Abstract
For a system of N bosons in one space dimension with two-body δ-interactions the Hamiltonian can be defined in terms of the usual closed semi-bounded quadratic form. We approximate this Hamiltonian in norm resolvent sense by Schrödinger operators with rescaled two-body potentials, and we estimate the rate of this convergence.
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Acknowledgements
The second author thanks Jacob Schach Møller for stimulating discussions and for the hospitality at Aarhus University. Our work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the Research Training Group 1838: Spectral Theory and Dynamics of Quantum Systems.
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Appendices
Appendix A: Properties of the Green’s Function
This appendix collects facts and estimates on the Green’s function of \(-{\Delta } + z: H^{2}(\mathbb {R}^{d}) \rightarrow L^{2}(\mathbb {R}^{d})\).
For \(d \in \mathbb {N}\) and \(z \in \mathbb {C}\) with Re(z) > 0, let the function \({G^{d}_{z}}: \mathbb {R}^{d} \rightarrow \mathbb {C}\) be defined by
Notice that \( {G^{d}_{z}}\) has a singularity at x = 0 unless d = 1. The proof of the following lemma is left as an exercise to the reader.
Lemma A.1
Let\(d \in \mathbb {N}\)and\(z \in \mathbb {C}\)with Re(z) > 0. Then\({G^{d}_{z}}\)defined by (A.1) has the following properties:
-
(i)
\({G^{d}_{z}} \in L^{1}(\mathbb {R}^{d})\)and\(\|{G^{d}_{z}} \|_{L^{1}}\leq \text {Re}(z)^{-1}\)
-
(ii)
The Fourier transform of\({G^{d}_{z}}\)is given by\(\widehat {{G^{d}_{z}}}(p)=(2\pi )^{-d/2}(p^{2}+z)^{-1}\)
-
(iii)
\({G^{d}_{z}}\)is the Green’s function of −Δ + z, i.e.\((-{\Delta } + z)^{-1}f = {G^{d}_{z}} \ast f\)holds for all\(f \in L^{2}(\mathbb {R}^{d})\)
-
(iv)
\({G^{d}_{z}} \in L^{2}(\mathbb {R}^{d})\)if and only if d ∈ {1, 2, 3}
-
(v)
\({G^{d}_{z}}\)is spherically symmetric, i.e. \({G^{d}_{z}}\)only depends on\(\left |x\right |\), and for\(z \in (0,\infty )\)it is positive and strictly monotonically decreasing both in\(\left |x\right |\)and z
-
(vi)
Let\(d_{1},d_{2}\in \mathbb {N}\)with d1 + d2 = d and let\(x=(x_{1},x_{2}) \in \mathbb {R}^{d_{1}} \times \mathbb {R}^{d_{2}}\).If x1 ≠ 0 or d1 = 1, then\({G^{d}_{z}}(x_{1},\cdot ) \in L^{1}(\mathbb {R}^{d_{2}})\)and the Fourier transform is
$$ \widehat{{G^{d}_{z}}}(x_{1},p_{2})=(2\pi)^{-d_{2}/2} G^{d_{1}}_{z+{p_{2}^{2}}}(x_{1}). $$(A.2)In particular, we have that
$$ \int\limits_{\mathbb{R}^{d_{2}}}^{} \mathrm{d}x_{2} {G^{d}_{z}}(x_{1},x_{2}) = G_{z}^{d_{1}}(x_{1}). $$(A.3)
The following lemma is one of our main tools for estimating differences of integral operators that depend on \({G^{d}_{z}}\):
Lemma A.2
Let\(d \in \mathbb {N}\)and\(z \in \mathbb {C}\)with Re(z) > 0. Then, for all\(x, \tilde {x} \in \mathbb {R}^{d}\)and Q ≥ 0, it holds that
Similarly, if d = d1 + d2for some\(d_{1},d_{2} \in \mathbb {N}\), then for all\( x_{1}, y_{1} \in \mathbb {R}^{d_{1}}\)and all\(x_{2} \in \mathbb {R}^{d_{2}}\),
Proof
To prove (A.5) we may assume, without loss of generality, that \(\left |x_{1}\right | \leq \left |y_{1}\right |\). Then,
The proof of (A.4) is very similar. □
Lemma A.3
Let d ≥ 2, s ∈ (0, 1) and\(z \in \mathbb {C}\)with Re(z) > 0. Then there exists a constant C = C(s, z) > 0 such that
Proof
Since \({G_{z}^{d}}(\cdot ,0) \in L^{1}(\mathbb {R}^{d-1})\) by Lemma A.1 (vi), the left side of (A.6) is bounded uniformly in \(y \in \mathbb {R}^{d-1}\). So it remains to prove (A.6) for |y|≤ 1, and to this end it suffices to show that there exists a constant C = C(z) > 0 such that
Using the integral representation (A.1) for \({G_{z}^{d}}\) and making the substitution \(x/\sqrt {4t}\to x\), we find
where u = Re(z). By applications of triangle inequality and the fundamental theorem of calculus, respectively,
Since
the desired estimate follows. □
Appendix B: The Konno-Kuroda Formula
Let \({\mathscr{H}}\) and \(\widetilde {{\mathscr{H}}}\) be arbitrary (complex) Hilbert spaces, let H0 ≥ 0 be a self-adjoint operator in \({\mathscr{H}}\) and let \(A:D(A)\subset {\mathscr{H}}\to \widetilde {{\mathscr{H}}}\) be densely defined and closed with \(D(A)\supset D(H_{0})\). Let \(J\in {\mathscr{L}}(\widetilde {{\mathscr{H}}})\) be a self-adjoint isometry and let B = JA.
Suppose that BD(H0) ⊂ D(A∗) and that A∗A and A∗B are H0-bounded with relative bound less than one. Then
is self-adjoint on D(H0) by Kato-Rellich. Operators of the more general form H = H0 − gA∗B with \(g\in \mathbb {R}\) can also be written in the form (B.1) by absorbing |g|1/2 in A and sgn(g) in J. For z ∈ ρ(H0), let \(\phi (z):D(A^{*})\subset \widetilde {{\mathscr{H}}}\to \widetilde {{\mathscr{H}}}\) be defined by
where R0(z) := (H0 + z)− 1. Note that \(D(A^{*})\subset \widetilde {{\mathscr{H}}}\) is dense because A is closed.
Theorem B.1
Let the above hypotheses be satisfied and let z ∈ ρ(H0). Then ϕ(z) is a bounded operator. The operator 1 − ϕ(z) is invertible if and only if z ∈ ρ(H0) ∩ ρ(H), and then
Remark 1
Note that (1 − ϕ(z))− 1 leaves D(A∗) invariant. This follows from (B.3) and from the assumption on B.
Proof
Step 1.A(H0 + c)− 1/2 is bounded for all c > 0, and A(H + c)− 1/2 is bounded for c > 0 large enough.
By assumption, the operator H0 − A∗A is bounded from below. This implies that
for all ψ ∈ D(H0), all c > 0, and some constant C. Since D(H0) is dense in \(D(H_{0}^{1/2})\) and since A is closed, this bound extends to all \(\psi \in D(H_{0}^{1/2})\) by an approximation argument. In particular, \(D(A)\supset D(H_{0}^{1/2})\). The second statement of Step 1 follows from the first and from the fact that H and H0 have equivalent form norms, which implies that \(D(H^{1/2}) = D(H_{0}^{1/2})\).
Step 2. If z ∈ ρ(H0), then ϕ(z) is a bounded operator, and if z ∈ ρ(H), then
is a bounded operator too. This easily follows from Step 1 and from the first resolvent identity.
Step 3. If z ∈ ρ(H0) ∩ ρ(H), then (1 − ϕ(z)) is invertible and 1 + Λ(z) = (1 − ϕ(z))− 1.
Both ϕ(z) and Λ(z) leave D(A∗) invariant and on this subspace, by straightforward computations using the second resolvent identity, (1 − ϕ(z))(1 + Λ(z)) = 1 = (1 + Λ(z))(1 − ϕ(z)).
Step 4. If z ∈ ρ(H0) and 1 − ϕ(z) is invertible, then z ∈ ρ(H), (1 − ϕ(z))− 1 leaves D(A∗) invariant and (B.2) holds.
By Step 3, (1 − ϕ(i))− 1 = 1 + Λ(i), which leaves D(A∗) invariant. Now suppose that z ∈ ρ(H0) and that 1 − ϕ(z) has a bounded inverse. Then
Since RanBR0(i) ⊂ D(A∗), it follows that (1 − ϕ(z))− 1 leaves D(A∗) invariant as well, and that
is well defined. Now it is a matter of straightforward computations to show that (H + z)R(z) = 1 on \({\mathscr{H}}\) and that R(z)(H + z) = 1 on D(H). □
Appendix C: Γ-Convergence
In this section we work in \(L^{2}(\mathbb {R}^{N})\) rather than the subspace \({\mathscr{H}}\) of symmetric wave functions. In fact, the results of this section are easily generalized to N distinct particles with masses m1,…,mN and potentials \(V_{ij}\in L^{1}(\mathbb {R})\) depending on the pair i < j of particles.
Let \(V\in L^{1}(\mathbb {R})\) with V (−r) = V (r), let \(g=\lim _{\varepsilon \to 0}g_{\varepsilon }\), and let \(\alpha =g{\int \limits } V(r)\mathrm {d}{r}\). Let q and qε denote the quadratic forms on \(H^{1}(\mathbb {R}^{N})\) defined by
where \(C\in \mathbb {R}\) and \(\gamma _{ij}:H^{1}(\mathbb {R}^{N})\to L^{2}(\mathbb {R}^{N-1})\) denotes the trace operator with
for \(\psi \in C_{0}^{\infty }(\mathbb {R}^{N})\). It is well known that this operator extends to a bounded operator from \(H^{1}(\mathbb {R}^{N})\) to \(H^{1/2}(\mathbb {R}^{N-1})\). By Lemma C.1, the quadratic forms q and qε are bounded below and closed. More precisely, we may choose C so large, that q ≥ 0 and qε ≥ 0 for all ε > 0.
We are going to prove weak and strong Γ-convergence qε → q as ε → 0. To this end, it is convenient to extend all quadratic forms to \(L^{2}(\mathbb {R}^{N})\) by setting \(q=q_{\varepsilon }=+\infty \) in \(L^{2}(\mathbb {R}^{N})\backslash H^{1}(\mathbb {R}^{N})\). The main ingredients of this section are the inequalities
for \(\psi \in H^{1}(\mathbb {R}^{N})\). They are obtained by applying to \(\varphi (r) = {\int \limits } |\psi (r,x)|^{2}\mathrm {d}{x}\) the elementary Sobolev inequalities
Lemma C.1
For all μ > 0 there exists\(C_{\mu }\in \mathbb {R}\)such that for all\(\psi \in H^{1}(\mathbb {R}^{N})\)and i, j ∈ {1, … , N}, i < j,
Proof
Inequality (C.3) follows from \(\|\gamma _{ij}\psi \| \leq C\|\psi \|_{H^{1}}\) by a scaling argument. To prove (C.4) for (i,j) = (1, 2), we set \(\tilde \psi (r,R,x) := \psi (R-\tfrac {r}{2},R+\tfrac {r}{2},x)\) and write
and we apply (C.1) to the H1-function \(\tilde \psi \). Then we use that \(\|\tilde \psi \| = \|\psi \|\) and \(\|\partial _{r}\tilde \psi \| \leq \|\nabla \psi \|\). □
Lemma C.1 and \(\|V_{\varepsilon }\|_{L^{1}} = \|V\|_{L^{1}}\) imply the following corollary:
Corollary C.2
Under the assumptions of this section, for every a > 0 there exists b > 0 such that for all ε > 0 and all\(\psi \in H^{1}(\mathbb {R}^{N})\),
Theorem C.3
Let\(V\in L^{1}(\mathbb {R})\)and\(\alpha =g{\int \limits } V(r)\mathrm {d}{r}\). Then qε → q in the sense of weak and strong Γ-convergence.
Proof
Due to the fact that all form domains are equal, it suffices to show that, see [5, 9], for all \(\psi \in H^{1}(\mathbb {R}^{N})\),
and for all \(\psi _{\varepsilon },\psi \in L^{2}(\mathbb {R})\),
We begin with the proof of (C.5). If \(\psi \in C_{0}^{\infty }(\mathbb {R}^{N})\), then it is a fairly straightforward application of Lebesgue dominated convergence to show that (C.5) holds. Now let \(\psi \in H^{1}(\mathbb {R}^{N})\) and let (ψn) be a sequence in \(C_{0}^{\infty }(\mathbb {R}^{N})\) with ψn → ψ in H1. Then, on the one hand,
because q is continuous w.r.t. its form norm, which is equivalent to the norm of \(H^{1}(\mathbb {R}^{N})\) by Corollary C.2. On the other hand,
uniformly in ε > 0. This also follows from Corollary C.2, which means that the interaction is H1-bounded uniformly in ε. Due to (C.7) and (C.8), the validity of (C.5) extends from \(C_{0}^{\infty }(\mathbb {R}^{N})\) to \(H^{1}(\mathbb {R}^{N})\).
Now we prove (C.6). Let \(\psi ,\psi _{\varepsilon }\in L^{2}(\mathbb {R}^{N})\) and suppose that \(\psi _{\varepsilon } \rightharpoonup \psi \) in \(L^{2}(\mathbb {R}^{N})\). To prove (C.6), we may assume that \(\liminf _{\varepsilon \to 0} q_{\varepsilon }(\psi _{\varepsilon })<\infty \). We choose a sequence εn → 0 so that \(\liminf _{\varepsilon \to 0} q_{\varepsilon }(\psi _{\varepsilon }) = \lim _{n\to \infty } q_{\varepsilon _{n}}(\psi _{\varepsilon _{n}})\). Then, by Corollary C.2 it follows that \(\psi _{\varepsilon _{n}}\) is bounded in \(H^{1}(\mathbb {R}^{N})\) uniformly in n. Therefore, after passing to a subsequence, we may assume that \(\psi _{\varepsilon _{n}} \rightharpoonup \tilde \psi \) in \(H^{1}(\mathbb {R}^{N})\). Since \(\psi _{\varepsilon _{n}} \rightharpoonup \psi \) in \(L^{2}(\mathbb {R}^{N})\) it follows that \(\psi =\tilde \psi \in H^{1}(\mathbb {R}^{N})\).
By the weak lower semicontinuity of positive quadratic forms we know that
On the right hand side we may replace \(q(\psi _{\varepsilon _{n}})\) by \(q_{\varepsilon _{n}}(\psi _{\varepsilon _{n}})\) if we can show that
uniformly in \(\psi \in H^{1}(\mathbb {R}^{N})\). To prove this, we first assume that V has compact support and we note that
where \(\alpha _{\varepsilon }:= g_{\varepsilon }{\int \limits } V(r)\mathrm {d}{r}\), and αε → α has been used. Applying (C.2) to \(\tilde \psi (r,R,x) := \psi (R-\tfrac {r}{2},R+\tfrac {r}{2},x)\), we see that the contribution of (i,j) = (1, 2) to (C.10) has the bound
Here, we used that \(\|\tilde \psi \|_{H^{1}}^{2} \leq C \|\psi \|_{H^{1}}^{2}\). Since all summands of (C.10) can be estimated in this way, (C.9) is true and the proof of the theorem is complete in the case where V has compact support.
It remains to prove (C.9) in the case of general \(V\in L^{1}(\mathbb {R})\). To this end we set, for \(k\in \mathbb {N}\),
and we define quadratic forms qk and qk,ε like q and qε with α and V replaced by αk and Vk, respectively. The constant C in the defining expressions for q and qε is left unchanged. Since Vk has compact support, we know from the proof above that (C.9) holds for Vk. That is, for each \(k\in \mathbb {N}\),
uniformly in \(\psi \in H^{1}(\mathbb {R}^{N})\). From Lemma C.1 and the fact that \(\|V_{\varepsilon }-V_{k,\varepsilon }\|_{L^{1}}=\|V-V_{k}\|_{L^{1}}\) we see that
uniformly in ε. Choosing first k large, then ε small we see that (C.11) and (C.12) combined prove (C.9). □
In view of Theorem 13.6 in [9], Theorem C.3 has the following corollary:
Corollary C.4
If\(V\in L^{1}(\mathbb {R})\)with V (r) = V (−r) and\(\lim _{\varepsilon \to 0}g_{\varepsilon }=g\), then Hε → H in the strong resolvent sense.
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Griesemer, M., Hofacker, M. & Linden, U. From Short-Range to Contact Interactions in the 1d Bose Gas. Math Phys Anal Geom 23, 19 (2020). https://doi.org/10.1007/s11040-020-09344-4
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DOI: https://doi.org/10.1007/s11040-020-09344-4