SOME PROPERTIES OF THE KMS-FUNCTION

Jan A. Van Casteren

Abstract. Let, for j = 0, 1, H_{j} = H_{j}^{∗} ≥ −ω_{j} > −∞, be a self-adjoint operator in the Hilbert spaces H_{j}. Let T : H_{1} → H_{0} be a linear operator with domain in H_{1} and range in H_{0}. Let V_{j}(t) = exp (−tH_{j}), t ≥ 0, be the strongly continuous semigroup generated by H_{j}, j = 0, 1. If the operators

(aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1} and (aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0})

are compact, (Hilbert-Schmidt, Trace class), then so is the operator

∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du.

The result is applicable if T = J H_{1} − H_{0}J, where J : H_{1} → H_{0} is a bounded linear (identification) operator. In this case ∫_{0}^{t_{0}} V_{0}(u)T V_{1}(t_{0} − u)du = V_{0} (t_{0}) J − J V_{1} (t_{0}); i.e. the difference of the semigroups. Some convergence and approximation results are presented as well. For example the operator ∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du is expressed in terms of the operator t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2).

1. Introduction, motivation

In [6] the authors proved a result of the following sort.

Theorem. Let V and W be Kato-Feller potentials on ℝ^{ν} for which W − V belongs to L^{1} (ℝ^{ν}). Then the scattering system (H_{0} ˙+V, H_{0} ˙+W ) is complete, in the sense that, for f ∈ Pac (H_{0} ˙+W ), the limits Ω_{±} (H_{0} ˙+V, H_{0} ˙+W ) f := lim_{t→±∞} e^{it(H_{0} ˙+V)}e^{−it(H_{0} ˙+W)}f exist in L^{2} (ℝ^{ν}).

In order to prove this we write, with H_{0} = −\frac{1}{2}Δ in L^{2} (ℝ^{ν}),

V_{0} (t_{0}) = exp (−t_{0} (H_{0} ˙+V )) , and V_{1} (t_{0}) = exp (−t_{0} (H_{0} ˙+W )) .

1991 Mathematics Subject Classification. 47D06, 47D07. Key words and phrases. compactness of operators, KMS-function, logistic variable, difference of

self-adjoint semigroups, trace class operators, Hilbert-Schmidt operators.

1991 Mathematics Subject Classification. 47D06, 47D07. Key words and phrases. compactness of operators, KMS-function, logistic variable, difference of

self-adjoint semigroups, trace class operators, Hilbert-Schmidt operators.

The author is obliged to the University of Antwerp (UIA) for its material support.

Typeset by AMS-TEX

2 PROPERTIES OF THE KMS-FUNCTION

From our assumption that W −V is an L^{1}-function it follows that the operator D (t_{0}) := V_{0} (t_{0}) − V_{1} (t_{0}) is compact (in fact it is Hilbert-Schmidt, but we only need its compactness). In addition, W − V being an L^{1}-function implies that the operator

t_{0}V_{0} (t_{0}) (W − V) V_{1} (t_{0}) is of trace class.

The latter property is even more important than the compactness of the operator D (t_{0}). The problem left out by this approach is establishing a direct relationship between t_{0}V_{0} (t_{0}) (W − V) V_{1} (t_{0}) and D (t_{0}). Such a link exists. To be specific, put Q (t_{0}) T = 𝔼^{logistic} [V_{0} (iU_{1}t_{0}) T V_{1} (−iU_{1}t_{0})], where U_{1} is a random variable, which is logistically

distributed: ℙ^{logistic} [U_{1} ∈ B] = \frac{π}{2} ∫_{B} \frac{1}{(cosh πτ)^{2}} dτ. Then

t_{0}V_{0} (t_{0}) (W − V) V_{1} (t_{0}) = Q (t_{0}) D (t_{0}) , and, more significantly,

D (t_{0}) = lim_{n→∞} ∑_{j=0}^{n} \binom{n + 1}{j + 1}(−1)^{j}Q (t_{0})^{j} (t_{0}V_{0} (t_{0}) (W − V) V_{1} (t_{0})) .

For this, see e.g. Proposition 4 and Theorem 11 below: the operator H_{0} in §2 may be any self-adjoint operator, which is bounded below. Duhamel’s formula shows that D (t_{0}) (W − V) = D (t_{0}) = V_{0} (t_{0}) − V_{1} (t_{0}).

2. Main results

In the following theorem we present the KMS-function, and some of its properties. Another interesting formula is the following; it is related to the KMS-condition in the theory of von Neumann-algebras: e.g. see Sunder [9, page 63 and following] and the remark prior to Theorem 2.

1. Theorem. (a) Suppose (τ, s) belongs to the strip ℝ × (−\frac{1}{2}, \frac{1}{2}), and let T : H_{1} → H_{0} be a linear operator with the property that the operators V_{0}(t_{0})T and V_{1}(t_{0})T^{∗} are densely defined. Then, in form sense, the following identity is true:

t_{0}V_{0} (−iτt_{0}) V_{0} ((\frac{1}{2} + s) t_{0}) T V_{1} ((\frac{1}{2} − s) t_{0}) V_{1} (iτt_{0})

= \frac{t_{0}}{2} ∫_{−∞}^{∞} \frac{cos πs}{cosh π(τ − σ) − sin πs}V_{0} (t_{0}) V_{0} (−iσt_{0}) T V_{1} (iσt_{0}) dσ

+ \frac{t_{0}}{2} ∫_{−∞}^{∞} \frac{cos πs}{cosh π(τ − σ) + sin πs}V_{0} (−iσt_{0}) T V_{1} (iσt_{0}) V_{1} (t_{0}) dσ.  (1)

(b) The following identity is valid as well:

D(t_{0})T = \frac{t_{0}}{2π} ∫_{−∞}^{∞} log (\frac{cosh πτ + 1}{cosh πτ − 1}) V_{0} (−iτt_{0}) \{V_{0} (t_{0}) T + T V_{1} (t_{0})\} V_{1} (iτt_{0}) dτ(2)

= \frac{t_{0}}{π} ∫_{−∞}^{∞} log |coth (\frac{1}{2}πτ)| V_{0} (−iτt_{0}) \{V_{0} (t_{0}) T + T V_{1} (t_{0})\} V_{1} (iτt_{0}) dτ. (3)

Proof. We outline a proof. This formula follows by virtue of the following observation. The function at the right hand side of formula (1) is harmonic on the strip ℝ × (−\frac{1}{2}, \frac{1}{2}) and it possesses boundary values

\{ \begin{matrix}t_{0}V_{0} (t_{0}) V_{0} (−iτt_{0}) T V_{1} (iτt_{0}) , & for s = \frac{1}{2}, and \\ t_{0}V_{0} (−iτt_{0}) T V_{1} (iτt_{0}) V_{1} (t_{0}) , & for s = −\frac{1}{2}. \end{matrix}  (4)

The left hand side is harmonic on the same strip (in fact it is holomorphic there), and has the same boundary values. The uniqueness part on the existence of solutions to the classical Dirichlet problem on a strip, yields the formula in (1).

(b) The identity in (2) follows upon integrating the previous identity with respect to s (between −\frac{1}{2} and \frac{1}{2}). The equality in (3) is a consequence of a very classical identity:

\frac{cosh πτ + 1}{cosh πτ − 1} = coth^{2} \frac{1}{2}πτ.

Remark 1. We consider the space H_{0} × H_{1} together with V (iτ) = ( \begin{matrix}V_{0}(iτ) & 0 \\ 0 & V_{1}(iτ) \end{matrix} ).

Define the flow A_{τ} on B (H_{0} × H_{1}, H_{0} × H_{1}) by A_{τ} (T) = V (−iτt_{0}) T V (iτt_{0}). Define for (f, g) ∈ H_{0} × H_{1}, and T ∈ B (H_{0} × H_{1}, H_{0} × H_{1}) the function F_{f,g}(τ + is), τ ∈ ℝ, −\frac{1}{2} ≤ s ≤ \frac{1}{2} by

F_{f,g}(τ + is) = t_{0} ⟨A_{τ} (V ((\frac{1}{2} + s) t_{0}) T V ((\frac{1}{2} − s) t_{0})) ( \begin{matrix}f \\ g \end{matrix} ) , ( \begin{matrix}f \\ g \end{matrix} )⟩ .

The function F_{f,g} is K(ubo)-M(artin)-S(chwinger)-admissible for the operators V (t_{0}) and T in the sense that it is continuous on the closed strip \{τ + is : τ ∈ ℝ, −\frac{1}{2} ≤ s ≤ \frac{1}{2}\}, and holomorphic on its interior. Moreover,

F_{f,g} (τ + \frac{1}{2}i) = t_{0} ⟨V (t_{0})A_{τ} (T) ( \begin{matrix}f \\ g \end{matrix} ) , ( \begin{matrix}f \\ g \end{matrix} )⟩;
F_{f,g} (τ − \frac{1}{2}i) = t_{0} ⟨A_{τ} (T)V (t_{0}) ( \begin{matrix}f \\ g \end{matrix} ) , ( \begin{matrix}f \\ g \end{matrix} )⟩ .

If T = T^{∗}, then F_{f,g} (τ − \frac{1}{2}i) = \overline{F_{f,g} (τ + \frac{1}{2}i)}.

Remark 2. The family of operators t ↦ ( \begin{matrix}V_{0}(t) & D(t)T \\ 0 & V_{1}(t) \end{matrix} ) is a strongly continuous

semigroup on the space H_{0} × H_{1}. Implicitly, this fact is used in the proof of equality (21).

2. Theorem. Let the hypotheses be as in Theorem 1. The following equalities are valid:

∫_{0}^{t_{0}} V_{0}(u)T V_{1}(t_{0} − u)du

= \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}H_{0}} (aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1} e^{−iτt_{0}H_{1}}dτ

+ \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}H_{0}} (aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0}) e^{−iτt_{0}H_{1}}dτ

+ i ∫_{−∞}^{∞} \frac{e^{iτt_{0}(aI+H_{0})} − I}{sinh πτ} V_{0} (t_{0}) T (aI + H_{1})^{−1} (e^{−iτt_{0}(aI+H_{1})} − I) dτ

− i ∫_{−∞}^{∞} (e^{iτt_{0}(aI+H_{0})} − I) (aI + H_{0})^{−1} T V_{1} (t_{0}) \frac{e^{−iτt_{0}(aI+H_{1})} − I}{sinh πτ} dτ

+ tanh (\frac{1}{2}t_{0} (aI + H_{0})) ((aI + H_{0})^{−1} T V_{1} (t_{0}) − V_{0} (t_{0}) T (aI + H_{1})^{−1})

− ((aI + H_{0})^{−1} T V_{1} (t_{0}) − V_{0} (t_{0}) T (aI + H_{1})^{−1}) tanh (\frac{1}{2}t_{0} (aI + H_{1}))  (5)

= \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}H_{0}} (aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1} e^{−iτt_{0}H_{1}}dτ

+ \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}H_{0}} (aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0}) e^{−iτt_{0}H_{1}}dτ

+ i ∫_{−∞}^{∞} \frac{e^{iτt_{0}(aI+H_{0})} − I}{sinh πτ} V_{0} (t_{0}) T (aI + H_{1})^{−1} e^{−iτt_{0}(aI+H_{1})}dτ

− i ∫_{−∞}^{∞} e^{iτt_{0}(aI+H_{0})} (aI + H_{0})^{−1} T V_{1} (t_{0}) \frac{e^{−iτt_{0}(aI+H_{1})} − I}{sinh πτ} dτ

+ tanh (\frac{1}{2}t_{0} (aI + H_{0})) (aI + H_{0})^{−1} T V_{1} (t_{0})

+ V_{0} (t_{0}) T (aI + H_{1})^{−1} tanh (\frac{1}{2}t_{0} (aI + H_{1})) .  (6)

Proof. For brevity we write D (t_{0}) T = ∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du, A_{0} = aI + H_{0}, A_{1} = aI + H_{1}. Then, from the KMS-formula (see Theorem 1 (b)) it follows that

D (t_{0}) T = \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}} (V_{0} (t_{0}) T + T V_{1} (t_{0})) e^{−iτt_{0}A_{1}}dτ

= −\frac{1}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}V_{0} (t_{0}) T A_{1}^{−1} \frac{∂}{∂(iτ)} (e^{−iτt_{0}A_{1}} − I) dτ

+ \frac{1}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1} \frac{∂}{∂(iτ)} (e^{iτt_{0}A_{0}} − I) A_{0}^{−1}T V_{1} (t_{0}) e^{−iτt_{0}A_{1}}dτ

(integration by parts)

= \frac{1}{2π} ∫_{−∞}^{∞} \frac{∂}{∂(iτ)} (log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}) V_{0} (t_{0}) T A_{1}^{−1} (e^{−iτt_{0}A_{1}} − I) dτ

− \frac{1}{2π} ∫_{−∞}^{∞} (e^{iτt_{0}A_{0}} − I) A_{0}^{−1}T V_{1} (t_{0}) \frac{∂}{∂(iτ)} (log \frac{cosh πτ + 1}{cosh πτ − 1}e^{−iτt_{0}A_{1}}) dτ

= \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}A_{0}V_{0} (t_{0}) T A_{1}^{−1} (e^{−iτt_{0}A_{1}} − I) dτ

+ i ∫_{−∞}^{∞} \frac{1}{sinh πτ} e^{iτt_{0}A_{0}}V_{0} (t_{0}) T A_{1}^{−1} (e^{−iτt_{0}A_{1}} − I) dτ

PROPERTIES OF THE KMS-FUNCTION 5

+ \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1} (e^{iτt_{0}A_{0}} − I) A_{0}^{−1}T A_{1}V_{1} (t_{0}) e^{−iτt_{0}A_{1}}dτ

− i ∫_{−∞}^{∞} \frac{1}{sinh πτ} (e^{iτt_{0}A_{0}} − I) A_{0}^{−1}T V_{1} (t_{0}) e^{−iτt_{0}A_{1}}dτ

= \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}A_{0}V_{0} (t_{0}) T A_{1}^{−1}e^{−iτt_{0}A_{1}}dτ

− \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}dτ A_{0}V_{0} (t_{0}) T A_{1}^{−1}

+ i ∫_{−∞}^{∞} \frac{1}{sinh πτ} (e^{iτt_{0}A_{0}} − I) V_{0} (t_{0}) T A_{1}^{−1} (e^{−iτt_{0}A_{1}} − I) dτ

+ V_{0} (t_{0}) T A_{1}^{−1} i ∫_{−∞}^{∞} \frac{1}{sinh πτ} (e^{−iτt_{0}A_{1}} − I) dτ

+ \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}A_{0}^{−1}T A_{1}V_{1} (t_{0}) e^{−iτt_{0}A_{1}}dτ

− A_{0}^{−1}T A_{1}V_{1} (t_{0}) \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{−iτt_{0}A_{1}}dτ

− i ∫_{−∞}^{∞} \frac{1}{sinh πτ} (e^{iτt_{0}A_{0}} − I) A_{0}^{−1}T V_{1} (t_{0}) (e^{−iτt_{0}A_{1}} − I) dτ

− i ∫_{−∞}^{∞} \frac{1}{sinh πτ} (e^{iτt_{0}A_{0}} − I) dτA_{0}^{−1}T V_{1} (t_{0}) .  (7)

From (7) together with the equalities:

−i ∫_{−∞}^{∞} \frac{e^{iτt_{0}A_{0}} − I}{sinh πτ} dτ = A_{0} \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}}dτ = tanh (\frac{1}{2}t_{0}A_{0});
i ∫_{−∞}^{∞} \frac{e^{−iτt_{0}A_{1}} − I}{sinh πτ} dτ = A_{1} \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{−iτt_{0}A_{1}}dτ = tanh (\frac{1}{2}t_{0}A_{1}) ,(8)

we obtain D (t_{0}) T

= \frac{t_{0}}{2π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}e^{iτt_{0}A_{0}} (A_{0}V_{0} (t_{0}) T A_{1}^{−1} + A_{0}^{−1}T A_{1}V_{1} (t_{0})) e^{−iτt_{0}A_{1}}dτ

+ i ∫_{−∞}^{∞} \frac{1}{sinh πτ} (e^{iτt_{0}A_{0}} − I) (V_{0} (t_{0}) T A_{1}^{−1} − A_{0}^{−1}T V_{1} (t_{0})) (e^{−iτt_{0}A_{1}} − I) dτ

+ tanh (\frac{1}{2}t_{0} (aI + H_{0})) (A_{0}^{−1}T V_{1} (t_{0}) − V_{0} (t_{0}) T A_{1}^{−1})

− (A_{0}^{−1}T V_{1} (t_{0}) − V_{0} (t_{0}) T A_{1}^{−1}) tanh (\frac{1}{2}t_{0} (aI + H_{1})) .  (9)

This proves equality (5) in Theorem 1. Another appeal to (8) yields equality (6). In the sequel we write D (t_{0}) T = ∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du.

3. Corollary. If the operators

(aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1} and (aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0})

are compact (Hilbert-Schmidt, trace class), then so is the operator D (t_{0}) T. Moreover,

∥∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du∥
≤ t_{0} (∥(aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1}∥ + ∥(aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0})∥)
+ ∥(aI + H_{0})^{−1} T V_{1} (t_{0})∥ + ∥V_{0} (t_{0}) T (aI + H_{1})^{−1}∥ .  (10)

Here ∥·∥ stands respectively for the usual operator norm (in the compact operator case), Hilbert-Schmidt norm (in the Hilbert-Schmidt case), or the trace norm (in the trace class situation).

Proof. This corollary is obtained from Theorem 2 (equality (6)) via the following observations:

\frac{1}{π} ∫_{−∞}^{∞} log \frac{cosh πτ + 1}{cosh πτ − 1}dτ = 2 ∫_{−∞}^{∞} \frac{τ}{sinh πτ} dτ = 1,

in conjunction with the operator equalities

e^{iτt_{0}A_{0}} − I = iτt_{0}A_{0} ∫_{0}^{1} e^{isτt_{0}A_{0}}ds, and I − e^{−iτt_{0}A_{1}} = iτt_{0}A_{1} ∫_{0}^{1} e^{−isτt_{0}A_{1}}ds,

and the inequalities:

∥tanh (\frac{1}{2}t_{0}A_{0})∥ ≤ 1 and ∥tanh (\frac{1}{2}t_{0}A_{1})∥ ≤ 1.

The result is applicable if T = J H_{1} − H_{0}J, where J : H_{1} → H_{0} is a bounded linear (identification) operator. In this case ∫_{0}^{t_{0}} V_{0}(u)T V_{1}(t_{0} − u)du = V_{0} (t_{0}) J − J V_{1} (t_{0}); i.e. the difference of the semigroups. Among other things, in the sequel identity (11) below is used substantially.

4. Proposition. In the presence of the hypotheses of Theorem 1, the following identities are true:

t_{0}V_{0}(t_{0}/2)T V_{1}(t_{0}/2) = 𝔼^{logistic} [V_{0} (it_{0}U_{1}) ∫_{0}^{t_{0}} V_{0}(u)T V_{1}(t_{0} − u)du V_{1} (−it_{0}U_{1})]

= \frac{π}{2} ∫_{−∞}^{∞} \frac{1}{(cosh πτ)^{2}} V_{0}(iτt_{0}) ∫_{0}^{t_{0}} V_{0}(u)T V_{1}(t_{0} − u)duV_{1}(−iτt_{0})dτ.  (11)

Here T is a linear operator from H_{1} to H_{0}, and U_{1} is a logistically distributed random variable:

ℙ^{logistic} [U_{1} ∈ B] = \frac{π}{2} ∫_{B} \frac{1}{(cosh πτ)^{2}} dτ = π^{2} ∫_{0}^{∞} \frac{sinh πτ}{(cosh πτ)^{3}} ∫_{−τ}^{τ} 1_{B}(σ)dσ dτ.

Remark. A not necessarily rigorous proof of Proposition 4 will be given in Proposition 10. The method used here is based in double Stieltjes operator integrals: [1, 2, 3, 4]. Some information on this topic can be found in Yafaev [14], page 225–228 as well.

Put ψ(σ) = \frac{π}{2} \frac{1}{(cosh πσ)^{2}} . Another relevant formula is the following one.

PROPERTIES OF THE KMS-FUNCTION 7

5. Proposition. Under the assumptions of Theorem 1, the following identity is true: t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2) − D (t_{0}) T = ∫_{0}^{1} ∫_{−∞}^{∞} exp (−isσt_{0} (aI + H_{0})) (aI + H_{0})^{−1} T V_{1} (t_{0})

\frac{∂}{∂σ} (σψ(σ)\frac{exp (isσt_{0} (aI + H_{1})) − I}{s} ) dσ ds

− ∫_{0}^{1} ∫_{−∞}^{∞} \frac{∂}{∂σ} (σψ(σ)\frac{I − exp (−isσt_{0} (aI + H_{0}))}{s})

V_{0} (t_{0}) T (aI + H_{1})^{−1} exp (isσt_{0} (aI + H_{1})) dσ ds

+ (I − \frac{1}{2} (aI + H_{0}) (sinh (\frac{1}{2} (aI + H_{0})))^{−1}) (aI + H_{0})^{−1} T V_{1} (t_{0})

− V_{0} (t_{0}) T (aI + H_{1})^{−1} (\frac{1}{2} (aI + H_{1}) (sinh (\frac{1}{2} (aI + H_{1})))^{−1} − I) . (12)

Proof. From equality (24) (with n = 1) in Proposition 10 below we obtain D (t_{0}) T − t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2) = it_{0}𝔼^{logistic} [U_{1} ∫_{0}^{1} V_{0} (isU_{1}t_{0}) (T V_{1}(t_{0}) − V_{0}(t_{0})T) V_{1} (−isU_{1}t_{0}) ds] = ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ)it_{0} exp (−isσt_{0} (aI + H_{0})) T V_{1} (t_{0})

exp (isσt_{0} (aI + H_{1})) dσ ds − ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ)it_{0} exp (−isσt_{0} (aI + H_{0})) V_{0} (t_{0}) T

exp (isσt_{0} (aI + H_{1})) dσ ds = it_{0} ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (−isσt_{0} (aI + H_{0})) T V_{1} (t_{0})

(exp (isσt_{0} (aI + H_{1})) − I) dσ ds − it_{0} ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) (exp (−isσt_{0} (aI + H_{0})) − I) V_{0} (t_{0}) T

exp (isσt_{0} (aI + H_{1})) dσ ds + it_{0} ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (−isσt_{0} (aI + H_{0})) dσ dsT V_{1} (t_{0}) − it_{0}V_{0} (t_{0}) T ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (isσt_{0} (aI + H_{1})) dσ ds

= ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ)\frac{∂}{∂σ} (\frac{− exp (−isσt_{0} (aI + H_{0}))}{s})

(aI + H_{0})^{−1} T V_{1} (t_{0}) (exp (isσt_{0} (aI + H_{1})) − I) dσ ds + ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) (exp (−isσt_{0} (aI + H_{0})) − I) V_{0} (t_{0}) T (aI + H_{1})^{−1}

\frac{∂}{∂σ} (\frac{− exp (isσt_{0} (aI + H_{1}))}{s}) dσ ds

+ it_{0} ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (−isσt_{0} (aI + H_{0})) dσ dsT V_{1} (t_{0})

− it_{0}V_{0} (t_{0}) T ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (isσt_{0} (aI + H_{1})) dσ ds

(integration by parts)

= ∫_{0}^{1} ∫_{−∞}^{∞} exp (−isσt_{0} (aI + H_{0})) (aI + H_{0})^{−1} T V_{1} (t_{0})

\frac{∂}{∂σ} (σψ(σ)\frac{exp (isσt_{0} (aI + H_{1})) − I}{s} ) dσ ds

− ∫_{0}^{1} ∫_{−∞}^{∞} \frac{∂}{∂σ} (σψ(σ)\frac{I − exp (−isσt_{0} (aI + H_{0}))}{s})

V_{0} (t_{0}) T (aI + H_{1})^{−1} exp (isσt_{0} (aI + H_{1})) dσ ds

+ it_{0} ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (−isσt_{0} (aI + H_{0})) dσ dsT V_{1} (t_{0})

− it_{0}V_{0} (t_{0}) T ∫_{0}^{1} ∫_{−∞}^{∞} σψ(σ) exp (isσt_{0} (aI + H_{1})) dσ ds

In what follows the sequence U_{1}, …, U_{n}, … denotes a sequence of independent identically distributed random variables, each distributed according to the logistic law. The operator Q(t_{0})(T) is defined by

Q (t_{0}) T = 𝔼^{logistic} [e^{it_{0}U_{1}H_{0}}T e^{−it_{0}U_{1}H_{1}}] .

From Proposition 4 it follows that

t_{0}V_{0}(t_{0}/2)T V_{1}(t_{0}/2) = Q (t_{0}) [D (t_{0}) T] .

6. Theorem. The following identity is true:

(I − Q (t_{0}))^{n+1} [∫_{0}^{a} V_{0}(s)T V_{1}(a − s)ds]

= it_{0} ∑_{j=0}^{n} \binom{n}{j}(−1)^{j}𝔼^{logistic}[exp (it_{0} (U_{1} + ⋯ + U_{j}) H_{0})

U_{n+1} ∫_{0}^{1} exp (iσt_{0}U_{n+1}H_{0}) (T V_{1}(a) − V_{0}(a)T) exp (−it_{0}U_{n+1}H_{1}) dσ
exp (−it_{0} (U_{1} + ⋯ + U_{j}) H_{1})].  (13)

Proof. For n = 0 the above equality coincides with (24) in Proposition 10. For general n employ induction.

7. Corollary. Let all operators of the form V_{0}(t)T V_{1}(t), t > 0, be compact (Hilbert-Schmidt, trace class). Suppose that, for t > 0,

lim_{{t_{2}>t_{1}>0, \atop t_{2}↓0}} ∥V_{0}(t)T ∫_{t_{1}}^{t_{2}} e^{−as}V_{1}(s)ds∥ = lim_{{t_{2}>t_{1}>0, \atop t_{2}↓0}} ∥∫_{t_{1}}^{t_{2}} e^{−as}V_{0}(s)dsT V_{1}(t)∥ = 0.

Then the operators D(t)T, t > 0, are compact.

A similar statement is valid in the Hilbert-Schmidt (trace class) situation: the Hilbert-Schmidt (trace class) norm replaces the operator norm.

8. Corollary. Suppose that the operator

S(t_{0}) := it_{0} ∫_{−∞}^{∞} τψ(τ) ∫_{0}^{1} e^{ist_{0}τH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−ist_{0}τH_{1}}ds dτ

= it_{0}𝔼^{logistic} [∫_{0}^{U_{1}} e^{ist_{0}H_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−ist_{0}H_{1}}ds]

= it_{0} \frac{1}{2} ∫_{−∞}^{∞} (sgn(τ) − tanh πτ) e^{it_{0}τH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−it_{0}τH_{1}}dτ

is Hilbert-Schmidt. Then, in Hilbert-Schmidt norm, the difference

∫_{0}^{t_{0}} V_{0}(s)T V_{1} (t_{0} − s) ds − ∑_{j=0}^{n} \binom{n + 1}{j + 1}(−1)^{j}Q (t_{0})^{j} (t_{0}V_{0} (\frac{t_{0}}{2} ) T V_{1} (\frac{t_{0}}{2} )) (14)

converges to zero if n tends to ∞.

Proof. From Theorem 6 and Proposition 4 we get

∫_{0}^{t_{0}} V_{0}(s)T V_{1} (t_{0} − s) ds − ∑_{j=0}^{n} \binom{n + 1}{j + 1}(−1)^{j}Q (t_{0})^{j} [t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2)]

= (I − Q (t_{0}))^{n} S (t_{0}) .  (15)

An argument in [13] yields the conclusion in Corollary 8.

9. Theorem. Suppose that the operators

(aI + H_{0})^{−1} T V_{1} (t_{0}) + V_{0} (t_{0}) T (aI + H_{1})^{−1} and  (16)
(aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0}) − (aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1}  (17)

are compact (Hilbert-Schmidt, trace class). Then the operator ∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du is compact (Hilbert-Schmidt, trace class) if and only if V_{0} (\frac{1}{2}t_{0}) T V_{1} (\frac{1}{2}t_{0}) is so. Moreover, if (16) and (17) are satisfied, then the Riemann approximation

\frac{t_{0}}{n} ∑_{j=0}^{n−1} V_{0} (\frac{2j + 1}{2n}t_{0}) T V_{1} (\frac{2n − 2j − 1}{2n}t_{0})

converges with respect to the operator norm (in the compact case), in the Hibert-Schmidt norm (in the Hilbert-Schmidt case), and in the trace norm (in the trace situation).

Remark. From the proof it will follow that (16) and (17) may be relaxed to the following assumption. For τ ∈ ℝ the operator

i ∫_{0}^{τ} e^{iσH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−iσH_{1}}dσ (integration by parts)

= e^{iτH_{0}} ((aI + H_{0})^{−1} T V_{1} (t_{0}) + V_{0} (t_{0}) T (aI + H_{1})^{−1}) e^{−iτH_{1}}

− ((aI + H_{0})^{−1} T V_{1} (t_{0}) + V_{0} (t_{0}) T (aI + H_{1})^{−1})

+ i ∫_{0}^{τ} e^{iσH_{0}} ((aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0}) − (aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1})

e^{−iσH_{1}}dσ  (18)

is compact (Hilbert-Schmidt, trace class), and converges to 0 in the operator norm (or in Hilbert-Schmidt norm, or in trace class norm, as the case may be), if τ tends to 0.

For the proof of Theorem 9 a number of identities are required. They are collected in the following proposition.

10. Proposition. Let the hypotheses be as in Theorem 9. Put

D (t_{0}) T = ∫_{0}^{t_{0}} V_{0}(u)T V_{1} (t_{0} − u) du;  (19)

D_{n} (t_{0}) T = \frac{t_{0}}{n} ∑_{j=0}^{n−1} V_{0} (\frac{2j + 1}{2n}t_{0}) T V_{1} (\frac{2n − 2j − 1}{2n}t_{0}).  (20)

The following equalities are valid:

∑_{j=0}^{n−1} V_{0} (\frac{j}{n}t_{0}) D (\frac{1}{n}t_{0}) T V_{1} (\frac{n − j − 1}{n}t_{0}) = D (t_{0}) T;  (21)

t_{0}V_{0} (\frac{1}{2}t_{0}) T V_{1} (\frac{1}{2}t_{0}) = ∫_{−∞}^{∞} ψ(τ)e^{iτt_{0}H_{0}}D (t_{0}) T e^{−iτt_{0}H_{1}}dτ;  (22)
D_{n} (t_{0}) T = n ∫_{−∞}^{∞} ψ(nτ)e^{iτt_{0}H_{0}}D (t_{0}) T e^{−iτt_{0}H_{1}}dτ;  (23)
e^{iτH_{0}}D (t_{0}) T e^{−iτH_{1}} − D (t_{0}) T = i ∫_{0}^{τ} e^{iσH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−iσH_{1}}dσ. (24)

Proof of Proposition 10. A proof of (21) reads as follows (see Remark 2 prior to Theorem 2 as well):

∑_{j=0}^{n−1} V_{0} (\frac{j}{n}t_{0}) D (\frac{1}{n}t_{0}) V_{1} (\frac{n − j − 1}{n}t_{0}) T

= ∑_{j=0}^{n−1} ∫_{0}^{t_{0}/n} V_{0} (\frac{j}{n}t_{0} + s) T V_{1} (\frac{n − j}{n} t_{0} − s) ds

= ∑_{j=0}^{n−1} ∫_{jt_{0}/n}^{(j+1)t_{0}/n} V_{0} (s) T V_{1} (t_{0} − s) ds = D (t_{0}) T.

An informal proof of (22) based on double Stieltjes operator integrals reads as follows:

\frac{π}{2} ∫_{−∞}^{∞} \frac{1}{(cosh πτ)^{2}} V_{0}(iτt_{0}) ∫_{0}^{t_{0}} V_{0}(u)T V_{1}(t_{0} − u)duV_{1}(−iτt_{0})dτ

= ∫ ∫ \frac{π}{2} ∫_{−∞}^{∞} \begin{matrix}exp & (−iτt_{0}(ξ − η)) \\ (cosh πτ) & 2 \end{matrix}dτ \frac{exp(−t_{0}η) − exp(−t_{0}ξ)}{ξ − η}E_{0}(dξ)T E_{1}(dη)

= ∫ ∫ \frac{\frac{1}{2}t_{0}(ξ − η)}{sinh (\frac{1}{2}(ξ − η)t_{0})} \frac{sinh (\frac{1}{2}t_{0}(ξ − η))}{\frac{1}{2}(ξ − η)} exp (−\frac{1}{2}t_{0}(ξ + η)) E_{0}(dξ)T E_{1}(dη)

= t_{0} ∫ ∫ exp (−\frac{1}{2}t_{0}(ξ + η)) E_{0}(dξ)T E_{1}(dη) = t_{0}V_{0}(t_{0}/2)T V_{1}(t_{0}/2).

A proof based on complex function theory is to found in [13] and in [11]. A proof of (23) proceeds as follows:

D_{n} (t_{0}) T = \frac{t_{0}}{n} ∑_{j=0}^{n−1} V_{0} (\frac{j}{n}t_{0}) V_{0} (\frac{1}{2n}t_{0}) T V_{1} (\frac{1}{2n}t_{0}) V_{1} (\frac{n − j − 1}{n}t_{0})

(employ (22) with t_{0}/n replacing t_{0})

= ∑_{j=0}^{n−1} V_{0} (\frac{j}{n}t_{0}) ∫_{−∞}^{∞} ψ(τ)e^{iτ(t_{0}/n)H_{0}}D (\frac{1}{n}t_{0}) T e^{−iτ(t_{0}/n)H_{1}}dτV_{1} (\frac{n − j − 1}{n}t_{0})

= n ∫_{−∞}^{∞} ψ(nτ)e^{iτt_{0}H_{0}}D (t_{0}) T e^{−iτt_{0}H_{1}}dτ.

(In the ultimate equality we employed equality (21).) This shows equality (23). In order to prove (24) we proceed as follows. The reasoning can be made rigorous upon replacing T by (aI + H_{0})^{−1} T (aI + H_{1})^{−1}. We write

e^{iτH_{0}}D (t_{0}) T e^{−iτH_{1}} − D (t_{0}) T

= i ∫_{0}^{τ} e^{iσH_{0}}D (t_{0}) (H_{0}T − T H_{1}) e^{−iσH_{1}}dσ

= i ∫_{0}^{τ} e^{iσH_{0}} ∫_{0}^{t_{0}} V_{0}(u) (H_{0}T − T H_{1}) V_{1} (t_{0} − u) du e^{−iσH_{1}}dσ

= i ∫_{0}^{τ} e^{iσH_{0}} ∫_{0}^{t_{0}} \{−\frac{∂}{∂u} (V_{0}(u)T V_{1} (t_{0} − u))\} due^{−iσH_{1}}dσ

= i ∫_{0}^{τ} e^{iσH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−iσH_{1}}dσ.

Proof of Theorem 9. The assertion in Theorem 9 follows from the identity:

D_{n} (t_{0}) T − D (t_{0}) T  (25)

= i n ∫_{−∞}^{∞} ψ(nτ) ∫_{0}^{τt_{0}} e^{iσH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−iσH_{1}}dσ dτ (see formula (18))

= n ∫_{−∞}^{∞} ψ(nτ) {e^{iτt_{0}H_{0}} ((aI + H_{0})^{−1} T V_{1} (t_{0}) + V_{0} (t_{0}) T (aI + H_{1})^{−1}) e^{−iτt_{0}H_{1}}

− ((aI + H_{0})^{−1} T V_{1} (t_{0}) + V_{0} (t_{0}) T (aI + H_{1})^{−1})} dτ

+ in ∫_{−∞}^{∞} ψ(nτ) ∫_{0}^{τt_{0}} e^{iσH_{0}}

((aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0}) − (aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1}) e^{−iσH_{1}}dσ dτ.

Our assumptions imply that the operator-valued function

τ ↦ e^{iτt_{0}H_{0}} ((aI + H_{0})^{−1} T V_{1} (t_{0}) + V_{0} (t_{0}) T (aI + H_{1})^{−1}) e^{−iτt_{0}H_{1}}

is continuous for the operator norm, and that the same is true for the function

τ ↦ ∫_{0}^{τt_{0}} e^{iσH_{0}} ((aI + H_{0})^{−1} T (aI + H_{1}) V_{1} (t_{0})
− (aI + H_{0}) V_{0} (t_{0}) T (aI + H_{1})^{−1}) e^{−iσH_{1}}dσ.

Consequently equality (25) yields lim_{n→∞} D_{n} (t_{0}) T = D (t_{0}) T for the operator norm. Another appeal to (25) for n = 1 yields the compactness of D (t_{0}) T. This follows because D_{1} (t_{0}) T = t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2). A similar argument works in the Hilbert-Schmidt and trace class context. The equality in (25) is a consequence of the equalities in Proposition 10:

D_{n} (t_{0}) T − D (t_{0}) T (see (23))
= n ∫_{−∞}^{∞} ψ(nτ)e^{iτt_{0}H_{0}}D (t_{0}) T e^{−iτt_{0}H_{1}}dτ − D (t_{0}) T
= n ∫_{−∞}^{∞} ψ(nτ) (e^{iτt_{0}H_{0}}D (t_{0}) T e^{−iτt_{0}H_{1}} − D (t_{0}) T ) dτ (see (24))
= i n ∫_{−∞}^{∞} ψ(nτ) ∫_{0}^{τt_{0}} e^{iσH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−iσH_{1}}dσ dτ.  (26)

The equality in (25) then follows from (18) and (26).

In the proof of Theorem 12 the following lemma will be used. The symbol ψ̂′(ξ) denotes the first derivative of ψ̂ with respect to ξ. A similar convention is employed for the higher derivatives of ψ̂.

11. Lemma. Put ψ(σ) = \frac{π}{2} \frac{1}{(cosh πσ)^{2}} . Then its Fourier transform ψ̂ is given by

ψ̂(ξ) = \frac{\frac{1}{2}ξ}{sinh \frac{1}{2}ξ}. In addition set C_{n} (ξ) = ξ (1 − (\frac{∂}{∂ξ} )^{2}) ((1 − ψ̂(ξ))^{n+1} \frac{1}{ξ}). Then

the following assertions are valid.

(a) The derivative of C_{n}(ξ) is given by

C′_{n}(ξ) = −(n + 1) (1 − ψ̂(ξ))^{n} ψ̂′(ξ)

+ \frac{4}{ξ^{3}} (n + 1) (1 − ψ̂(ξ))^{n} ψ̂′(ξ) + \frac{4}{ξ^{3}} (1 − ψ̂(ξ))^{n+1}

− 3(n + 1)n (1 − ψ̂(ξ))^{n−1} ψ̂′(ξ)ψ̂′′(ξ)

+ (n + 1) (1 − ψ̂(ξ))^{n} ψ̂′′′(ξ)

+ (n + 1)n(n − 1) (1 − ψ̂(ξ))^{n−2} ψ̂′(ξ)^{3}

+ \frac{1}{ξ^{2}} (n + 1)n (1 − ψ̂(ξ))^{n−1} ψ̂′(ξ)^{2}.

(b) Moreover, there exists a constant C such that, for n ≥ 2,

|C′_{n}(ξ)| ≤ C \frac{n}{2} (1 − ψ̂(ξ))^{\frac{1}{2} n−1} |ψ′(ξ)| + \frac{4}{ξ^{3}} (1 − ψ̂(ξ))^{n+1} .

(c) Consequently, for τ > 0, and n ≥ 2

∫_{0}^{∞} |C′_{n}(ξ)| \frac{1}{τξ + 1}dξ ≤ (C + \frac{2}{3}) ∫_{0}^{∞} (1 − ψ̂(ξ))^{\frac{1}{2} n} \frac{τ}{(τξ + 1)^{2}} dξ.

(d) For τ > 0 the following inequalities are valid:

|∫_{0}^{∞} C_{n}(ξ)\frac{sin τξ}{ξ} dξ| ≤ ∫_{0}^{∞} |C′_{n}(ξ)| |∫_{τξ}^{∞} \frac{sin η}{η} dη| dξ ≤ (\frac{π}{2} + 2) ∫_{0}^{∞} |C′_{n}(ξ)| \frac{1}{1 + τξ} dξ.

Proof. (a) The expression for C′_{n}(ξ) is obtained after a tedious, but straight-forward calculation.

(b) We note that the functions ψ̂ and its first, second and third derivative obey the following inequalities:

0 ≤ −ψ̂′(ξ) ≤ \frac{1}{2}ψ̂(ξ), −\frac{1}{12}ψ̂(ξ) ≤ ψ̂′′(ξ) ≤ \frac{1}{4}ψ̂(ξ), and,
−\frac{7}{20} (−ψ̂′(ξ)) ≤ −ψ̂′′′(ξ) ≤ \frac{1}{4} (−ψ̂′(ξ)) .

So the constants C_{1} = \frac{1}{2}, C_{2} = \frac{1}{4}, and C_{3} = \frac{7}{20} respectively satisfy

|ψ̂′(ξ)| ≤ C_{1}ψ̂(ξ), |ψ̂′′(ξ)| ≤ C_{2}ψ̂(ξ), |ψ̂′′′(ξ)| ≤ C_{3} |ψ̂′(ξ)| .

The existence of these constants is guaranteed by the following arguments. Put f(ξ) =

−\frac{1}{ξ} + \frac{1}{2} coth \frac{1}{2}ξ. Then 0 ≤ f(ξ) ≤ min (\frac{1}{2}, \frac{1}{12}ξ), and −\frac{4}{15} ≤ \frac{6}{ξ^{2}} − \frac{1}{2f(ξ)ξ} ≤ 0. We

also have

−ψ̂′(ξ) = ψ̂(ξ)f(ξ),
ψ̂′′(ξ) = ψ̂(ξ) (2f(ξ)^{2} + 2\frac{f(ξ)}{ξ} − \frac{1}{4}) = ψ̂(ξ) \{\frac{1}{4} − \frac{sinh ξ − ξ}{ξ (cosh ξ − 1)}\}, and, finally,
ψ̂′′′(ξ) = ψ̂′(ξ)\{6f(ξ)^{2} + 12\frac{f(ξ)}{ξ} + \frac{6}{ξ^{2}} − \frac{1}{2ξf(ξ)} − \frac{5}{4}\}

= ψ̂′(ξ) \{\frac{1}{4} − \frac{\frac{1}{2} (cosh 2ξ + 1) + 4 cosh ξ − 5 − 3ξ sinh ξ}{\frac{1}{2}ξ sinh 2ξ − ξ sinh ξ − cosh 2ξ + 4 cosh ξ − 3}\} .

Moreover we have

0 ≤ \frac{sinh ξ − ξ}{ξ (cosh ξ − 1)} ≤ \frac{1}{3}, and
0 ≤ \frac{\frac{1}{2} (cosh 2ξ + 1) + 4 cosh ξ − 5 − 3ξ sinh ξ}{\frac{1}{2}ξ sinh 2ξ − ξ sinh ξ − cosh 2ξ + 4 cosh ξ − 3} ≤ \frac{3}{5}.

Since, in addition,

0 ≤ 1 − ψ(ξ) ≤ min (\frac{1}{24}ξ^{2}, 1), and for 0 < y < 1,

(n + 1) (1 − y)^{\frac{1}{2} n} y ≤ 2, (n + 1) (1 − y)^{\frac{1}{2} n−1} y ≤ 6, and

(n + 1) (n − 1) (1 − y)^{\frac{1}{2} n−1} y^{2} ≤ 16,

the inequality in (b) follows from (a).

(c) This inequality follows from (b) in conjunction with integration by parts. The constant C may be chosen to be

C = 3 + \frac{1}{3} \frac{1}{\sqrt{6}} + 12C_{2} + 3C_{3} + 32C_{1}^{2} + \frac{1}{2}C_{1}.

The choice C_{1} = \frac{1}{2}, C_{2} = \frac{1}{4}, and C_{3} = \frac{7}{20} yields C < 15.5.

(d) The first inequality in (d) is obtained upon integration by parts. The second one is a consequence of

|∫_{ρ}^{∞} \frac{sin η}{η} dη| ≤ (\frac{π}{2} + 2) \frac{1}{1 + ρ}, ρ > 0,

which in turn can also be proved upon employing integration by parts.

12. Theorem. Suppose that the operator T V_{1} (t_{0}) − V_{0} (t_{0}) T is bounded (Hilbert-Schmidt, trace class). Then, in operator norm (Hilbert-Schmidt norm, trace class norm) sense, the difference

∫_{0}^{t_{0}} V_{0}(s)T V_{1} (t_{0} − s) ds − ∑_{j=0}^{n} \binom{n + 1}{j + 1}(−1)^{j}Q (t_{0})^{j} (t_{0}V_{0} (\frac{t_{0}}{2} ) T V_{1} (\frac{t_{0}}{2} ))

converges to zero if n tends to ∞.

Proof. Put χ(τ) = \frac{1}{2} (sgn(τ) − tanh πτ). Then its Fourier transform is given by

χ̂(ξ) = \frac{1}{ξ} (1 − ψ̂(ξ)) .  (27)

First we notice (see Corollary 8):

∫_{−∞}^{∞} τψ(τ) ∫_{0}^{1} e^{ist_{0}τH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−ist_{0}τH_{1}}ds dτ
= ∫_{−∞}^{∞} ψ(τ) ∫_{0}^{τ} e^{ist_{0}H_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−ist_{0}H_{1}}ds dτ

(apply Fubini’s theorem)

= \frac{1}{2} ∫_{−∞}^{∞} (sgn(τ) − tanh πτ) e^{it_{0}τH_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−it_{0}τH_{1}}dτ.  (28)

Put F (τ) = exp (it_{0}τH_{0}) (T V_{1} (t_{0}) − V_{0} (t_{0}) T) exp (−it_{0}τH_{1}). We also write m_{ψ}(B) = ∫_{B} ψ(τ)dτ = ℙ^{logistic} [U_{1} ∈ B], if B is a Borel subset of ℝ. With δ we denote the Dirac measure at 0, and with f ∗ μ(τ) = ∫ f(τ − σ)dμ(σ) the convolution product of the function f and the Borel measure μ. From (27) and (28) in conjunction with (15) we infer

∫_{0}^{t_{0}} V_{0}(s)T V_{1} (t_{0} − s) ds − ∑_{j=0}^{n} \binom{n + 1}{j + 1}(−1)^{j}Q (t_{0})^{j} [t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2)]

= it_{0} (I − Q (t_{0}))^{n} 𝔼^{logistic} [∫_{0}^{U_{1}} e^{ist_{0}H_{0}} (T V_{1} (t_{0}) − V_{0} (t_{0}) T) e^{−ist_{0}H_{1}}ds]

= it_{0} ∑_{j=0}^{n} \binom{n}{j}(−1)^{j}

𝔼^{logistic} [exp (it_{0} (U_{1} + ⋯ + U_{j}) H_{0}) ∫_{0}^{U_{n+1}} F (τ)dτ exp (−it_{0} (U_{1} + ⋯ + U_{j}) H_{1})]

= it_{0} ∫_{−∞}^{∞} [χ ∗ (δ − m_{ψ})^{∗n}] (τ)F (τ)dτ

= \frac{it_{0}}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} ; [χ ∗ (δ − m_{ψ})^{∗n}](ξ)e^{iτξ}dξ F (τ)dτ

= \frac{it_{0}}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} (1 − ψ̂(ξ))^{n+1} \frac{1}{ξ}e^{iτξ}dξ F (τ)dτ

= −\frac{t_{0}}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} (1 − ψ̂(ξ))^{n+1} \frac{1}{ξ} sin(τξ)dξ F (τ)dτ

= −\frac{t_{0}}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} (1 − ψ̂(ξ))^{n+1} \frac{1}{ξ} (1 − (\frac{∂}{∂ξ} )^{2}) \frac{sin(τξ)}{1 + τ^{2}}F (τ)dξ dτ

(integration by parts)

= −\frac{t_{0}}{2π} ∫_{−∞}^{∞} ∫_{−∞}^{∞} [ξ (1 − (\frac{∂}{∂ξ} )^{2}) ((1 − ψ̂(ξ))^{n+1} \frac{1}{ξ})] \frac{sin(τξ)}{ξ}dξ \frac{1}{1 + τ^{2}} F (τ)dτ

= −\frac{t_{0}}{π} ∫_{0}^{∞} ∫_{0}^{∞} C_{n}(ξ)\frac{sin(τξ)}{ξ}dξ \frac{1}{1 + τ^{2}} (F (τ) − F (−τ)) dτ

= −\frac{t_{0}}{π} ∫_{0}^{∞} (∫_{0}^{∞} C_{n}(ξ)\frac{sin(τξ)}{ξ}dξ) \frac{1}{1 + τ^{2}} (F (τ) − F (−τ)) dτ

(integration by parts)

= −\frac{t_{0}}{π} ∫_{0}^{∞} (∫_{0}^{∞} C′_{n}(ξ) ∫_{τξ}^{∞} \frac{sin η}{η} dη dξ) \frac{1}{1 + τ^{2}} (F (τ) − F (−τ)) dτ,  (29)

where, as in Lemma 11, C_{n} (ξ) = ξ (1 − (\frac{∂}{∂ξ} )^{2}) ((1 − ψ̂(ξ))^{n+1} \frac{1}{ξ}). Then by

Lemma 11

∥∫_{0}^{t_{0}} V_{0}(s)T V_{1} (t_{0} − s) ds − ∑_{j=0}^{n} \binom{n + 1}{j + 1}(−1)^{j}Q (t_{0})^{j} [t_{0}V_{0} (t_{0}/2) T V_{1} (t_{0}/2)]∥

≤ \frac{t_{0}}{π} ∫_{0}^{∞} |∫_{0}^{∞} C′_{n}(ξ) ∫_{τξ}^{∞} \frac{sin η}{η} dη dξ| \frac{1}{1 + τ^{2}} ∥(F (τ) − F (−τ))∥ dτ

≤ \frac{t_{0}}{π} C_{0} ∫_{0}^{∞} ∫_{0}^{∞} (1 − ψ̂(ξ))^{\frac{1}{2} n} \frac{τ}{(1 + τξ)^{2}} dξ \frac{1}{1 + τ^{2}} ∥(F (τ) − F (−τ))∥ dτ,  (30)

where C_{0} = (\frac{π}{2} + 2) (C + \frac{2}{3}) < 83.2. By Lebesgue’s dominated convergence theorem,

the final expression in (30) tends to zero, as n tends to ∞. This proves Theorem 12.

References

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2. M.Sh. Birman and M.Z. Solomyak, Double Stieltjes operator integrals II, Problemy Mat. Fyz. 2 (1967), 26-60.

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Department of Mathematics & Computer Science, University of Antwerp (UIA), Universiteitsplein 1, 2610 Antwerp, Belgium

Telephone: +32 3 820 24 02 (department), +32 3 239 77 33 (home); Fax: +32 3 820 24 21 (department). E-mail address: vcaster@uia.ua.ac.be