The Resolvent of the Nelson Hamiltonian Improves Positivity

Abstract

We give a new proof that the resolvent of the renormalised Nelson Hamiltonian at fixed total momentum P improves positivity in the (momentum) Fock-representation, for every P. The argument is based on an explicit representation of the renormalised operator and its domain using interior boundary conditions, which allows us to avoid the intermediate steps of regularisation and renormalisation used in other proofs of this result.

Appendix A: Technical Lemmas

Here we reprove the key Lemmas of [12, 20] for the special case of the (massive or massless) Nelson model at fixed momentum.

Lemma A.1

The family of operators G_λ has the following properties:

• a) For every λ > 0, the operator G_λ is bounded;

• b) ${ran} G_{\lambda } \subset D({L_{P}^{s}})$ for any 0 ≤ s < 1/4, and for all λ₀ > 0

  $$ \sup_{\lambda\geq \lambda_{0}} \|(L_{P}+\lambda)^{s} G_{\lambda} \|_{\mathscr{B}(\mathscr{H}_{P})} <\infty; $$

• c) G_λ maps D(dΓ(ω)^1/2) to itself and there exists C > 0 so that for all λ > 0 and ψ ∈ D(dΓ(ω)^1/2)

  $$ \|\mathrm{d} {\Gamma}(\omega)^{1/2} G_{\lambda} \psi\|_{\mathscr{H}_{P}} \leq C \lambda^{-1/4} \|\mathrm{d} {\Gamma}(\omega)^{1/2} \psi\|_{\mathscr{H}_{P}}; $$

• d) There exists λ₀ so that for all λ > λ₀ the operator 1 − G_λ is boundedly invertible on ℋ_P and D(dΓ(ω)^1/2), and

  $$ \sup_{\lambda>\lambda_{0}}(\|(1-G_{\lambda})^{-1}\|_{\mathscr{B}(\mathscr{H}_{P})} + \|(1-G_{\lambda})^{-1}\|_{\mathscr{B}(D(\mathrm{d} {\Gamma}(\omega)^{1/2})} )<\infty. $$

Proof

For a) and b) it is sufficient to prove that

$$ -a(v)(L_{P}+\lambda)^{s-1} $$

(56)

defines a bounded operator on ℋ_P, uniformly in λ. To prove this, we insert a factor of $\sqrt {\omega (\xi )/\omega (\eta )}$ and its inverse, and then use that ab ≤ (a² + b²)/2 as well as the symmetry in ξ,η, to obtain for n ≥ 1

$$ \begin{array}{@{}rcl@{}} &&\|a(v)(L_{P}+\lambda)^{s-1}\psi\ensuremath{^{(n)}}\|^{2}_{\mathscr{H}_{P}\ensuremath{^{(n-1)}}} \\ &&=n \int\limits_{\mathbb{R}^{3(n-1)}} \mathrm{d} Q \int\limits_{\mathbb{R}^{3}} \mathrm{d} \xi \int\limits_{\mathbb{R}^{3}} \mathrm{d} \eta \frac{v(\eta)\psi\ensuremath{^{(n)}}(Q,\xi) \omega(\xi)^{1/2}}{(L_{P}(Q,\eta)+\lambda)^{1-s}\omega(\eta)^{1/2}} \frac{v(\xi) \overline{\psi}\ensuremath{^{(n)}}(Q,\eta)\omega(\eta)^{1/2}}{(L_{P}(Q,\xi)+\lambda)^{1-s}\omega(\xi)^{1/2}}\\ &&\leq n \int\limits_{\mathbb{R}^{3(n-1)}} \mathrm{d} Q \int\limits_{\mathbb{R}^{3}} \mathrm{d} \xi \frac{\omega(\xi) |\psi\ensuremath{^{(n)}}(Q,\xi)|^{2}}{L_{P}(Q,\xi)+\lambda} \int\limits_{\mathbb{R}^{3}} \mathrm{d} \eta \frac{|v(\eta)|^{2}}{(L_{P}(Q,\eta)+\lambda)^{1-2s}\omega(\eta)}. \end{array} $$

(57)

By the Hardy-Littlewood rearrangement inequality we have for s < 1/4

$$ \begin{array}{@{}rcl@{}} {\int}_{\mathbb{R}^{3}} \frac{|v(\eta)|^{2} \mathrm{d} \eta}{(L_{P}(Q,\eta)+\lambda)^{1-2s} \omega(\eta)} &\leq& g^{2} {\int}_{\mathbb{R}^{3}} \frac{\mathrm{d} \eta }{(\eta^{2} + m^{2})((P-\eta-{\sum}_{j=1}^{n-1}k_{j})^{2}+\lambda)^{1-2s}} \\ & \leq& g^{2} {\int}_{\mathbb{R}^{3}} \frac{\mathrm{d} \tau }{|\tau|^{2} (\tau^{2}+ \lambda)^{1-2s}}, \end{array} $$

(58)

which is uniformly bounded for λ ≥ λ₀. Together with (57) and the symmetry of ψ(n) this gives

$$ \begin{array}{@{}rcl@{}} \|a(v)(L_{P}+\lambda)^{s-1}\psi\ensuremath{^{(n)}}\|^{2}_{\mathscr{H}_{P}\ensuremath{^{(n-1)}}} &\leq& n C\int\limits_{\mathbb{R}^{3(n-1)}} \mathrm{d} Q \int\limits_{\mathbb{R}^{3}} \mathrm{d} \xi \frac{\omega(\xi) |\psi\ensuremath{^{(n)}}(Q,\xi)|^{2}}{L_{P}(Q,\xi)+\lambda} \\ & =& C \int\limits_{\mathbb{R}^{3n}} \mathrm{d} K \frac{ \sum\limits_{j=1}^{n} \omega(k_{j}) |\psi\ensuremath{^{(n)}}(K)|^{2}}{L_{P}(K)+\lambda} \\ &\leq& C \| \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}}^{2}. \end{array} $$

(59)

To prove A.1, we proceed as in (57) to obtain (denoting ${\Omega }(K)={\sum }_{j=1}^{n} \omega (k_{j})$)

$$ \begin{array}{@{}rcl@{}} &&\|a(v)(L_{P}+\lambda)^{-1}\mathrm{d} {\Gamma}(\omega)^{1/2}\psi\ensuremath{^{(n)}}\|^{2}_{\mathscr{H}_{P}\ensuremath{^{(n-1)}}} \\ & &\leq 2n \int\limits_{\mathbb{R}^{3(n-1)}} \mathrm{d} Q \int\limits_{\mathbb{R}^{3}} \mathrm{d} \xi \omega(\xi) |\psi\ensuremath{^{(n)}}(Q,\xi)|^{2} \int\limits_{\mathbb{R}^{3}} \mathrm{d} \eta \frac{\Omega(Q, \eta)|v(\eta)|^{2} }{(L_{P}(Q,\eta)+\lambda)^{2}\omega(\eta)}. \end{array} $$

(60)

By scaling and rearrangement, we have

$$ {\int}_{\mathbb{R}^{3}} \frac{|v(\eta)|^{2} \mathrm{d} \eta}{(L_{P}(Q,\eta)+\lambda)^{2}} \leq \lambda^{-1}g^{2}{\int}_{\mathbb{R}^{3}} \frac{\mathrm{d} \tau}{|\tau| (\tau^{2} +1)^{2}}, $$

(61)

and

$$ {\int}_{\mathbb{R}^{3}} \frac{|v(\eta)|^{2} \mathrm{d} \eta}{(L_{P}(Q,\eta)+\lambda)^{2}\omega(\eta)} \leq ({\Omega}(Q)+\lambda)^{-3/2} g^{2}{\int}_{\mathbb{R}^{3}} \frac{\mathrm{d} \tau}{|\tau|^{2} (\tau^{2} +1)^{2}}. $$

(62)

Using symmetry as in (59) this implies

$$ \begin{array}{@{}rcl@{}} \|a(v)(L_{P}+\lambda)^{-1}\mathrm{d} {\Gamma}(\omega)^{1/2}\psi\ensuremath{^{(n)}}\|^{2}_{\mathscr{H}_{P}\ensuremath{^{(n-1)}}} \leq C \lambda^{-1/2} \|\mathrm{d} {\Gamma}(\omega)^{1/2} \psi\ensuremath{^{(n)}}\|^{2}_{\mathscr{H}_{P}\ensuremath{^{(n)}}}, \end{array} $$

(63)

and proves c) by taking adjoints.

To prove d) observe that b) and c) imply that

$$ \|G_{\lambda}\|_{\mathscr{B}(\mathscr{H}_{P})} + \|G_{\lambda}\|_{\mathscr{B}(D(\mathrm{d} {\Gamma}(\omega)^{1/2}))} \leq C \lambda^{-s} $$

(64)

for s < 1/4. Thus for large enough λ the inverse of 1 − G_λ in both spaces exists and is given by the Neumann series, whose norm is bounded by $(1-C\lambda _{0}^{-s})^{-1}$. □

Lemma A.2

For any ε > 0 there exists C > 0 such that for all λ > 0, n ∈ ℕ₀ and K ∈ ℝ³ⁿ

$$ |T_{\mathrm{d},\lambda}(K)| \leq C (L_{P}(K) + \lambda)^{\varepsilon}. $$

Proof

We treat only the case n > 0, the case n = 0 is similar but simpler. We have (with ${\Omega }(K)={\sum }_{j=1}^{n} \omega (k_{j})$)

$$ \begin{array}{@{}rcl@{}} |T_{\mathrm{d},\lambda}(K)| &= & \left|{\int}_{\mathbb{R}^{3}} |v(\xi)|^{2} \frac{(P-\sum\limits_{j=1}^{n}k_{j} - \xi)^{2} - \xi^{2} + {\Omega}(K) + \lambda}{(\xi^{2} +\omega(\xi))(L_{P}(K, \xi)+ \lambda)} \mathrm{d} \xi\right| \end{array} $$

$$ \begin{array}{@{}rcl@{}} &\leq & {\int}_{\mathbb{R}^{3}} |v(\xi)|^{2} \frac{(P-\sum\limits_{j=1}^{n}k_{j})^{2} + 2 |\xi||P-\sum\limits_{j=1}^{n}k_{j}|}{(\xi^{2} +\omega(\xi))(L_{P}(K, \xi)+ \lambda)}\mathrm{d} \xi \end{array} $$

(65)

$$ \begin{array}{@{}rcl@{}} &&+ {\int}_{\mathbb{R}^{3}} |v(\xi)|^{2} \frac{\Omega(K) + \lambda}{(\xi^{2} +\omega(\xi))(L_{P}(K, \xi)+ \lambda)} \mathrm{d} \xi . \end{array} $$

(66)

To simplify the notation, we set $p:=P-{\sum }_{j=1}^{n}k_{j}$. The first term in (65) and the term (66) are bounded by almost identical arguments, so we only give the details for one of them.

Using that (ξ² + ω(ξ)) ≥ ξ^{2 − 4ε}ω(ξ)^2ε ≥|ξ|^{2 − 2ε} (for 1 ≥ ε > 0) in (66), to avoid a singularity in ξ = 0, and then scaling out $\sqrt {\Omega (K)+\lambda }$, we obtain by rearrangement

$$ \begin{array}{@{}rcl@{}} (66) &\leq & ({\Omega}(K) + \lambda)^{\varepsilon} g^{2}\int\limits_{\mathbb{R}^{3}} \frac{\mathrm{d} \xi}{\xi^{3-2\varepsilon}\Big(\Big(\frac{p}{\sqrt{\Omega(K)+\lambda}}-\xi\Big)^{2} +1\Big)} \\ & \leq & ({\Omega}(K) + \lambda)^{\varepsilon} g^{2}{\int}_{\mathbb{R}^{3}} \frac{\mathrm{d} \xi}{\xi^{3-2\varepsilon}(\xi^{2} +1)}. \end{array} $$

(67)

For the second term in (65) we have

$$ \begin{array}{@{}rcl@{}} {\int}_{\mathbb{R}^{3}} |v(\xi)|^{2} \frac{2 |\xi||p|}{(\xi^{2} +\omega(\xi))((p-\xi)^{2} +{\Omega}(k) +\omega(\xi)+ \lambda)} \mathrm{d} \xi\\ \leq g^{2} {\int}_{\mathbb{R}^{3}} \frac{2 |p|}{\xi^{2}((p-\xi)^{2} +\lambda)} \mathrm{d} \xi. \end{array} $$

(68)

Scaling by |p|≠ 0 then yields

$$ {\int}_{\mathbb{R}^{3}} \frac{2 |p|}{\xi^{2}((p-\xi)^{2} + \lambda)} \mathrm{d} \xi \leq {\int}_{\mathbb{R}^{3}}\frac{2}{\xi^{2}(\frac{p}{|p|}-\xi)^{2}}\mathrm{d} \xi=C, $$

(69)

where C is independent of p since the last integral is invariant by rotations. Combining these bounds proves the claim. □

Lemma A.3 (cf. [19, Lem.3.8])

There is C > 0 so that the inequality

$$ \| T_{{\mathrm{od}}, \lambda} \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}} \leq C \| \mathrm{d} {\Gamma}(\omega)^{1/2} \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}} $$

holds for all λ > 0 and n ∈ ℕ.

Proof

We may write

$$ \begin{array}{@{}rcl@{}} T_{{\mathrm{od}}, \lambda} \psi\ensuremath{^{(n)}}(K)= -\sum\limits_{j=1}^{n} {\int}_{\mathbb{R}^{3}}\frac{v(k_{j})\omega(\xi)^{1/2}\psi\ensuremath{^{(n)}}(\hat K_{j}, \xi)}{(L_{P}(K, \xi) + \lambda)^{1/2} \omega(k_{j})^{1/2}} \frac{v(\xi)\omega(k_{j})^{1/2}\mathrm{d} \xi}{(L_{P}(K, \xi) + \lambda)^{1/2} \omega(\xi)^{1/2}}. \end{array} $$

(70)

By the Cauchy-Schwarz inequality we obtain

$$ \begin{array}{@{}rcl@{}} |T_{{\mathrm{od}}, \lambda} \psi\ensuremath{^{(n)}}(K)|^{2} &\leq &\left( \sum\limits_{j=1}^{n} {\int}_{\mathbb{R}^{3}}\frac{|v(k_{j})|^{2} \omega(\xi)|\psi\ensuremath{^{(n)}}(\hat K_{j}, \xi)|^{2} }{(L_{P}(K, \xi) + \lambda) \omega(k_{j})}\mathrm{d} \xi \right) \end{array} $$

(71)

$$ \begin{array}{@{}rcl@{}} & &\times \left( \sum\limits_{\ell=1}^{n} {\int}_{\mathbb{R}^{3}}\frac{|v(\eta)|^{2}\omega(k_{\ell})}{(L_{P}(K, \eta) + \lambda) \omega(\eta)}\mathrm{d} \eta\right). \end{array} $$

(72)

The factor (72) is bounded by (writing ${\Omega }(K)={\sum }_{j=1}^{n} \omega (k_{j})$ and $p=P-{\sum }_{j=1}^{n} k_{j}$)

$$ \begin{array}{@{}rcl@{}} && \sum\limits_{\ell=1}^{n} {\int}_{\mathbb{R}^{3}}\frac{|v(\eta)|^{2}\omega(k_{\ell})}{(L_{P}(K, \eta) + \lambda) \omega(\eta)}\mathrm{d} \eta \\ &&\leq g^{2} \sum\limits_{\ell=1}^{n} \omega(k_{\ell}) ({\Omega}(K)+\lambda)^{-1/2} {\int}_{\mathbb{R}^{3}}\frac{1}{\Big(\Big(\frac{p}{\sqrt{\Omega(K)+\lambda}} - \eta\Big)^{2} +1\Big) |\eta|^{2}}\mathrm{d} \eta\\ &&\leq g^{2} \sum\limits_{\ell=1}^{n} \omega(k_{\ell}) {\Omega}(K)^{-1/2} {\int}_{\mathbb{R}^{3}}\frac{1}{(\eta^{2} +1) |\eta|^{2}}\mathrm{d} \eta, \end{array} $$

(73)

where in the final step we have used the Hardy-Littlewood inequality on the integral.

We use this to estimate the integral of |T_od,λψ(n)(K)|² over K and obtain

$$ \begin{array}{@{}rcl@{}} && \| T_{{\mathrm{od}}, \lambda} \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}}^{2} \\ & \leq& C \sum\limits_{j, \ell=1}^{n}{\int}_{\mathbb{R}^{3n}} {\int}_{\mathbb{R}^{3}}\frac{ \omega(k_{\ell}) {\Omega}(K)^{-1/2} |v(k_{j})|^{2} \omega(\xi)|\psi\ensuremath{^{(n)}}(\hat K_{j}, \xi)|^{2} }{(L_{P}(K, \xi) + \lambda) \omega(k_{j})} \mathrm{d} \xi \mathrm{d} K. \end{array} $$

(74)

By renaming the variables k_j = η, ξ = k_j in the j-th integral, and using the symmetry of ψ(n), this becomes

$$ \begin{array}{@{}rcl@{}} \| T_{{\mathrm{od}}, \lambda} \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}}^{2} &\leq& \sum\limits_{j\neq \ell=1}^{n}{\int}_{\mathbb{R}^{3n}} {\int}_{\mathbb{R}^{3}}\frac{\omega(k_{\ell}) {\Omega}(\hat K_{j}, \eta)^{-1/2} |v(\eta)|^{2} \omega(k_{j})|\psi\ensuremath{^{(n)}}(K)|^{2} }{(L_{P}(K, \eta) + \lambda) \omega(\eta)} \mathrm{d} \eta \mathrm{d} K \end{array} $$

(75)

$$ \begin{array}{@{}rcl@{}} &&+ \sum\limits_{j=1}^{n}{\int}_{\mathbb{R}^{3n}} {\int}_{\mathbb{R}^{3}}\frac{ {\Omega}(\hat K_{j}, \eta)^{-1/2} |v(\eta)|^{2} \omega(k_{j})|\psi\ensuremath{^{(n)}}(K)|^{2} }{(L_{P}(K, \eta) + \lambda) } \mathrm{d} \eta \mathrm{d} K. \end{array} $$

(76)

The first term is bounded by

$$ \begin{array}{@{}rcl@{}} (75) &\leq & \sum\limits_{j=1}^{n} {\int}_{\mathbb{R}^{3n}} {\int}_{\mathbb{R}^{3}}\frac{\Omega(\hat K_{j})^{1/2} |v(\eta)|^{2} \omega(k_{j})|\psi\ensuremath{^{(n)}}(K)|^{2} }{(L_{P}(K, \eta) + \lambda) \omega(\eta)} \mathrm{d} \eta \mathrm{d} K \\ &\leq & C {\int}_{\mathbb{R}^{3n}} \sum\limits_{j=1}^{n} \omega(k_{j}){\Omega}(K)^{-1/2} {\Omega}(\hat K_{j})^{1/2}|\psi\ensuremath{^{(n)}}(K)|^{2} \mathrm{d} K \\ &\leq & C {\int}_{\mathbb{R}^{3n}} {\Omega}(K)|\psi\ensuremath{^{(n)}}(K)|^{2} \mathrm{d} K, \end{array} $$

(77)

by the same scaling argument as before.

By the same reasoning, the second term satisfies the bound

$$ \begin{array}{@{}rcl@{}} (76) &\leq &\sum\limits_{j=1}^{n}{\int}_{\mathbb{R}^{3n}} {\int}_{\mathbb{R}^{3}}\frac{ |v(\eta)|^{2} \omega(k_{j})|\psi\ensuremath{^{(n)}}(K)|^{2} }{(L_{P}(K, \eta) + \lambda) \omega(\eta)^{1/2} } \mathrm{d} \eta \mathrm{d} K \\ & \leq & C {\int}_{\mathbb{R}^{3n}} {\Omega}(K) |\psi\ensuremath{^{(n)}}(K)|^{2} \mathrm{d} K \end{array} $$

(78)

and this proves the claim. □

Lemma A.4

There is a C > 0 so that the inequality

$$ \| S_{{\mathrm{od}}, \lambda} \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}} \leq C \| \mathrm{d} {\Gamma}(\omega)^{1/2} \psi\ensuremath{^{(n)}} \|_{\mathscr{H}_{P}\ensuremath{^{(n)}}} $$

holds for all λ > 0 and n ∈ ℕ.

Proof

As τ_+,λ(K) ≥ 0, the proof is identical to that of Lemma A.3. □