One is that Z(J) and similar integrals appear in the sum rules that we study here. In Chapter 3 of our thesis, we will address this issue and extend all the sum rules to full generality. This is another reason why we extend the sum rules to full generality.
In Chapter 3, we extend the sum rules to general Jacobi matrices, in particular to non-sequential classes, and prove Theorem 1.1.
Jacobi Matrices, Spectral Measures, and Orthogonal Polynomials
Finally, section 2.5 contains some basic facts about the m-function, the Borel transform of the spectral measure. We have just seen that there is a bijection between matrices and measures, so that bounded Jacobi matrices and probability measures with bounded infinite support are essentially two manifestations of the same object. For future reference, we note that according to (2.5) the leading coefficient of the polynomial Pn is γn = (a1. . an)−1, and thus according to Gram-Schmidt.
We close this section with an estimate of the J−J0 difference rate in the Schatten Ip classes [29].
Bound States
The second result is the Lieb-Thirring inequality for Jacobi matrices [12], which limits the distance of the eigenvalues from the essential spectrum in terms of a certain Ip norm of the difference J−J0.
Convergence of Jacobi Matrices
The induction step for Pn then follows from the same facts, (2.7), and from the convergence ahead, which has just been established. 2.6) then proves the convergence for bn. This sequence will play an important role in proving one inequality in the sum rules in Chapter 3. This property will prove useful in obtaining the opposite inequality in the sum rules in Chapter 3.
In the following, we assume that f is a continuous function on R, so that f is even, f(x) = 0 for|x| ≤2 and f increases monotonically on [2,∞).
The Szeg˝o Integral and Its Siblings — a Semi-Continuous Family
The first question to be answered at this point is the convergence of the above integrals. In the remainder of this section, we will prove a result of Killip-Simon [13], who established that integrals such as Z(J) are only negative entropies. Inequality (2.36) is the basis of the transition from stepwise sum rules to “full size” sum rules (see Chapter 3) and plays an important role in the proof of the main results of Chapter 4.
Since the infimum of continuous functions above is semicontinuous, the second statement of the theorem from the first follows.
In light of Theorem 2.14, the following lemma connects the spectral measure and the behavior of 2-eigenfunctions at the origin (cf. Theorem 4.25). In this mapping, the upper half-plane is mapped to the lower half-plane, and the lower half-plane is mapped to the upper half-plane. Note that K(z) is a harmonic function in D, so it could be rewritten in terms of its limit values using a Poisson kernel.
In Chapter 3 the sum rules in these terms are stated and proven using the properties of the limit in (2.64). The problem with this identity is that the measure µ# within the integral has a sign and can be infinite, as shown in (2.66). To prove the sum rules for Z`±, we will have to calculate the Taylor coefficients of lnf.
In Section 3.1, we prove a technical result on the continuity of the limits of the M-function presented in Section 2. In this section, we will prove a general continuity result on the limits of M-functions satisfying (2.71). Note that (3.1) does not contradict (2.53) because the logarithm destroys the singular part of the measure under consideration, which affects M(reiϕ) but not M(eiϕ).
We will prove Theorem 3.2 using the dominated convergence theorem and standard maximum function techniques. As already noted, Fatou's lemma implies that the lim inf of the left side of (3.8) is bounded from below by the right side, so it is sufficient to prove that.
The Step-by-Step Sum Rules
In this case, stepwise sum rules can be developed and used to simplify the proof of one's rules, as we will simplify the proof of the Killip-SimonP2 rule (our Z2− rule) in the next section. It is only necessary to repeat the proofs in the previous section with d(ϕ) as in the note for (3.4). Step-by-step sum rules were introduced in Killip-Simon, which first considers r < 1 (in our language below), then takes n → ∞, and then, with some technical hurdles, r ↑ 1.
It is possible that this conjecture is only generally true if J −J0 is only assumed to be compact or only assumed to be Hilbert-Schmidt. Ideas in this work would prove this conjecture if one can prove a result of the following form. The conjecture would be provable with the methods in this work (using the stepwise sum rule to remove the first n pieces of J and then replace them with the first n pieces of ˜J) if one had a limit on shape.
If it held with a limit that depended only on kJk, the conjecture would hold in general.
Our goal in this section is to prove the following three theorems, the proof of which is deferred until after all statements. Our proof that Z(J) < ∞ is essentially the same as theirs, but our proof of the sum rules is much easier. In Section 3.5 we will examine (iii), which is the most striking of these results, since its contrapositive gives very general conditions under which the Szeg˝o condition fails.
The Hilbert-Schmidt condition in (i) and (iv) can be replaced by the somewhat weaker condition. Our proof that Z2−(J) +E2(J)≤A2(J) is identical to that in [13], but our proof of the reverse inequality is somewhat efficient. But from (3.27), the existence of limnζ`(n)(J) is exactly the existence of the conditional trace of B`(J), and they are equal. while in the same way. 3.79).
Shohat’s Theorem with an Eigenvalue Estimate
In the case of orthogonal polynomials on the unit circle, Szeg˝'s theorem states that Z < ∞ if and only if κn is bounded if and only if P∞.
Necessary Conditions for Z(J) < ∞
Consider Theorem 3.14 for the weight wp and the corresponding Szeg˝o-type integral instead of Z(J).
Appendix to Chapter 3
First note that at (2.19), the Chebyshev polynomial of the second kind Un(x2) from (2.11) is a multiple of the characteristic polynomial of J0,n;F. Note that if A is a triangular matrix, then (Am)k,j depends only on elements of A with distance at most m−1 from the position (k, j) (in the metric |k1−k2|+ |j1−j2| ). Using this lemma, we will now calculate the Taylor coefficients of ln(M(z)/z) in terms of J.
The content of the first four sections of this chapter follows, often verbatim, the content of [42] although, of course, we change the numbering to suit. We cannot replace the last two terms bycn|dn+1−dn|because we get positive parts of the sums in (4.2). We will be able to pass to some infinite-order perturbations of J by representing them as bounds of finite-order perturbations and using (2.48).
To do this, we will need to monitor the change in Ej± under these perturbations in order to estimate the sum of the eigenvalues in (3.47) (or, more precisely, in (4.22) below). In Section 4.3, we use these tools to prove Theorem 4.1 and related results (including Theorem 1.4). In Section 4.4, we complete the picture described in Chapter 1 by providing sufficient conditions for Z1±(J)<∞.
We will closely follow their presentation and introduce an additional twist that will provide this improvement. This is because in δn (and not in δ0n) the contribution of the positive |bn+1−bn| terms can be canceled by a fall inan.
Control of Change of Eigenvalues under Perturbations
By Theorem 2.2(i), p(j) changes signj−1 times, so Ej+ will increase as we decrease the corresponding ans. This means that decreasing an and an+2 results in decreasing all but the finite number |Ej±|. Before we start troubling the bns, here's another result with the same flavor.
If you decrease bn, it is clear that all Ej+ decrease, but all Ej− also decrease. To ensure that, we have to counteract the unwanted movement of Ej± by reducing an's.
Sufficient Conditions for Z(J) < ∞
A natural question here is what happens if we allow errors more general than just O(n−1−ε), but still small compared to the leading term of the perturbation. As for the Z(J) = ∞ result in Theorem 4.16, it certainly holds for such errors, as can be seen from the arguments in the proof. One can easily see that we need strong hypotheses at 2α =|β|, the boundary of the “Szeg˝o” region.
Since Theorem 3.14 does not distinguish between the unbound state and E0(J)<∞, we can extend the above result to this case, but we need to restrict it only to δ-minorized perturbations an (eg, decreasing an by en ↓ 0 ). We cannot object to this, but let us consider that in the case J = J0+Hilbert-Schmidt (i.e. J −J0 ∈ I2), which we will consider here, these two definitions coincide. As in the previous section, the main tool for dealing with class perturbations will be the following inequality, which is obtained from (3.48) just as we obtained (4.22) from (3.47) (with the same ˜Jn).
When exciting the ans as in Section 4.2, we must be careful with negative bound states. This problem can be overcome if the contribution of Ej−(J) to this sum is finite. But before we can apply this idea to deal with certain tracking-class perturbations, as described in Section 4.3, we must first find something, bn, to be perturbed.
To prove the next result, we will return to the methods of Section 4.1. i) Let {an} be finitely strictly monotone and an−an−1. ii) Let {bn} be finitely strictly monotone and bn−bn−1. We call the matrix with these new an, bn again J. Now we consider the same ˜Jn as in the proof of Theorem 4.1. Results related to the density of the absolutely continuous part of the spectral measurement and the asymptotics of solutions of difference (or differential) equations, under the assumption of finite variation of the potential, go back to Weidmann [40, 41].
Denisov, On the coexistence of absolutely continuous and single continuous components of the spectral measure for some Sturm-Liouville operators with square summable potentials, J.
