independent of n and is called the (discrete) Wronskian of u and v. One always has W (P (λ), Q(λ)) ≡ 1. The spectrum of every Jacobi matrix is simple and the vector e1 from the standard basis is the generating vector [3]. Let dE be the operatorvalued spectral measure associated with J . Polynomials of the first kind are orthogonal with respect to the measure dρ := (dEe1 , e1 ), which is also called the spectral measure [2]. For non-real values of λ solutions of (1) that belong to l2 are proportional to Qn (λ)+m(λ)Pn (λ) [2], where Z dρ(x) m(λ) := , λ ∈ C\R x−λ R
is the Weyl function. By Fatou’s Theorem [6], 1 ρ0 (λ) = Im m(λ + i0), π for a.a. λ ∈ R. In the present paper we consider small√perturbations of the operator ∞ J0 , which is defined by the sequences { n}∞ n=1 and {0}n=1 : 0 1 √0 · · · 1 0 2 ··· J0 = 0 √ 2 0 · · · . .. .. .. . . . . . . Let us call J0 the ”free” Hermite operator. We will call (following [11]) J the Hermite operator, if it can be considered close to J0 in some sense. Let us call J the ”small” perturbation of J0 , if J √ is defined by ∞ ∞ sequences {an }n=1 and {bn }n=1 such that (let cn := an − n) (2) ¶ ∞ µ X √ |cn | |cn+1 − cn | + |bn | √ + cn = o( n) as n → ∞ and < ∞. n n n=1
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
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Denote the following expression by Λ: for any sequence {un }∞ n=1 , (3)
(Λu)n := cn−1 un−1 + bn un + cn bn+1 , n ≥ 2, (Λu)1 := b1 u1 + c1 u2 .
Although Λ is not a Jacobi matrix, we will write J = J0 + Λ. The spectrum of J0 is purely absolutely continuous on R with the spectral density λ2
e− 2 ρ00 (λ) = √ . 2π As it will be shown, the spectrum of J is also purely absolutely continuous under assumption (2). Our goal in the present paper is to study the spectral density of J using the asymptotic analysis of generalized eigenvectors of J (i.e., solutions of the spectral equation (1)). The method is based upon the comparison of solutions of (1) to solutions of the spectral equation for the free Hermite operator, √ √ (4) n − 1un−1 + nun+1 = λun , n ≥ 2. This is analogous to the Weyl-Titchmarsh theory for the Schr¨odinger operator on the half-line with the summable potential. The following results will be proven (Theorem 1 in Section 2 and Theorem 2 in Section 3). Let w be the standard error function [1] Z Z 2 2 1 e−ζ dζ 1 e−ζ dζ (5) w(z) := =− , πi Γ−z ζ − z πi Γ+−z ζ + z where the contours Γ± z are shown on Figure 1. Function w is entire.
Γz+
z z
−
Γz
Figure 1. Contours Γ± z Theorem 1. For every λ ∈ C equation (4) has a basis of solutions ³ ´ 2 (n−1) √λ n−1 λ2 (−1) e w 2 p In+ (λ) := n+1 (n − 1)!2
4
SERGEY SIMONOV
and
³ ´ λ2 e 2 w(n−1) − √λ2 In− (λ) := p , (n − 1)!2n+1
which have the following asymptotics as n → ∞: √ n
λ2
In± (λ)
(∓i)n−1 e 4 ±iλ = (8πn)1/4
µ µ ¶¶ 1 1+O √ . n
These asymptotics are uniform with respect to λ in every bounded set in C. Polynomials of the first kind for J0 are related to In± in the following way: P0 n (λ) = In+ (λ) + In− (λ). Theorem 2. Let the conditions (2) hold for J . Then 1. For every λ ∈ C+ there exists √
F (λ) := 1 + i 2πe
2
− λ2
∞ X
(ΛI + (λ))n Pn (λ)
n=1
(the Jost function), which is analytic function in C+ and continuous in C+ . 2. Polynomials of the first kind have the following asymptotics as n → ∞: • For λ ∈ C+ , µ Pn (λ) =
F (λ)In− (λ)
+o
√
eImλ n n1/4
¶ as n → ∞,
• For λ ∈ R, 1
Pn (λ) = F (λ)In− (λ) + F (λ)In+ (λ) + o(n− 4 ) as n → ∞. 3. The spectrum of J is purely absolutely continuous, and for a.a. λ∈R λ2
e− 2 ρ0 (λ) = √ 2π|F (λ)|2 (the Weyl-Titchmarsh type formula). The idea of the Weyl-Titchmarsh type formula is the relation between the spectral density and the behavior of Pn (λ) for large values of n. We can formulate this in the form of the corollary.
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
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Corollary 1. Let the conditions (2) hold for J . Then the spectrum of J is purely absolutely continuous and the spectral density equals for a.a. λ ∈ R 1 1 , ρ0 (λ) = lim √ 2 π n→∞ n(Pn2 (λ) + Pn+1 (λ)) the right-hand side being finite and non-zero for every λ ∈ R. Theorem 2 can be proven by another method, based on the Levinsontype analytical and smooth theorem, cf. [5] and papers of BernzaidLutz [8], Janas-Moszy´ nski [13] and Silva [16], [17]. None of their results is directly applicable here, and the approach of the present paper is different. The considered situation is parallel to the Weyl-Titchmarsh theory for Schr¨odinger operator on the half-line with summable potential. Let q be a real-valued function on R+ and q ∈ L1 (R+ ). Consider the Schr¨odinger operator on R+ d2 + q(x) dx2 with the Dirichlet boundary condition. The purely absolutely continuous spectrum of L coincides with R+ [7]. Let ϕ(x, λ) be a solution of the spectral equation for L, L=−
−u00 (x, λ) + q(x)u(x, λ) = λu(x, λ), such that ϕ(0, λ) ≡ 0, ϕ0 (0, λ) ≡ 1 (satisfying the boundary condition). The following result holds [7]. Proposition 1. If q ∈ L1 (R+ ), then for every k > 0 there exist a(k) and b(k) such that ϕ(x, k 2 ) = a(k) cos(kx) + b(k) sin(kx) + o(1) as x → +∞, and for a.a. λ > 0 1 √ √ ρ0 (λ) = √ π λ(a2 ( λ) + b2 ( λ)) (the classical Weyl-Titchmarsh formula). Solutions In+ (λ) and In− (λ) are the direct analogues to the solutions −ikx and e−2ik of the spectral equation for ”free” Schr¨odinger operator, 2ik
eikx
−u00 (x, k 2 ) = k 2 u(x, k 2 ). The main technical difficulty of our problem is non-triviality of solu±ikx tions In± (λ) compared to e±2ik . The model of the Hermite operator was studied in the paper of Brown-Naboko-Weikard [11], but solutions In± (λ) were not introduced there.
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SERGEY SIMONOV
2. The free Hermite operator In this section we study asymptotic properties of generalized eigenvectors for J0 and prove Theorem 1. Let us give its formulation again. Theorem. For every λ ∈ C equation (4) has a basis of solutions ³ ´ λ2 (−1)n−1 e 2 w(n−1) √λ2 p (6) In+ (λ) := (n − 1)!2n+1 and (7)
³ ´ λ2 e 2 w(n−1) − √λ2 In− (λ) := p , (n − 1)!2n+1
which have the following asymptotics as n → ∞: √ µ µ ¶¶ λ2 1 (∓i)n−1 e 4 ±iλ n ± 1+O √ (8) In (λ) = . (8πn)1/4 n These asymptotics are uniform with respect to λ in every bounded set in C. Polynomials of the first kind for J0 are related to In± in the following way: (9)
P0n (λ) = In+ (λ) + In− (λ).
Proof. The spectral equation (4) for J0 , √ √ n − 1un−1 + nun+1 = λun , n ≥ 2, can be transformed to the recurrence relation (10)
2nvn−1 (x) + vn+1 (x) = 2xvn (x), n ≥ 1 √ if one takes vn := 2n n!un+1 and x := √λ2 . Equation (10) is satisfied by Hermite polynomials [1], and this means (together with initial values: H0 (x) ≡ 1 and H1 (x) = 2x), that the polynomials of the first kind for J0 equal (11)
P0 n (λ) = p
Hn−1 ( √λ2 ) 2n−1 (n − 1)!
.
Equation (10) has two other linearly independent solutions, w(n) (−x) and (−1)n w(n) (x) [1]. This can be checked by substituting them into (10) using the formula Z 2 e−ζ dζ n! (n) w (z) = πi Γ−z (ζ − z)n+1
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
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and integrating by parts. From the integral representation for Hermite polynomials [1], I 2xz−z2 n! e Hn (x) = dz 2πi 0 z n+1 ÃZ ! Z 2 2 2 n!ex e−z dz e−z dz = − 2πi (z + x)n+1 (z + x)n+1 Γ− Γ+ −x −x 2
2
ex (n) ex = w (−x) + (−1)n w(n) (x), 2 2 where the contour Γ+ z is shown on Figure 1. Correspondingly, equation (4) has two linearly independent solutions of the form (6) and (7) and relation (9) holds. Asymptotics of these solutions follow immediately from Corollary 2 from the appendix. ¤ In what follows we will need to know the Wronskian of the solutions. Lemma 1. λ2
e2 W (I (λ), I (λ)) = i √ . 2π +
−
Proof. One has from (6) and (7): W (I + (λ), I − (λ)) = I1+ (λ)I2− (λ) − I2+ (λ)I1− (λ) µ ¶ µ ¶ µ ¶ µ ¶¶ λ2 2 µ 2 eλ λ λ λ λ e = √ w √ w0 − √ + w − √ w0 √ = i√ , 4 2 2 2 2 2 2π using the following properties of the error function [1]: w0 (z) = −2zw(z) + √2iπ , 2 w(z) + w(−z) = 2e−z . ¤ 3. The perturbed Hermite operator In this section, we consider the Hermite operator J with ”small” perturbation, i.e., satisfying conditions (2), and prove Theorem 2. We study asymptotics of polynomials of the first and second kind using the Volterra-type equation and derive from these asymptotics a formula for the Weyl function. The desired Weyl-Titchmarsh type formula follows from this. We start with proving a formula of variation of parameters. Remind that Pn (λ) are polynomials of the first kind for J , P0 n (λ) are
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SERGEY SIMONOV
polynomials of the first kind for J0 , Λ is the expression given by (3), √ an = n + cn . Let us denote W (λ) := W (I + (λ), I − (λ)). Lemma 2. For n ≥ 2, (12) n−1 X an−1 (ΛI + (λ))k In− (λ) − In+ (λ)(ΛI − (λ))k √ Pn (λ) = P0 n (λ)− Pk (λ). W (λ) n−1 k=1 Proof. Let us omit the dependence on λ everywhere. First let us prove that n−1 + − X Ik In − Ik− In+ (13) Pn = un − (ΛP )k , n ≥ 3, W k=2 where u is the solution of (4) such that u1 = P1 and u2 = P2 . Let us denote n−1 P Ik+ In− −Ik− In+ u − (ΛP )k , n ≥ 3, n W Pen := k=2 Pn , n = 1, 2 I fact, one has to check that √ √ n − 1Pen−1 − λPen + nPen+1 = −(ΛP )n , n ≥ 2, (this non-homogeneous equation has only one solution with fixed two first values, so Pe should coincide with P ). Since u, I + and I − are solutions to (4) and n − + √ X Ik+ In+1 − In+1 Ik− n (ΛP )k = (ΛP )n , W k=n
the previous is equivalent to n−1 n−1 − + X √ X Ik+ In+1 − In+1 Ik− Ik+ In− − In+ Ik− −λ (ΛP )k + n (ΛP )k = 0 W W k=n−1 k=n−1 √ √ ± ± for n ≥ 3. The latter is true, because −λIn± + nIn+1 = − n − 1In−1 . After shifting indices in different parts of the sum in (13) one obtains:
I1+ In− − In+ I1− (b1 P1 + c1 P2 ) Pn = un + W n−1 X (ΛI + )k In− − In+ (ΛI − )k cn−1 − Pk − √ Pn . W n − 1 k=1
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
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Since un =
I1+ P2 − I2+ P1 − I1− P2 − I2− P1 + λ − b1 In − In , P1 = 1, P2 = , W W a1
one has: I1+ In− − In+ I1− λI1+ − I2+ − λI1− − I2− + un + (b1 P1 + c1 P2 ) = In − In W W W I + P0 − I2+ P0 1 − I2+ P0 1 − I1+ P0 2 + = 1 2 In − In = P0 n . W W Therefore n−1 X a (ΛI + )k In− − In+ (ΛI − )k √ n−1 Pn = P0 n − Pk . W n−1 k=1
¤ Equation (12) is of Volterra type. We need the following standard lemma to deal with it. Consider the Banach space ¶ ¾ ½ µ |un |n1/4 ∞ B := {un }n=1 : sup |Im λ|√n < ∞ e n with the norm
µ kukB := sup n
|un |n1/4 √ e|Im λ| n
¶
(we omit the dependence on λ in the notation for B). Let V be the expression ½ 0, n = 1 (14) (Vu)n := Pn−1 k=1 Vnk uk , n ≥ 2 for any sequence {u}∞ n=1 . Let (15)
ν := sup n>1
n−1 X k=1
|Vnk |e
√ √ |Im λ|( k− n)
³ n ´1/4 k
.
Lemma 3. If ν < ∞, then V is a bounded operator in B, (I − V)−1 exists and kVkB ≤ ν, k(I − V)−1 kB ≤ eν .
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SERGEY SIMONOV
Proof. By definition of the operator norm we have to check the finiteness of the following:
sup u6=0
kVukB = sup kukB u6=0
|
sup
Pn−1
|n1/4
k=1 Vnk uk √ e|Im λ| n
n>1
1/4
n |n √ sup e|u|Im λ| n
n
sup ≤ sup
n−1 P
n>1 k=1
1/4
k |k √ |Vnk | e|u|Im λ| k
¡ n ¢1/4 k
e|Im λ|(
√ √ k− n)
.
1/4
n |n √ sup e|u|Im λ| n
u6=0
n
n1/4 √
Denoting u en := un eImλ
sup u6=0
kVukB ≤ sup kukB u e6=0
n
sup
, we have: n−1 P
n>1 k=1
|Vnk ||e uk |
¡ n ¢1/4 k
√
e|Im λ|(
√ k− n)
sup |e un | n
≤ sup n>1
n−1 X k=1
|Vnk |
³ n ´1/4 k
√
e|Im λ|(
√ k− n)
,
hence V is bounded. Quite similarly, ¯ ¯ ¯³ ´ n−1 ¯ X X ¯ ¯ n 1/4 |Im λ|(√k−√n) l ¯ ¯ kV kB ≤ sup V e V ...V nk1 k1 k2 kl−1 k ¯ ¯ k n>1 ¯ ¯ k=1 1≤k1 0, Z +∞ 2 2 (26) tα e−βt dt = O(xα+1 e−βx ) as x → +∞, x
! 2
e−ns ds.
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SERGEY SIMONOV
one has:
ÃZ
Z
−n−3/8
+∞
!
+
e
−ns2
n−3/8
−∞
µ ¶ 1 ds = O as n → ∞. n
3
Let |s| < min{n− 8 ; δ0 } and |µ| < δ0 . Then n|h(s, µ) + s2 | < C1 n|s|3
N1 , |s| < n−3/8 and |µ| < δ0 , 2 then |en(h(s,µ)+s ) − 1| < 2C1 n|s|3 . Hence we arrive at the following (uniform for |µ| < δ0 ) estimate: ¯Z −3/8 ¯ Z −3/8 ¯ n ¯ n 2 2 ¯ ¯ nh(s,µ) −ns2 (e −e )ds¯ ≤ e−ns |en(h(s,µ)+s ) − 1|ds ¯ ¯ −n−3/8 ¯ −n−3/8 µ ¶ Z n−3/8 1 3 −ns2 |s| e ds = O < 2C1 n as n → ∞ n −n−3/8 from (26). 2. The following is an immediate consequence of (25) and (26): for |µ| < δ0 , ¯ÃZ −3/8 Z ¯ ! Z δ0 ¯ ¯ −n δ0 ns2 ¯ ¯ nh(s,µ) + e ds¯ < 2 e− 2 ds ¯ ¯ −δ0 ¯ n−3/8 n−3/8 µ ¶ Z +∞ 2 2 1 − t2 21 and | arg γ(µ)| < ϕ0 . Then |i + γ(µ)s| > 1. Let θ := 31 . By the choice of µ1 we can also ensure that if |µ| < µ1 , then ½ Re a2 (µ) > 12 , Re[a(µ)(ϕ(µ) − 2µ)] > − δ02θ γ(µ) :=
and hence
1 Reh(s, µ) < − (s2 − 2sδ0 θ) 4 for every real s such that |s| > δ0 . One has: ¯·Z −δ0 Z +∞ ¸ ¯ Z +∞ ¯ ¯ n 2 nh(s,µ) ¯ ¯ + e ds¯ < 2 e− 4 (s −2sδ0 θ) ds ¯ −∞ δ0 δ0 µ ¶ Z +∞ ´ ³ n 2 2 n 2 n 2 1 δ θ s δ (2θ−1) − = 2e 4 0 =O as n → ∞ e 4 ds = O e 4 0 n δ0 (1−θ) uniformly with respect to µ. This completes the proof of the lemma. ¤ ¤ As a corollary we have asymptotics of w(n−1) (z) as n → ∞ for fixed z.
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SERGEY SIMONOV
Corollary 2. r
µ µ ¶¶ n 2 (n − 1)!in−1 n +iz√2n− z2 1 2 √ w (z) = e2 1+O √ as n → ∞ n n 2π uniformly with respect to z in every bounded set in C. (n−1)
Proof. We just need to substitute µ = √z2n into (24) and go through tedious calculation, using that iz 2 ϕ(z) = i + z − + O(z 4 ) as z → 0. 2 ¤ 5. Acknowledgement The author expresses his deep gratitude to Dr. A.V. Kiselev for valuable discussions of the problem and to Prof. S.N. Naboko for his constant attention to this work and for many fruitful discussions of the subject. The author also wishes to thank Prof. V.S. Buslaev for constructive criticism. The work was supported by grants RFBR-06-0100249 and INTAS-05-1000008-7883, and also by Vladimir Deich prize. References [1] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1964. [2] N.I. Akhiezer, The classical moment problem and some related questions in analysis, Oliver & Boyd, 1965. [3] Yu.M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators. (Russian), Naukova Dumka, Kiev, 1965. [4] M.S. Birman , M.Z. Solomyak, Spectral Theory of self-adjoint operators in Hilbert space, Reidel, 1987. [5] E.A. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. [6] P. Koosis, Introduction to Hp spaces, Cambridge University Press, Cambridge, 1980. [7] E.C. Titchmarsh, Eigenfunction expantions associated with second-order differential equations, 2, Clerandon Press, Oxford, 1958. [8] Z. Benzaid, D.A. Lutz, Asymptotic representation of solutions of perturbed systems of linear difference equations, Studies Appl. Math. 77, 1987. [9] B.M. Brown, M.S.P. Eastham, D.K.R. McCornack, Spectral concentration and rapidly decaying potentials, J. Comput. Appl. Math. 81, 1997. [10] B.M. Brown, M.S.P. Eastham, D.K.R. McCornack, Spectral concentration and perturbed discrete spectra, J. Comput. Appl. Math. 86, 1997. [11] B.M. Brown, S. Naboko, R. Weikard, The inverse resonance problem for Hermite operators(preprint). [12] D.J. Gilbert, D.B. Pearson, On subordinacy and analysis of the spectrum of one dimensional Schr¨ odinger operators, J. Math. Anal. Appl. 128, 1987.
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[13] J. Janas, M. Moszy´ nski, Spectral properties of Jacobi matrices by asymptotic analysis, J. Approx. Theory 120, 2003. [14] S. Khan, D.B. Pearson, Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta 65, 1992. [15] M. Plancherel, W. Rotach, Sur ³ les valeurs asymptotiques des polynomes ´ x2
n
x2
d d’Hermite Hn (x) = (−1)n e 2 dx e− 2 (French) Commentarii Math. Heln vetici 1, 1929. [16] L.O. Silva, Uniform Levinson type theorems for discrete linear systems, Oper. Theory Adv. Appl., Birkhauser-Verlag 154, 2004. [17] L.O. Silva, Uniform and smooth Benzaid-Lutz type theorems and applications to Jacobi matrices, Oper. Theory Adv. Appl., Birkhauser-Verlag 174, 2007.
Department of Mathematical Physics, Institute of Physics, St. Petersburg University, Ulianovskaia 1, 198904, St. Petergoff, St. Petersburg, Russia E-mail address: sergey [email protected]
1. Introduction ∞ Let {an }∞ n=1 be a sequence of positive numbers and {bn }n=1 be a sequence of real numbers, {en }∞ n=1 be the canonical basis in the space l2 (N) (i.e., each vector en has zero components except the n-th which is 1), let also lf in be the linear set of sequences with finite number of non-zero components. One can define an operator J in l2 , which acts in lf in by the rule
(J u)n = an−1 un−1 + bn un + an un+1 , n ≥ 2, (J u)1 = b1 u1 + a1 u2 . The operator is first defined on lf in and then the closure is considered. ∞ P 1 Then J is self-adjoint in l2 provided = ∞ [3] (Carleman conan dition), and it has the following the canonical basis: b1 a1 J = 0 .. .
n=0
matrix representation with respect to a1 0 b2 a 2 a2 b3 .. .. . .
··· ··· ··· . ...
1991 Mathematics Subject Classification. 47A10, 47B36. Key words and phrases. Jacobi matrices, Absolutely continuous spectrum, Subordinacy theory, Weyl-Titchmarsh theory. 1
2
SERGEY SIMONOV
Consider the spectral equation for J : (1)
an−1 un−1 + bn un + an un+1 = λun , n ≥ 2.
1 Solution Pn (λ) of (1) such that P1 (λ) ≡ 1, P2 (λ) = λ−b is a polynomial a1 in λ of degree n − 1 and is called the polynomial of the first kind. Correspondingly the solution Qn (λ) such that Q1 (λ) ≡ 0, Q2 (λ) ≡ a11 is a polynomial of degree n − 2 and is called the polynomial of the second kind. For two solutions of (1) un and vn , the expression
∞ W (u, v) := W ({un }∞ n=1 , {vn }n=1 ) := an (un vn+1 − un+1 vn )
is independent of n and is called the (discrete) Wronskian of u and v. One always has W (P (λ), Q(λ)) ≡ 1. The spectrum of every Jacobi matrix is simple and the vector e1 from the standard basis is the generating vector [3]. Let dE be the operatorvalued spectral measure associated with J . Polynomials of the first kind are orthogonal with respect to the measure dρ := (dEe1 , e1 ), which is also called the spectral measure [2]. For non-real values of λ solutions of (1) that belong to l2 are proportional to Qn (λ)+m(λ)Pn (λ) [2], where Z dρ(x) m(λ) := , λ ∈ C\R x−λ R
is the Weyl function. By Fatou’s Theorem [6], 1 ρ0 (λ) = Im m(λ + i0), π for a.a. λ ∈ R. In the present paper we consider small√perturbations of the operator ∞ J0 , which is defined by the sequences { n}∞ n=1 and {0}n=1 : 0 1 √0 · · · 1 0 2 ··· J0 = 0 √ 2 0 · · · . .. .. .. . . . . . . Let us call J0 the ”free” Hermite operator. We will call (following [11]) J the Hermite operator, if it can be considered close to J0 in some sense. Let us call J the ”small” perturbation of J0 , if J √ is defined by ∞ ∞ sequences {an }n=1 and {bn }n=1 such that (let cn := an − n) (2) ¶ ∞ µ X √ |cn | |cn+1 − cn | + |bn | √ + cn = o( n) as n → ∞ and < ∞. n n n=1
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
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Denote the following expression by Λ: for any sequence {un }∞ n=1 , (3)
(Λu)n := cn−1 un−1 + bn un + cn bn+1 , n ≥ 2, (Λu)1 := b1 u1 + c1 u2 .
Although Λ is not a Jacobi matrix, we will write J = J0 + Λ. The spectrum of J0 is purely absolutely continuous on R with the spectral density λ2
e− 2 ρ00 (λ) = √ . 2π As it will be shown, the spectrum of J is also purely absolutely continuous under assumption (2). Our goal in the present paper is to study the spectral density of J using the asymptotic analysis of generalized eigenvectors of J (i.e., solutions of the spectral equation (1)). The method is based upon the comparison of solutions of (1) to solutions of the spectral equation for the free Hermite operator, √ √ (4) n − 1un−1 + nun+1 = λun , n ≥ 2. This is analogous to the Weyl-Titchmarsh theory for the Schr¨odinger operator on the half-line with the summable potential. The following results will be proven (Theorem 1 in Section 2 and Theorem 2 in Section 3). Let w be the standard error function [1] Z Z 2 2 1 e−ζ dζ 1 e−ζ dζ (5) w(z) := =− , πi Γ−z ζ − z πi Γ+−z ζ + z where the contours Γ± z are shown on Figure 1. Function w is entire.
Γz+
z z
−
Γz
Figure 1. Contours Γ± z Theorem 1. For every λ ∈ C equation (4) has a basis of solutions ³ ´ 2 (n−1) √λ n−1 λ2 (−1) e w 2 p In+ (λ) := n+1 (n − 1)!2
4
SERGEY SIMONOV
and
³ ´ λ2 e 2 w(n−1) − √λ2 In− (λ) := p , (n − 1)!2n+1
which have the following asymptotics as n → ∞: √ n
λ2
In± (λ)
(∓i)n−1 e 4 ±iλ = (8πn)1/4
µ µ ¶¶ 1 1+O √ . n
These asymptotics are uniform with respect to λ in every bounded set in C. Polynomials of the first kind for J0 are related to In± in the following way: P0 n (λ) = In+ (λ) + In− (λ). Theorem 2. Let the conditions (2) hold for J . Then 1. For every λ ∈ C+ there exists √
F (λ) := 1 + i 2πe
2
− λ2
∞ X
(ΛI + (λ))n Pn (λ)
n=1
(the Jost function), which is analytic function in C+ and continuous in C+ . 2. Polynomials of the first kind have the following asymptotics as n → ∞: • For λ ∈ C+ , µ Pn (λ) =
F (λ)In− (λ)
+o
√
eImλ n n1/4
¶ as n → ∞,
• For λ ∈ R, 1
Pn (λ) = F (λ)In− (λ) + F (λ)In+ (λ) + o(n− 4 ) as n → ∞. 3. The spectrum of J is purely absolutely continuous, and for a.a. λ∈R λ2
e− 2 ρ0 (λ) = √ 2π|F (λ)|2 (the Weyl-Titchmarsh type formula). The idea of the Weyl-Titchmarsh type formula is the relation between the spectral density and the behavior of Pn (λ) for large values of n. We can formulate this in the form of the corollary.
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
5
Corollary 1. Let the conditions (2) hold for J . Then the spectrum of J is purely absolutely continuous and the spectral density equals for a.a. λ ∈ R 1 1 , ρ0 (λ) = lim √ 2 π n→∞ n(Pn2 (λ) + Pn+1 (λ)) the right-hand side being finite and non-zero for every λ ∈ R. Theorem 2 can be proven by another method, based on the Levinsontype analytical and smooth theorem, cf. [5] and papers of BernzaidLutz [8], Janas-Moszy´ nski [13] and Silva [16], [17]. None of their results is directly applicable here, and the approach of the present paper is different. The considered situation is parallel to the Weyl-Titchmarsh theory for Schr¨odinger operator on the half-line with summable potential. Let q be a real-valued function on R+ and q ∈ L1 (R+ ). Consider the Schr¨odinger operator on R+ d2 + q(x) dx2 with the Dirichlet boundary condition. The purely absolutely continuous spectrum of L coincides with R+ [7]. Let ϕ(x, λ) be a solution of the spectral equation for L, L=−
−u00 (x, λ) + q(x)u(x, λ) = λu(x, λ), such that ϕ(0, λ) ≡ 0, ϕ0 (0, λ) ≡ 1 (satisfying the boundary condition). The following result holds [7]. Proposition 1. If q ∈ L1 (R+ ), then for every k > 0 there exist a(k) and b(k) such that ϕ(x, k 2 ) = a(k) cos(kx) + b(k) sin(kx) + o(1) as x → +∞, and for a.a. λ > 0 1 √ √ ρ0 (λ) = √ π λ(a2 ( λ) + b2 ( λ)) (the classical Weyl-Titchmarsh formula). Solutions In+ (λ) and In− (λ) are the direct analogues to the solutions −ikx and e−2ik of the spectral equation for ”free” Schr¨odinger operator, 2ik
eikx
−u00 (x, k 2 ) = k 2 u(x, k 2 ). The main technical difficulty of our problem is non-triviality of solu±ikx tions In± (λ) compared to e±2ik . The model of the Hermite operator was studied in the paper of Brown-Naboko-Weikard [11], but solutions In± (λ) were not introduced there.
6
SERGEY SIMONOV
2. The free Hermite operator In this section we study asymptotic properties of generalized eigenvectors for J0 and prove Theorem 1. Let us give its formulation again. Theorem. For every λ ∈ C equation (4) has a basis of solutions ³ ´ λ2 (−1)n−1 e 2 w(n−1) √λ2 p (6) In+ (λ) := (n − 1)!2n+1 and (7)
³ ´ λ2 e 2 w(n−1) − √λ2 In− (λ) := p , (n − 1)!2n+1
which have the following asymptotics as n → ∞: √ µ µ ¶¶ λ2 1 (∓i)n−1 e 4 ±iλ n ± 1+O √ (8) In (λ) = . (8πn)1/4 n These asymptotics are uniform with respect to λ in every bounded set in C. Polynomials of the first kind for J0 are related to In± in the following way: (9)
P0n (λ) = In+ (λ) + In− (λ).
Proof. The spectral equation (4) for J0 , √ √ n − 1un−1 + nun+1 = λun , n ≥ 2, can be transformed to the recurrence relation (10)
2nvn−1 (x) + vn+1 (x) = 2xvn (x), n ≥ 1 √ if one takes vn := 2n n!un+1 and x := √λ2 . Equation (10) is satisfied by Hermite polynomials [1], and this means (together with initial values: H0 (x) ≡ 1 and H1 (x) = 2x), that the polynomials of the first kind for J0 equal (11)
P0 n (λ) = p
Hn−1 ( √λ2 ) 2n−1 (n − 1)!
.
Equation (10) has two other linearly independent solutions, w(n) (−x) and (−1)n w(n) (x) [1]. This can be checked by substituting them into (10) using the formula Z 2 e−ζ dζ n! (n) w (z) = πi Γ−z (ζ − z)n+1
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
7
and integrating by parts. From the integral representation for Hermite polynomials [1], I 2xz−z2 n! e Hn (x) = dz 2πi 0 z n+1 ÃZ ! Z 2 2 2 n!ex e−z dz e−z dz = − 2πi (z + x)n+1 (z + x)n+1 Γ− Γ+ −x −x 2
2
ex (n) ex = w (−x) + (−1)n w(n) (x), 2 2 where the contour Γ+ z is shown on Figure 1. Correspondingly, equation (4) has two linearly independent solutions of the form (6) and (7) and relation (9) holds. Asymptotics of these solutions follow immediately from Corollary 2 from the appendix. ¤ In what follows we will need to know the Wronskian of the solutions. Lemma 1. λ2
e2 W (I (λ), I (λ)) = i √ . 2π +
−
Proof. One has from (6) and (7): W (I + (λ), I − (λ)) = I1+ (λ)I2− (λ) − I2+ (λ)I1− (λ) µ ¶ µ ¶ µ ¶ µ ¶¶ λ2 2 µ 2 eλ λ λ λ λ e = √ w √ w0 − √ + w − √ w0 √ = i√ , 4 2 2 2 2 2 2π using the following properties of the error function [1]: w0 (z) = −2zw(z) + √2iπ , 2 w(z) + w(−z) = 2e−z . ¤ 3. The perturbed Hermite operator In this section, we consider the Hermite operator J with ”small” perturbation, i.e., satisfying conditions (2), and prove Theorem 2. We study asymptotics of polynomials of the first and second kind using the Volterra-type equation and derive from these asymptotics a formula for the Weyl function. The desired Weyl-Titchmarsh type formula follows from this. We start with proving a formula of variation of parameters. Remind that Pn (λ) are polynomials of the first kind for J , P0 n (λ) are
8
SERGEY SIMONOV
polynomials of the first kind for J0 , Λ is the expression given by (3), √ an = n + cn . Let us denote W (λ) := W (I + (λ), I − (λ)). Lemma 2. For n ≥ 2, (12) n−1 X an−1 (ΛI + (λ))k In− (λ) − In+ (λ)(ΛI − (λ))k √ Pn (λ) = P0 n (λ)− Pk (λ). W (λ) n−1 k=1 Proof. Let us omit the dependence on λ everywhere. First let us prove that n−1 + − X Ik In − Ik− In+ (13) Pn = un − (ΛP )k , n ≥ 3, W k=2 where u is the solution of (4) such that u1 = P1 and u2 = P2 . Let us denote n−1 P Ik+ In− −Ik− In+ u − (ΛP )k , n ≥ 3, n W Pen := k=2 Pn , n = 1, 2 I fact, one has to check that √ √ n − 1Pen−1 − λPen + nPen+1 = −(ΛP )n , n ≥ 2, (this non-homogeneous equation has only one solution with fixed two first values, so Pe should coincide with P ). Since u, I + and I − are solutions to (4) and n − + √ X Ik+ In+1 − In+1 Ik− n (ΛP )k = (ΛP )n , W k=n
the previous is equivalent to n−1 n−1 − + X √ X Ik+ In+1 − In+1 Ik− Ik+ In− − In+ Ik− −λ (ΛP )k + n (ΛP )k = 0 W W k=n−1 k=n−1 √ √ ± ± for n ≥ 3. The latter is true, because −λIn± + nIn+1 = − n − 1In−1 . After shifting indices in different parts of the sum in (13) one obtains:
I1+ In− − In+ I1− (b1 P1 + c1 P2 ) Pn = un + W n−1 X (ΛI + )k In− − In+ (ΛI − )k cn−1 − Pk − √ Pn . W n − 1 k=1
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
9
Since un =
I1+ P2 − I2+ P1 − I1− P2 − I2− P1 + λ − b1 In − In , P1 = 1, P2 = , W W a1
one has: I1+ In− − In+ I1− λI1+ − I2+ − λI1− − I2− + un + (b1 P1 + c1 P2 ) = In − In W W W I + P0 − I2+ P0 1 − I2+ P0 1 − I1+ P0 2 + = 1 2 In − In = P0 n . W W Therefore n−1 X a (ΛI + )k In− − In+ (ΛI − )k √ n−1 Pn = P0 n − Pk . W n−1 k=1
¤ Equation (12) is of Volterra type. We need the following standard lemma to deal with it. Consider the Banach space ¶ ¾ ½ µ |un |n1/4 ∞ B := {un }n=1 : sup |Im λ|√n < ∞ e n with the norm
µ kukB := sup n
|un |n1/4 √ e|Im λ| n
¶
(we omit the dependence on λ in the notation for B). Let V be the expression ½ 0, n = 1 (14) (Vu)n := Pn−1 k=1 Vnk uk , n ≥ 2 for any sequence {u}∞ n=1 . Let (15)
ν := sup n>1
n−1 X k=1
|Vnk |e
√ √ |Im λ|( k− n)
³ n ´1/4 k
.
Lemma 3. If ν < ∞, then V is a bounded operator in B, (I − V)−1 exists and kVkB ≤ ν, k(I − V)−1 kB ≤ eν .
10
SERGEY SIMONOV
Proof. By definition of the operator norm we have to check the finiteness of the following:
sup u6=0
kVukB = sup kukB u6=0
|
sup
Pn−1
|n1/4
k=1 Vnk uk √ e|Im λ| n
n>1
1/4
n |n √ sup e|u|Im λ| n
n
sup ≤ sup
n−1 P
n>1 k=1
1/4
k |k √ |Vnk | e|u|Im λ| k
¡ n ¢1/4 k
e|Im λ|(
√ √ k− n)
.
1/4
n |n √ sup e|u|Im λ| n
u6=0
n
n1/4 √
Denoting u en := un eImλ
sup u6=0
kVukB ≤ sup kukB u e6=0
n
sup
, we have: n−1 P
n>1 k=1
|Vnk ||e uk |
¡ n ¢1/4 k
√
e|Im λ|(
√ k− n)
sup |e un | n
≤ sup n>1
n−1 X k=1
|Vnk |
³ n ´1/4 k
√
e|Im λ|(
√ k− n)
,
hence V is bounded. Quite similarly, ¯ ¯ ¯³ ´ n−1 ¯ X X ¯ ¯ n 1/4 |Im λ|(√k−√n) l ¯ ¯ kV kB ≤ sup V e V ...V nk1 k1 k2 kl−1 k ¯ ¯ k n>1 ¯ ¯ k=1 1≤k1 0, Z +∞ 2 2 (26) tα e−βt dt = O(xα+1 e−βx ) as x → +∞, x
! 2
e−ns ds.
18
SERGEY SIMONOV
one has:
ÃZ
Z
−n−3/8
+∞
!
+
e
−ns2
n−3/8
−∞
µ ¶ 1 ds = O as n → ∞. n
3
Let |s| < min{n− 8 ; δ0 } and |µ| < δ0 . Then n|h(s, µ) + s2 | < C1 n|s|3
N1 , |s| < n−3/8 and |µ| < δ0 , 2 then |en(h(s,µ)+s ) − 1| < 2C1 n|s|3 . Hence we arrive at the following (uniform for |µ| < δ0 ) estimate: ¯Z −3/8 ¯ Z −3/8 ¯ n ¯ n 2 2 ¯ ¯ nh(s,µ) −ns2 (e −e )ds¯ ≤ e−ns |en(h(s,µ)+s ) − 1|ds ¯ ¯ −n−3/8 ¯ −n−3/8 µ ¶ Z n−3/8 1 3 −ns2 |s| e ds = O < 2C1 n as n → ∞ n −n−3/8 from (26). 2. The following is an immediate consequence of (25) and (26): for |µ| < δ0 , ¯ÃZ −3/8 Z ¯ ! Z δ0 ¯ ¯ −n δ0 ns2 ¯ ¯ nh(s,µ) + e ds¯ < 2 e− 2 ds ¯ ¯ −δ0 ¯ n−3/8 n−3/8 µ ¶ Z +∞ 2 2 1 − t2 21 and | arg γ(µ)| < ϕ0 . Then |i + γ(µ)s| > 1. Let θ := 31 . By the choice of µ1 we can also ensure that if |µ| < µ1 , then ½ Re a2 (µ) > 12 , Re[a(µ)(ϕ(µ) − 2µ)] > − δ02θ γ(µ) :=
and hence
1 Reh(s, µ) < − (s2 − 2sδ0 θ) 4 for every real s such that |s| > δ0 . One has: ¯·Z −δ0 Z +∞ ¸ ¯ Z +∞ ¯ ¯ n 2 nh(s,µ) ¯ ¯ + e ds¯ < 2 e− 4 (s −2sδ0 θ) ds ¯ −∞ δ0 δ0 µ ¶ Z +∞ ´ ³ n 2 2 n 2 n 2 1 δ θ s δ (2θ−1) − = 2e 4 0 =O as n → ∞ e 4 ds = O e 4 0 n δ0 (1−θ) uniformly with respect to µ. This completes the proof of the lemma. ¤ ¤ As a corollary we have asymptotics of w(n−1) (z) as n → ∞ for fixed z.
20
SERGEY SIMONOV
Corollary 2. r
µ µ ¶¶ n 2 (n − 1)!in−1 n +iz√2n− z2 1 2 √ w (z) = e2 1+O √ as n → ∞ n n 2π uniformly with respect to z in every bounded set in C. (n−1)
Proof. We just need to substitute µ = √z2n into (24) and go through tedious calculation, using that iz 2 ϕ(z) = i + z − + O(z 4 ) as z → 0. 2 ¤ 5. Acknowledgement The author expresses his deep gratitude to Dr. A.V. Kiselev for valuable discussions of the problem and to Prof. S.N. Naboko for his constant attention to this work and for many fruitful discussions of the subject. The author also wishes to thank Prof. V.S. Buslaev for constructive criticism. The work was supported by grants RFBR-06-0100249 and INTAS-05-1000008-7883, and also by Vladimir Deich prize. References [1] M. Abramowitz, I.A. Stegun, Handbook of mathematical functions, Dover, New York, 1964. [2] N.I. Akhiezer, The classical moment problem and some related questions in analysis, Oliver & Boyd, 1965. [3] Yu.M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators. (Russian), Naukova Dumka, Kiev, 1965. [4] M.S. Birman , M.Z. Solomyak, Spectral Theory of self-adjoint operators in Hilbert space, Reidel, 1987. [5] E.A. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. [6] P. Koosis, Introduction to Hp spaces, Cambridge University Press, Cambridge, 1980. [7] E.C. Titchmarsh, Eigenfunction expantions associated with second-order differential equations, 2, Clerandon Press, Oxford, 1958. [8] Z. Benzaid, D.A. Lutz, Asymptotic representation of solutions of perturbed systems of linear difference equations, Studies Appl. Math. 77, 1987. [9] B.M. Brown, M.S.P. Eastham, D.K.R. McCornack, Spectral concentration and rapidly decaying potentials, J. Comput. Appl. Math. 81, 1997. [10] B.M. Brown, M.S.P. Eastham, D.K.R. McCornack, Spectral concentration and perturbed discrete spectra, J. Comput. Appl. Math. 86, 1997. [11] B.M. Brown, S. Naboko, R. Weikard, The inverse resonance problem for Hermite operators(preprint). [12] D.J. Gilbert, D.B. Pearson, On subordinacy and analysis of the spectrum of one dimensional Schr¨ odinger operators, J. Math. Anal. Appl. 128, 1987.
WEYL-TITCHMARSH TYPE FORMULA FOR HERMITE OPERATOR
21
[13] J. Janas, M. Moszy´ nski, Spectral properties of Jacobi matrices by asymptotic analysis, J. Approx. Theory 120, 2003. [14] S. Khan, D.B. Pearson, Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta 65, 1992. [15] M. Plancherel, W. Rotach, Sur ³ les valeurs asymptotiques des polynomes ´ x2
n
x2
d d’Hermite Hn (x) = (−1)n e 2 dx e− 2 (French) Commentarii Math. Heln vetici 1, 1929. [16] L.O. Silva, Uniform Levinson type theorems for discrete linear systems, Oper. Theory Adv. Appl., Birkhauser-Verlag 154, 2004. [17] L.O. Silva, Uniform and smooth Benzaid-Lutz type theorems and applications to Jacobi matrices, Oper. Theory Adv. Appl., Birkhauser-Verlag 174, 2007.
Department of Mathematical Physics, Institute of Physics, St. Petersburg University, Ulianovskaia 1, 198904, St. Petergoff, St. Petersburg, Russia E-mail address: sergey [email protected]
