{"id":"4d5d3ce1-1834-4000-882f-8ec1eec1e166","arxiv_id":"1908.04514","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Fourier transform interchanges the Schrödinger representation with a dual representation, yielding dual equivalence bimodules and a conditional correspondence between self-dual and anti-self-dual solitons.","lead":"This mathematics paper constructs a dual version of Rieffel's Heisenberg modules by applying the Fourier transform to the Schrödinger representation, and studies how solitons on noncommutative tori behave under this transform. The paper is useful for readers who want the Fourier-transformed picture of Gabor analysis and noncommutative torus modules, though the main applications rest on a strong assumption.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.4 rests on the unproved and imprecisely stated intertwining relation (25); without a proof or explicit verification for the standard Heisenberg-module examples, the self-dual/anti-self-dual soliton claim is only conditional.","rationale":"The reader's weakest-assumption analysis identifies equation (25) as the load-bearing input, and the full text confirms this: without (25), the proof of Proposition 3.3 cannot turn a self-dual eigenvector into an anti-self-dual eigenvector, and Theorem 3.4 inherits that failure. My closer reading adds two specifications. First, (25) is not merely unproved; it is also under-specified, because 'up to a scalar' must mean the same scalar in both components for the holomorphic/anti-holomorphic conversion to work. Second, the natural connection on S(R) can in fact be chosen so that the literal signed relation holds, so the concern is not that the statement is obviously false, but that the paper does not establish it for the lattice modules it actually uses. The rest of the paper, including the bimodule construction in Theorem 2.5, is standard and appears sound, so the appropriate verdict remains conditional: the soliton conclusions are valid if (25) is supplied, but the paper as written does not supply it. Since the reader already issued a conditional verdict, no change is needed.","tokens_in":15965,"tokens_out":16359,"duration_ms":163904,"concrete_test":"Take Γ=R and Λ=Z^2. Let E=S(R) with the left A(Λ,σ)-action and right action (20), and define ∇1ξ=2πix·ξ, ∇2ξ=dξ/dx. Let E^∘=S(Rhat) with the Fourier-conjugated connections ∇′j=F∇jF^{-1}. Compute F∘∇1 and F∘∇2 explicitly and check whether F∘∇1=∇′2∘F and F∘∇2=∇′1∘F, with the same scalar in both identities. Then repeat the check for the standard lattice module over A(Λ,σ) using the T^2-derivations. If either identity fails with a common scalar, Theorem 3.4, Theorem 3.10, and Proposition 3.13 lose their support as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 introduces the module connections only by assumption. Equation (25), F∘∇i ≡ ∇_{i+1}∘F (mod 2) for i=1,2, is exactly the input that converts a ∇-eigenvector of ψ into a ∇-eigenvector of ψhat in Proposition 3.3 and Theorem 3.4. This is not derived from Theorem 2.5, and no definition of ∇1,∇2 on S(Γ) and S(Γhat) is given. Moreover, the symbol '≡' is not made precise: the proof of Proposition 3.3 needs the equalities F∘∇1 = ∇2∘F and F∘∇2 = ∇1∘F with the same scalar for i=1,2. If F∘∇1 = c1∇2F and F∘∇2 = c2∇1F with c1≠c2, then F(∇1+i∇2)ψ is not a scalar multiple of (∇1−i∇2)Fψ, and the self-dual/anti-self-dual transfer fails. For the standard Heisenberg module over the noncommutative torus, one can choose explicit connections such as ∇1=2πix and ∇2=d/dx and Fourier-conjugated dual connections so that (25) holds, but this verification is absent from the paper. Since Theorem 3.4, Theorem 3.10, and Proposition 3.13 all rely on (25), the advertised soliton result is as written conditional on a nontrivial geometric fact that the paper neither proves nor localizes in the literature.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Fourier transform as an intertwiner between the Schrödinger representation π of the twisted group algebra L^1(Γ̂×Γ,σ) and a dual representation π∘∘ι on L^2(Γ̂). It proves several representation-theoretic identities (Theorem 1.5, Propositions 1.9, 1.13) and uses them to construct a Fourier-dual Heisenberg equivalence bimodule (Theorems 2.3 and 2.5). In the final section, the paper claims that, under a commutation relation between Fourier transform and covariant derivatives (Eq. (25)), a self-dual noncommutative soliton ψ maps to an anti-self-dual soliton ψ̂, that the relevant generalized eigenvector conditions are preserved (Theorem 3.10), and that the Connes–Chern number changes sign (Proposition 3.13). The advertised soliton results, however, rest on Eq. (25), which is assumed without proof or an explicit verification in the standard examples.","tokens_in":16312,"tokens_out":8022,"duration_ms":80597,"significance":"If the main results hold, the paper provides a clean account of the Fourier-transformed Schrödinger representation and a useful dictionary between Gabor frames over a lattice Λ and the dual lattice Λ∘. The explicit proof of Theorem 1.5 and the Moyal-type identities in Section 1 are genuine strengths, as are the careful statements of the associativity conditions for the dual equivalence bimodule. The potential significance of Section 3 is high: a Fourier duality for noncommutative solitons would connect Gabor analysis with noncommutative geometry and reproduce known soliton phenomena in a more conceptual way. However, the central soliton conclusions are conditional on an unproved commutation relation, Eq. (25), which is introduced immediately before the applications. That relation is load-bearing for Propositions 3.3, Theorems 3.4, 3.9, 3.10, and Proposition 3.13. Until it is either proved for the standard Heisenberg modules or explicitly identified as an additional hypothesis that is checked in examples, the paper's advertised claim about Fourier transforms of solitons is not established.","major_comments":[{"comment":"Equation (25), F∘∇i ≡ ∇_{i+1}∘F (mod 2) for i=1,2, is the central assumption of the soliton section, but it is neither proved nor derived from the bimodule structure of Section 2. The symbol ≡ is not defined: Proposition 3.3 and Theorem 3.4 require equalities of the form F∘∇1 = ∇2∘F and F∘∇2 = ∇1∘F with the same scalar factors for the complex structure to transfer. If instead F∘∇1 = c1∇2∘F and F∘∇2 = c2∇1∘F with c1 ≠ c2, then the displayed computation in Proposition 3.3 is invalid and the self-dual/anti-self-dual correspondence can fail. Moreover, the connections ∇1 and ∇2 on S(Γ) and S(Γ̂) are never defined, so the reader cannot check (25) or the Leibniz compatibility conditions (26)–(27). Since Propositions 3.3, Theorem 3.4, Proposition 3.8, Theorem 3.9, Theorem 3.10, and Proposition 3.13 all use (25), the advertised soliton results are conditional on a nontrivial geometric fact that must be stated as an explicit hypothesis of those results and verified for the standard Heisenberg modules over noncommutative tori or the Moyal plane.","section":"Section 3, Theorem 3.4"},{"comment":"The proof of Theorem 3.4 is not a complete derivation. The displayed chain F(A⟨ψ,ψ⟩·∇ψ) = A∘⟨ψhat,ψhat⟩·(F∘∇ψ) requires that ∇ψ lie in the Schwartz space S(Γ) and that Proposition 2.1 extend from ψ to ∇ψ; this is not stated. More importantly, the notation ∇ changes between the anti-holomorphic connection used in the hypothesis and the holomorphic connection on the Fourier side; without an explicit index-by-index application of (25), the equality of the resulting operators cannot be checked. The proof should spell out which of ∇1 and ∇2 appears at each step and should state the scalar factors in (25) that make the two equalities hold with the same sign.","section":"Section 3, Theorem 3.4"},{"comment":"Theorem 3.10 is presented as a proof that ψ is a generalized eigenvector for ∇ if and only if ψhat is a generalized eigenvector for ∇, but the proof passes from F(∇ψ) to ∇(ψhat) by invoking the same unproved commutation relation (25) without quantifying the scalars. The map J is defined, but the key identity J(⟨ψ,ψ⟩B) = ⟨ψhat,ψhat⟩B∘ is asserted rather than proved from the definitions (22)–(23). As written, this theorem is largely a restatement of the assumptions encoded in (25), not an independent result.","section":"Section 3, Theorem 3.10"},{"comment":"Proposition 3.13 derives the sign change of the Connes–Chern number from the computation F12ψhat = ∇1∇2Fψ − ∇2∇1Fψ = −F(F12ψ). This computation again uses (25) with equal scalar factors, which is not established. If the scalar constants in the two parts of (25) differ, the conclusion c1(pψhat) = −c1(pψ) can fail. The displayed formula also contains a typographical error, `−∇2∇2`, which should read `−∇2∇1`; after that correction the argument is still conditional on the unproved commutation relation.","section":"Section 3, Proposition 3.13"}],"minor_comments":[{"comment":"The last sentence of Theorem 2.5 says that S(Γ) is an equivalence bimodule between A∘∞ and B∘∞, but the first sentence of the theorem and the actions (14)–(15) and (22)–(23) are defined on S(Γ̂); the domain should be S(Γ̂), not S(Γ).","section":"Section 2, Theorem 2.5"},{"comment":"Equation (24) is written as Tr_A(∂_i K) = 0, j = 1,2; the index on the left is i while the tag says j. This should be ∂_j for j=1,2.","section":"Section 3, Eq. (24)"},{"comment":"The notation ci(pψ) in the proposition statement should be c1(pψ), matching the definition in the same section.","section":"Section 3, Proposition 3.13"},{"comment":"The symbols ∇, ∇̅, ∂, and ∂̅ are used with inconsistent descriptions of holomorphic versus anti-holomorphic connections, and Proposition 3.1 mixes them in a way that makes it hard to determine which equation (28) or (29) is being transferred in Theorem 3.4. A consistent convention should be fixed throughout the section.","section":"Section 3 (soliton equations)"},{"comment":"The notation π∗(λ∘) is used for the right action of the dual lattice before it is defined; the authors should explicitly state that π∗(g) = π(g)^* or give the representation of Λ∘ used to define B and B∘.","section":"Section 2, Eq. (20)"},{"comment":"The paper uses S(Γ) and S(Γ̂) as Schwartz spaces for a general locally compact abelian group Γ, but such spaces are not defined for arbitrary LCA groups. The text should specify the class of groups (for example, elementary groups in the sense of Bruhat, or Γ = R^n × Z^q) or define the relevant rapidly decreasing functions explicitly.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear X,\n\nMy read of 1908.04514: the first two sections are a competent, mostly expository account of the Fourier transform as an intertwiner for the Schrödinger representation and of the resulting dual Heisenberg module. Theorem 2.5 is the genuinely new part: it states the Fourier-transformed version of Rieffel's equivalence bimodule, and the proof, reducing associativity to Rieffel via F, works. Theorem 1.5 is standard time-frequency analysis; the proof is fine but it is not a new result.\n\nThe soft spot is Section 3. Equation (25) is introduced without proof, without defining the covariant derivatives on both modules, and without checking any standard example. It is also written with a vague '≡ up to a scalar.' Proposition 3.3 needs the specific equalities F∘∇1=∇2∘F and F∘∇2=∇1∘F; if the scalars differ, the self-dual/anti-self-dual transfer does not follow. The proofs of Theorems 3.4, 3.9, 3.10, and Proposition 3.13 all depend on (25). So the advertised claim that the Fourier transform of a self-dual soliton is an anti-self-dual soliton is, as written, conditional on a nontrivial geometric fact that the paper neither proves nor verifies in the standard Γ=R case. I would not call this circular—the author is not assuming the conclusion—but the assumption is exactly the mechanism that produces the result, and presenting it in the form of unconditional theorems is an overstatement. There are also typos (Theorem 2.5 says S(Γ) where it should be S(Γ̂); Proposition 3.13 has ∇2∇2) and several proofs are sketched.\n\nThe right fix is straightforward: either prove (25) for the standard Heisenberg module, or state all soliton claims as conditional and provide a concrete example where (25) holds. The first option is clearly better and probably plausible for Γ=R with ∇1=2πix, ∇2=d/dx and their Fourier conjugates.\n\nWho should read this: people working at the interface of noncommutative tori and Gabor analysis. With the Section 3 gap addressed, it would be a useful note. As is, I would send it to a serious referee—the gap is fixable and the bimodule part has value—but with a clear instruction to either prove (25) or make the conditional status explicit. I would not build on the soliton results until that is sorted.","headline":"A correct but modest dual-bimodule construction in Sections 1-2, with the advertised soliton duality in Section 3 resting on an unproved and imprecisely stated commutation assumption that must be justified or explicitly declared conditional.","tokens_in":16820,"tokens_out":4568,"would_cite":false,"duration_ms":49775,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["58B20","35C08","58B16","58J05","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Fourier transform turns self-dual solitons on noncommutative tori into anti-self-dual solitons, via a dual Schrödinger representation and a dual Heisenberg equivalence bimodule.","keywords":["Schrödinger representation","Heisenberg modules","noncommutative tori","Gabor frames","solitons","Fourier transform","Morita equivalence","self-duality equation"],"falsifier":"Compute both sides of equation (25) for an explicit tight Gabor atom, for instance $\\psi(t)=e^{-\\pi t^2}$ on $\\Gamma=\\mathbb R$ with the standard noncommutative torus connection, using the paper's formulas for $\\nabla_1,\\nabla_2$; if $F(\\nabla_i\\psi)$ and $\\nabla_{i+1}(F\\psi)$ are not equal as specified, then Proposition 3.3 and Theorem 3.4 collapse.","tokens_in":15735,"feed_emoji":"🔄","tokens_out":11182,"duration_ms":96525,"temperature":0.7,"pith_summary":"The paper asks whether the Fourier transform preserves the class of solitons over noncommutative tori and answers with a duality: a self-dual soliton maps to an anti-self-dual soliton. The key device is a dual Schrödinger representation, obtained by swapping the order of translation and modulation, which the Fourier transform intertwines with the original representation. Using this intertwining, the paper constructs a dual Heisenberg module: the Schwartz space on the Pontryagin dual group is an equivalence bimodule pairing the dual noncommutative torus algebras. It then shows that a tight Gabor frame solving the self-duality equation has Fourier image that is again a tight Gabor frame solving the anti-self-duality equation, and that the topological charge changes sign. The transfer runs through an intertwining identity for the covariant derivatives that the paper assumes.","feed_headline":"Fourier transform turns self-dual solitons into anti-self-dual ones","feed_subtitle":"Fourier images of self-dual Gabor frames solve the anti-self-duality equation, and topological charge flips sign.","key_machinery":"The load-bearing object is the dual Schrödinger representation $\\pi^\\circ\\iota$, defined on the phase space $G=\\hat\\Gamma\\times\\Gamma$ by $\\pi^\\circ\\iota(\\gamma,t)=T_\\gamma M_{-t}$ (translation first, then modulation), together with the Fourier intertwining identity $F\\,\\pi(\\gamma,t)=\\pi^\\circ\\iota(\\gamma,t)\\,F$, which is Theorem 1.5. This identity makes the Fourier transform a unitary equivalence between the original and dual projective representations, and it converts the Heisenberg module $S(\\Gamma)$ into the dual module $S(\\hat\\Gamma)$: $F({}_A\\langle\\xi,\\eta\\rangle\\cdot\\psi)={}_{A^\\circ}\\langle\\hat\\xi,\\hat\\eta\\rangle\\cdot\\hat\\psi$, with the right action transported by $\\pi^*\\mapsto(\\pi^\\circ\\iota)^*$. On the soliton side, the same intertwining is assumed to hold for the covariant derivatives as equation (25), $F\\circ\\nabla_i\\equiv\\nabla_{i+1}\\circ F \\pmod 2$ for $i=1,2$; this is what flips $\\bar\\nabla=\\nabla_1+i\\nabla_2$ into $\\nabla=\\nabla_1-i\\nabla_2$ up to a scalar $-i$, thereby reversing duality and reversing the curvature $F_{12}\\hat\\psi=-F(F_{12}\\psi)$, which drives the sign change of the topological charge.","core_discovery":"On the paper's own terms, the central claim is Theorem 3.4: for a lattice $\\Lambda$ in $G=\\hat\\Gamma\\times\\Gamma$, let $A_\\infty$, $A_\\infty^\\circ$, $B_\\infty$, and $B_\\infty^\\circ$ be the smooth noncommutative torus algebras defined by the Schrödinger representation $\\pi$ and its dual $\\pi^\\circ\\iota$. If $\\psi\\in S(\\Gamma)$ is a tight Gabor frame, so that $\\langle\\psi,\\psi\\rangle_B=1_B$, and the associated projection $p_\\psi={}_A\\langle\\psi,\\psi\\rangle$ satisfies the self-duality equation $(\\bar\\partial p_\\psi)p_\\psi=0$, then $\\hat\\psi=F(\\psi)$ satisfies $\\langle\\hat\\psi,\\hat\\psi\\rangle_{B^\\circ}=1_{B^\\circ}$ and the anti-self-duality equation $(\\partial p_{\\hat\\psi})p_{\\hat\\psi}=0$. Thus the Fourier transform sends the soliton condition 'self-dual' to its mirror 'anti-self-dual'. In the continuous case (Proposition 3.3) the same transfer holds with the right algebra $\\mathbb C$, and Theorem 3.10 says that being a generalized eigenvector of the anti-holomorphic connection is preserved under Fourier transform. The topological charge of the projection is reversed, $c_1(p_\\psi)=-c_1(p_{\\hat\\psi})$ (Proposition 3.13).","pith_inferences":["An extension the paper leaves implicit: one could verify equation (25) for $\\Gamma=\\mathbb R^n$ with arbitrary symplectic lattices, turning the assumed intertwining into a theorem and widening the result to higher-rank noncommutative tori.","Because the proof only needs the intertwining of $\\nabla$ and $\\bar\\nabla$, the same mechanism likely works for any pair of dual representations linked by a unitary that shifts the connection index, not only the Fourier transform; this suggests a broader 'duality of solitons' under other unitary transforms.","The sign flip in the topological charge may have a physical reading in noncommutative sigma models: the Fourier transform acts like a charge-conjugation symmetry, exchanging instantons and anti-instantons. This is not stated in the paper and would need a separate analysis of the action functional.","One could use Theorem 3.10 as a solution-generating technique: starting from a known generalized eigenvector of $\\nabla$, its Fourier transform solves the same equation in the dual module, yielding new examples of Gabor frames and projections."],"forward_implications":["If Theorem 3.4 is right, every tight Gabor frame satisfying the self-duality equation produces, by Fourier transform, a tight Gabor frame satisfying the anti-self-duality equation, so soliton solutions come in Fourier-paired dual pairs.","The sign reversal $c_1(p_\\psi)=-c_1(p_{\\hat\\psi})$ means the Fourier transform is a duality that exchanges positive and negative topological charge, so a soliton and its Fourier image live in opposite instanton sectors.","The dual Heisenberg module $S(\\hat\\Gamma)$ gives a second Morita equivalence between the dual noncommutative torus algebras, so projective modules and their gauge-theoretic data have Fourier-dual counterparts.","In Gabor analysis, the Riesz-sequence property for the dual lattice is preserved under Fourier transform (Proposition 3.7), so frame duality and Fourier duality are compatible.","When $\\Gamma\\cong\\hat\\Gamma$ and $\\psi=\\hat\\psi$ is a self-dual tight frame, Corollary 3.11 forces $\\langle\\nabla_j\\psi,\\psi\\rangle=0$, a concrete constraint on any Fourier-invariant soliton."],"supporting_citations":[{"why":"Supplies the standard construction of Heisenberg modules as equivalence bimodules between noncommutative torus algebras; Theorem 2.4 is quoted from it and Theorem 2.5 follows its method.","marker":"[21]"},{"why":"Supplies the general criterion that a projection $p_\\psi$ solves a self-duality equation iff $\\psi$ is a generalized eigenvector of the corresponding connection, used in Propositions 3.1 and 3.3.","marker":"[6]"},{"why":"Supplies the characterization of solitons as generalized eigenvectors and the gauge-action framework used in Theorem 3.10 and the trace computations.","marker":"[16]"},{"why":"Connects noncommutative torus projections with Gabor frames, motivating the tight-frame formulation in Theorems 3.4 and 3.9.","marker":"[17]"},{"why":"Shows projective modules over noncommutative tori are multi-window Gabor frames; provides the abstract framework for the module-theoretic part in Section 2.","marker":"[18]"},{"why":"Provides the standard treatment of the Heisenberg group and Schrödinger representation that Example 1.6 and the intertwining Theorem 1.5 rely on.","marker":"[12]"},{"why":"Source of the duality principle for Gabor frames (Theorem 3.5), used to characterize invertibility of $\\langle\\psi,\\psi\\rangle_B$ and Riesz sequences.","marker":"[14]"}],"fun_headline_variants":["Fourier flips soliton duality","Fourier turns self-dual solitons anti-self-dual","Fourier transform swaps topological charge","Self-dual solitons become anti-self-dual via Fourier"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is equation (25), assumed without proof, that the Fourier transform intertwines the two covariant derivatives up to a swap and a scalar, $F\\circ\\nabla_i \\equiv \\nabla_{i+1}\\circ F \\pmod{2}$; if this intertwining fails for the standard Heisenberg modules over noncommutative tori, the self-dual-to-anti-self-dual soliton transfer does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Fourier flips soliton duality","Fourier turns self-dual solitons anti-self-dual","Fourier transform swaps topological charge","Self-dual solitons become anti-self-dual via Fourier"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000414,"raw_usage":{"total_tokens":2151,"prompt_tokens":970,"completion_tokens":1181,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":586,"completion_tokens_details":{"reasoning_tokens":1116}},"tokens_in":586,"tokens_out":1181,"duration_ms":9560,"temperature":1.0,"reasoning_tokens":1116,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:40:26.843276+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of equation (25) for an explicit tight Gabor atom, for instance $\\psi(t)=e^{-\\pi t^2}$ on $\\Gamma=\\mathbb R$ with the standard noncommutative torus connection, using the paper's formulas for $\\nabla_1,\\nabla_2$; if $F(\\nabla_i\\psi)$ and $\\nabla_{i+1}(F\\psi)$ are not equal as specified, then Proposition 3.3 and Theorem 3.4 collapse.","supporting_citations":[{"cited_title":"Rieffel, Projective modules over higher-dimensional non-commutat ive tori , Canad","cited_arxiv_id":null,"evidence_quote":"Supplies the standard construction of Heisenberg modules as equivalence bimodules between noncommutative torus algebras; Theorem 2.4 is quoted from it and Theorem 2.5 follows its method."},{"cited_title":"Landi, and F Luef, Sigma-model solitons on noncommutative spaces , Lett","cited_arxiv_id":null,"evidence_quote":"Supplies the general criterion that a projection $p_\\psi$ solves a self-duality equation iff $\\psi$ is a generalized eigenvector of the corresponding connection, used in Propositions 3.1 and 3.3."},{"cited_title":"2, DOI 10.1142/S023902571850008X","cited_arxiv_id":null,"evidence_quote":"Supplies the characterization of solitons as generalized eigenvectors and the gauge-action framework used in Theorem 3.10 and the trace computations."},{"cited_title":"Luef, Projections in noncommutative tori and Gabor frames , Proc","cited_arxiv_id":null,"evidence_quote":"Connects noncommutative torus projections with Gabor frames, motivating the tight-frame formulation in Theorems 3.4 and 3.9."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows projective modules over noncommutative tori are multi-window Gabor frames; provides the abstract framework for the module-theoretic part in Section 2."},{"cited_title":"Howe, On the role of the Heisenberg group in harmonic analysis , Bull","cited_arxiv_id":null,"evidence_quote":"Provides the standard treatment of the Heisenberg group and Schrödinger representation that Example 1.6 and the intertwining Theorem 1.5 rely on."},{"cited_title":"Jakobsen and J","cited_arxiv_id":null,"evidence_quote":"Source of the duality principle for Gabor frames (Theorem 3.5), used to characterize invertibility of $\\langle\\psi,\\psi\\rangle_B$ and Riesz sequences."}],"review_version":1}