{"id":"08e5bfc6-1c72-4951-b3a7-8633a4033c90","arxiv_id":"2505.15100","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves several existence theorems for L2-normalized solutions of nonlinear Dirac equations on noncompact metric graphs with localized nonlinearities.","lead":"This paper proves existence of fixed-mass (L2-normalized) solitary wave solutions for nonlinear Dirac equations on noncompact metric graphs, with the nonlinearity confined to a compact core, across subcritical, critical/supercritical, and negative-sign regimes. It is the first study of normalized solutions for this class of equations on graphs, and it offers a variational method that avoids Fourier transforms.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The c=1 normalization is the load-bearing gap: retaining c in the Section 4 estimate gives a threshold with c^{4-p}, not the c^{(4-p)/2} stated in (1.16), so the general-c theorems are unproved.","rationale":"The reader's weakest assumption identified the c=1 normalization in Section 2.2; my stress test agrees and sharpens it: at general c, the paper's own non-existence estimate produces a different c-power in the threshold, so the stated c-dependence of Theorem 1.3 and the later theorems is not established. This is the most load-bearing concern because the central claim is an existence theorem for the Dirac operator with physical constants m and c, and every quantitative threshold in the paper depends on these parameters. The c=1 variational argument itself appears coherent: the reduction, mountain-pass geometry in Lemma 2.2, the test-function estimate in Lemma 2.3, and the contradiction in Section 4 are internally plausible, and the topological argument in Section 5 (Lemma 5.2) is a genuine contribution. The issue is therefore one of missing parameter handling rather than a fatal internal contradiction at c=1. A concrete rescaling check would settle whether the stated theorems hold for general c; because the paper is conditional on just such a check, I would keep the reader's conditional verdict rather than reject.","tokens_in":29649,"tokens_out":28905,"duration_ms":225392,"concrete_test":"Recompute the estimate in Section 4 with c kept arbitrary: replace every use of ∥u∥² ≥ m∥u∥₂² by ∥u∥² ≥ mc²∥u∥₂² and check whether the resulting smallness condition on a is m^{(4-p)/2} c^{4-p}/(2C_{p,K}) rather than (1.16). Separately, substitute y=cx and w(y)=c^{-1/2}u(y/c) into the c=1 solution and solve for the coefficient and frequency that solve (1.9) on the original graph; if the transformed threshold does not equal (1.16), the general-c theorem needs a new proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.2 fixes c=1 and the entire variational proof runs at c=1. In the non-existence estimate of Section 4, the only role of c is through the spectral inequality ∥u∥² ≥ mc²∥u∥₂²; at c=1 this yields (4.1) with m^{-(4-p)/2}. Repeating the estimates in (4.2) without setting c=1 gives a threshold a < m^{(4-p)/2} c^{4-p}/(2C_{p,K}), not the c^{(4-p)/2} appearing in (1.16). Moreover, the natural change of variables y=cx maps a c=1 solution on the scaled graph cG to a solution on G with a different nonlinearity coefficient; it does not simply restore the constants in (1.16)-(1.17). Since no such scaling argument is supplied, the theorems as stated for general m,c>0 are not consequences of the proofs as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies normalized solutions (with L^2 mass equal to 1) of the nonlinear Dirac equation D u - ω u = a χ_K |u|^{p-2}u on noncompact metric graphs with nonempty compact core, where D is the Dirac operator with Kirchhoff-type vertex conditions. The authors adapt the reduction/perturbation method of Ding, Yu and Zhao [19] to the metric-graph setting, where the Fourier transform is not available. The main results are: for 2<p<4, a dichotomy theorem (Theorem 1.2) and existence of normalized solutions for 0<a<a_0 (Theorem 1.3); existence for all a>0 under a topological assumption on the compact core (Theorem 1.4); for p≥4, existence for a above a threshold (Theorem 1.5); the construction of graphs admitting normalized solutions for 4≤p<6 (Theorem 1.7 and Corollary 1.8); extensions to the negative-coefficient case and to the case where -mc^2 is an eigenvalue (Section 8); and an appendix on non-existence thresholds that depend on m and c.","tokens_in":29830,"tokens_out":8744,"duration_ms":77100,"significance":"If the results are valid, this is the first systematic treatment of normalized solutions of nonlinear Dirac equations on metric graphs, opening a genuinely new direction. The paper gives explicit thresholds, constructs test functions without Fourier transform, and covers several regimes (subcritical, critical/supercritical, negative coefficient, eigenvalue case). The proof is detailed in many places and the constants are explicit. However, the blanket normalization c=1 in Section 2.2, together with the importation of the reduction map from [19] without a proof, makes the general-c theorems not consequences of the written arguments.","major_comments":[{"comment":"The statement 'For the remainder of the paper, we set c=1, as its value does not affect the results' is not justified. The theorem statements and the constants a_0 and a^{*,0} in (1.16) and (1.17) are given for general m,c>0, and Appendix A explicitly discusses the dependence of thresholds on c. The proof, however, uses the c=1 spectral inequality \\|u\\|^2 ≥ m\\|u\\|_2^2 in (4.1)-(4.2). Repeating the non-existence estimate without setting c=1 replaces m by mc^2 and changes the threshold by a power of c. The natural rescaling y=cx does not preserve the normalization ∫_G |u|^2 dx = 1 and also changes the coefficient a, so it is not a free normalization. The general-c theorems are therefore not consequences of the proofs as written; the authors should either carry c through the estimates or supply a correct scaling identity that restores the constants in (1.16)-(1.17).","section":"Section 2.2"},{"comment":"The existence, regularity, and critical-point correspondence of the reduction maps h and h_{r,μ} are imported from [19] (and [15]) without proof. Those references concern the Euclidean setting R^N, whereas the present problem has metric-graph vertex conditions and no Fourier transform. Since Lemma 2.4 and Theorem 3.1 depend on h_{r,μ} being a C^1 map from Y^+ ∩ U_μ into Y with the stated one-to-one correspondence, the paper should either prove the reduction in this setting or state an abstract theorem and verify its hypotheses explicitly.","section":"Section 2.2, reduction map"},{"comment":"In the limit passage of Step 2, the proof asserts that 0 < c_{r_1,μ_1} ≤ (1/2)\\|u_0^+\\|^2 - (1/2)\\|u_0^-\\|^2 - Ψ(u_0) = c_∞. Weak lower semicontinuity gives at most an inequality in one direction; the equality with c_∞ is not established. The dichotomy (mass 1 or mass <1) may suffice for some existence conclusions, but the stated energy identity in Theorem 3.1 is not supported by the written argument. The authors should either justify the equality or remove it from the statement.","section":"Theorem 3.1, Step 2"},{"comment":"The proof invokes 'similar to Lemma 6.1' to obtain c_{s,∞} < (m-s)/2 for all s∈(-m,m), but Lemma 6.1 is stated only for p≥4 and for a above a threshold. In the present setting 2<p<4 and a is arbitrary, so the needed bound is not proved. Since this bound is used to obtain the boundedness of {u_s} and the limiting solution u_{-m}, the missing argument should be supplied.","section":"Proof of Theorem 1.4"}],"minor_comments":[{"comment":"Integrals over G are sometimes written as integrals over K, for example '∫_K |u_s|^2 dx = 1' in the proof of Theorem 1.2 and '∫_K |u_s|^2 dx < 1' in the proof of Theorem 1.4; these should be ∫_G to match the mass constraint.","section":"Statements and proofs of Theorem 1.2 and Theorem 1.4"},{"comment":"There are several typos: 'Thoerem' in the introduction, 'defiened' in Section 1, 'by by' in the definition of Sobolev spaces, and 'relying' in the introduction. Remark 5.3 states '±m are not the eigenvalue' and should read 'not eigenvalues'.","section":"Throughout"},{"comment":"Lemmas 8.5 and 8.6 again state 'we still set c=1 for simplicity', which repeats the normalization issue raised above; the c-dependent theorem statements in Section 8 require a consistent treatment of c.","section":"Section 8"},{"comment":"Lemma A.2 refers to 'Lemma (A.1)' instead of Lemma A.1, and the notation \\|u\\|_{H^1} is used without redefinition in the appendix; a local definition would improve readability.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a new and potentially interesting problem, and the overall strategy is coherent. The main risk is the c=1 normalization: if the authors can either carry c through the estimates or prove a valid scaling identity, the central existence theorems may become correct. The second risk is the unproved reduction map imported from [19]; this should be addressed directly rather than by citation. I would not recommend rejection on novelty grounds, but the general-c theorems are not proved as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — quick read on He–Ji, arXiv:2505.15100.\n\nThe genuinely new thing: this is the first paper to treat prescribed-mass (normalized) solutions of nonlinear Dirac equations on metric graphs, and the first to adapt the Ding–Yu–Zhao reduction/penalization machinery to a graph setting without Fourier transform. The explicit graph test functions in Lemma 2.3 are a real workaround for the absence of Fourier analysis on graphs, and Lemma 5.2 plus the comparison argument are honest graph-theoretic input. If the c=1 proofs are taken at face value, the paper does establish existence for 2<p<4 with small a, gives a topological condition for all a>0, and makes a serious attempt at p≥4 under extra assumptions. The a<0 extensions and the eigenvalue-based Theorem 8.3 are plausible and well-motivated.\n\nSoft spots, in increasing seriousness.\n\nMinor: the reduction map h and h_{r,μ} are imported from [19] rather than re-proved. That is acceptable practice, but in a new setting the hypotheses should have been checked explicitly. I do not read this as fatal.\n\nMore serious: Step 2 of Theorem 3.1 has a gap. The energy of the limit u0 is asserted to equal c∞, but the passage from c_{r_n,μ_n} to c∞ along r_n↗∞, μ_n↘0 needs a monotonicity argument the paper does not supply. Fixable, but not fully written.\n\nLoad-bearing: the c=1 normalization in Section 2.2. The paper says \"its value does not affect the results\" and then runs the entire variational proof at c=1, while the theorem statements, the constants (1.16)–(1.17), and the appendix retain general m,c. No scaling argument is provided. Repeating the Section 4 non-existence estimate without c=1 gives a threshold proportional to m^{(4-p)/2} c^{4-p}, which matches (1.16) as printed — so the stress-test note's specific claim of a c^{(4-p)/2} mismatch misreads the formula — but the proof still never handles c≠1. A rescaling y=cx changes the graph lengths, the L² normalization, and the nonlinearity coefficient simultaneously; the paper needs to show how a c=1 solution on a suitably scaled graph produces a general-c solution and what condition on a results. Without that, the theorems as stated for general m,c are not consequences of the proofs.\n\nBottom line: the c=1 existence results and the graph ideas are worth refereeing. This opens a small but active direction, and the gaps are fixable rather than fatal. A serious referee should ask for a proof of the scaling claim or a restatement of the theorems for c=1 only. I would accept for peer review, but I would not cite the general-c statements until the gap is closed.","headline":"First normalized-solution results for nonlinear Dirac equations on metric graphs, but the general-c theorems rest on an unproved normalization and need a real scaling argument before they are usable as stated.","tokens_in":30357,"tokens_out":6082,"would_cite":false,"duration_ms":48377,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R02","35Q40","35A15","34B45"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes the first existence theory of $L^2$-normalized solutions for nonlinear Dirac equations on noncompact metric graphs: unit-mass spinors for subcritical powers with small nonlinearity strength, for every $a>0$ under a…","keywords":["normalized solutions","nonlinear Dirac equations","metric graphs","localized nonlinearities","variational methods","strongly indefinite functional","perturbation argument","Gagliardo-Nirenberg inequality"],"falsifier":"Repeat the estimate of Lemma 2.3 and of inequality (4.2) with the general operator $D=-ic\\,\\sigma_1\\tfrac{d}{dx}+mc^2\\sigma_3$. If the resulting threshold is not $a_0=m^{(4-p)/2}c^{(4-p)/2}/(2C_{p,K})$ — for instance if an extra power of $c$ appears from $\\|u\\|_2^2\\le (mc^2)^{-1}\\|u\\|^2$ in the $L^p(K)$ estimates — then the theorems as stated hold only at $c=1$. A concrete check: take the half-line test function $\\varphi_b$ of Lemma 2.3 for a single half-line with general $c$ and compute whether $c_\\infty<\\tfrac{mc^2}{2}$ still follows with the same choice $b\\to0$; the calculation in (2.6) would acquire factors of $c$.","tokens_in":29415,"feed_emoji":"📐","tokens_out":11445,"duration_ms":92744,"temperature":0.7,"pith_summary":"The paper asks whether the nonlinear Dirac equation $Du-\\omega u = a\\chi_K |u|^{p-2}u$ on a noncompact metric graph, with the nonlinearity switched on only on the compact core $K$, admits solutions whose $L^2$ mass is fixed in advance (normalized solutions). It proves the first existence results of this kind: for subcritical powers $2<p<4$ and sufficiently small coefficient $a>0$, a normalized solution exists on every such graph; for $p\\ge 4$, normalized or sub-normalized solutions exist for large $a$; and under a tree-core topological assumption the normalized solution exists for every $a>0$. The proof is variational, combining a perturbation/reduction method for the strongly indefinite Dirac energy with a test-function construction that does not use the Fourier transform, which is unavailable on general graphs. If correct, the theorems open the study of mass-constrained relativistic solitons on networks, a setting previously treated only for nonlinear Schrödinger equations.","feed_headline":"Dirac equations on metric graphs admit normalized solutions","feed_subtitle":"A perturbation argument yields unit-mass spinor solutions, covering subcritical and higher powers.","key_machinery":"The argument is carried by the strongly indefinite energy functional $I_\\omega(u)=\\tfrac12\\|u^+\\|^2-\\tfrac12\\|u^-\\|^2-\\tfrac{\\omega}{2}\\|u\\|_2^2-\\Psi(u)$ on the form domain $Y=Y^+\\oplus Y^-$, where the splitting comes from the positive/negative spectral subspaces of $D$. Because $Y$ embeds compactly only into $L^p(K,\\mathbb{C}^2)$, the nonlinearity is localized on $K$. The paper adapts the perturbation method of [19]: a penalization term $H_{r,\\mu}(u)=f_r((T_\\mu u,u)_2)$ with $T_\\mu=I+\\mu|D|$ replaces the hard constraint $\\|u\\|_2=1$, the reduced functional $J_{r,\\mu}$ on $Y^+\\cap U_\\mu$ has mountain-pass geometry, and the minimax value $c_\\infty=\\sup_{r,\\mu}c_{r,\\mu}$ is shown to lie below $\\tfrac{m}{2}$ by a new test function $\\varphi_b=(\\varphi_b^1,0)$ that is identically 1 on $K$ and decays linearly on the half-lines. The threshold $c_\\infty<m/2$ is what forces the limiting solution to have unit mass. A Gagliardo-Nirenberg-Sobolev inequality on graphs (Lemma 2.1) converts energy bounds into mass bounds, and a graph-theoretic unique-continuation lemma (Lemma 5.2) excludes degenerate sub-normalized solutions on tree cores.","core_discovery":"The central claim is that the spectral-gap problem $Du-\\omega u = a\\chi_K|u|^{p-2}u$, $\\int_G|u|^2=1$, with $D=-ic\\,\\sigma_1\\tfrac{d}{dx}+mc^2\\sigma_3$, has nontrivial solutions for suitable $\\omega\\in(-mc^2,mc^2)$. The main route is a dichotomy: for $2<p<4$, either a normalized solution exists, or a family of sub-normalized solutions (mass $<1$) exists at the bottom of the spectral gap; Theorem 1.3 rules out the second alternative when $a<a_0$, so a normalized solution exists. For $p\\ge 4$ a similar dichotomy holds for large $a$ (Theorem 1.5), and for $4\\le p<6$ the paper shows that lengthening a half-line (adding a vertex) forces the normalized alternative (Theorem 1.7 and Corollary 1.8). In the sign-flipped equation with $a<0$, all these results survive, and if $-mc^2$ is an eigenvalue of $D$, normalized solutions exist for every $p>2$ (Theorem 8.3). The paper also states that this is the first study of normalized solutions of nonlinear Dirac equations on metric graphs.","pith_inferences":["The plateau-linear-decay test function suggests a general principle: on graphs, spectral-gap upper bounds for strongly indefinite problems can be obtained from functions that are constant on the compact core and decay linearly on half-lines, without Fourier analysis; this could transfer to Dirac operators with other vertex conditions or to fractional Dirac-type operators.","The dichotomy 'unit mass versus sub-normalized with $\\omega$ at the edge of the gap' hints at a mass-threshold effect: the sub-normalized solutions might concentrate on $K$ as $a\\to0$ or $\\omega\\to-mc^2$, producing spike profiles analogous to NLS ground states on graphs; this is not explored in the paper.","Because the proof is written at $c=1$, the stated dependence of thresholds on the speed of light ($a_0\\sim m^{(4-p)/2}c^{(4-p)/2}$) is not actually established; a scaling analysis that tracks $\\ell$, $a$, and the Gagliardo-Nirenberg constants would settle whether the nonrelativistic limit $c\\to\\infty$ is governed by these constants, connecting to the bound-state limit in [10].","The unique-continuation lemma restricts Theorem 1.4 to tree cores; if the 'at most one leaf' condition could be relaxed to graphs with cycles, Theorem 1.4 would extend to the graphs covered by Remark 8.4, where $-mc^2$ is an eigenvalue."],"forward_implications":["For $2<p<4$ and $0<a<a_0$, every noncompact metric graph with a non-empty compact core carries a normalized solution: $\\|u\\|_2=1$ and $Du-\\omega u = a\\chi_K|u|^{p-2}u$ for some $\\omega\\in(-mc^2,mc^2)$.","When the compact core is a tree with at most one leaf incident with no half-line, the normalized solution exists for every $a>0$, not just small $a$ (Theorem 1.4).","For $p\\ge4$, if $a$ is large enough, either a normalized solution or a sub-normalized family exists; the sub-normalized alternative is excluded when the energy stays below $\\tfrac{mc^2}{2}$ (Lemma 7.2), yielding genuine normalized solutions in the ranges covered by Theorems 1.5, 1.7, and Remark 1.6.","For $4\\le p<6$, attaching a sufficiently long segment to a half-line produces a new metric graph on which a normalized solution exists, for any $a>0$ (Corollary 1.8).","For the sign-flipped equation with negative coefficient, the same existence theory holds; if $-mc^2\\in\\sigma_p(D)$ (e.g., the core contains a simple cycle), then for every $p>2$ either a normalized or sub-normalized solution exists, and for small $a$ a normalized one (Theorem 8.3)."],"supporting_citations":[{"why":"It supplies the definition of the Dirac operator on metric graphs with Kirchhoff-type vertex conditions, its spectrum, the form domain $Y$, and the regularity result that weak solutions lie in $\\mathrm{dom}(D)$.","marker":"[10]"},{"why":"It supplies the perturbation method: the penalized functional $I_{r,\\mu}$, the reduction $h_{r,\\mu}$, the mountain-pass geometry, and the limiting argument that the paper adapts from $\\mathbb{R}^3$ to graphs.","marker":"[19]"},{"why":"It provides the variational reduction $J(v)=I_0(v+h(v))$ for strongly indefinite functionals and the normalization results the paper extends; its missing spectrum results motivate Lemma 5.2.","marker":"[15]"},{"why":"It gives the Gagliardo-Nirenberg inequality for $H^{1/2}$-type spaces that Lemma 2.1 turns into the graph estimate $\\int_K|u|^p\\le C_{p,K}\\|u\\|^{p-2}\\|u\\|_2^2$.","marker":"[14]"},{"why":"It justifies through the nonlinear superposition principle that the critical points of the reduced functional $J$ are in one-to-one correspondence with critical points of $I_0$.","marker":"[1]"},{"why":"It supplies the definition of a metric graph and quantum-graph conventions (edges, vertices, compact core, half-lines) used throughout.","marker":"[8]"},{"why":"It is used to pass from weak solutions to $C^1$ spinors on each edge, a step needed in the non-existence arguments and in Lemma 5.2.","marker":"[13]"},{"why":"It provides the physical motivation of stationary Dirac solitons on networks, the starting point for considering nonlinear Dirac equations on graphs.","marker":"[31]"}],"fun_headline_variants":["First normalized Dirac solutions on metric graphs","Normalized Dirac equations solved on metric graphs","Graph Dirac problems admit normalized solutions","Existence of normalized Dirac solutions on graphs","Dirac spinors with fixed mass on metric graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the speed of light is 1 and claims that this does not affect the results, but gives no scaling that would restore a general speed of light; the theorems' stated dependence on mass and light speed therefore rests on an unproved normalization.","fun_headline_variants_meta":{"raw":{"variants":["First normalized Dirac solutions on metric graphs","Normalized Dirac equations solved on metric graphs","Graph Dirac problems admit normalized solutions","Existence of normalized Dirac solutions on graphs","Dirac spinors with fixed mass on metric graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00031,"raw_usage":{"total_tokens":1831,"prompt_tokens":1074,"completion_tokens":757,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":690,"completion_tokens_details":{"reasoning_tokens":692}},"tokens_in":690,"tokens_out":757,"duration_ms":7662,"temperature":1.0,"reasoning_tokens":692,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:26:10.638993+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the estimate of Lemma 2.3 and of inequality (4.2) with the general operator $D=-ic\\,\\sigma_1\\tfrac{d}{dx}+mc^2\\sigma_3$. If the resulting threshold is not $a_0=m^{(4-p)/2}c^{(4-p)/2}/(2C_{p,K})$ — for instance if an extra power of $c$ appears from $\\|u\\|_2^2\\le (mc^2)^{-1}\\|u\\|^2$ in the $L^p(K)$ estimates — then the theorems as stated hold only at $c=1$. A concrete check: take the half-line test function $\\varphi_b$ of Lemma 2.3 for a single half-line with general $c$ and compute whether $c_\\infty<\\tfrac{mc^2}{2}$ still follows with the same choice $b\\to0$; the calculation in (2.6) would acquire factors of $c$.","supporting_citations":[{"cited_title":"Borrelli, R","cited_arxiv_id":null,"evidence_quote":"It supplies the definition of the Dirac operator on metric graphs with Kirchhoff-type vertex conditions, its spectrum, the form domain $Y$, and the regularity result that weak solutions lie in $\\mathrm{dom}(D)$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the perturbation method: the penalized functional $I_{r,\\mu}$, the reduction $h_{r,\\mu}$, the mountain-pass geometry, and the limiting argument that the paper adapts from $\\mathbb{R}^3$ to graphs."},{"cited_title":"Buffoni, M","cited_arxiv_id":null,"evidence_quote":"It provides the variational reduction $J(v)=I_0(v+h(v))$ for strongly indefinite functionals and the normalization results the paper extends; its missing spectrum results motivate Lemma 5.2."},{"cited_title":"Brezis, P","cited_arxiv_id":null,"evidence_quote":"It gives the Gagliardo-Nirenberg inequality for $H^{1/2}$-type spaces that Lemma 2.1 turns into the graph estimate $\\int_K|u|^p\\le C_{p,K}\\|u\\|^{p-2}\\|u\\|_2^2$."},{"cited_title":"Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schr¨ odinger equations, J","cited_arxiv_id":null,"evidence_quote":"It justifies through the nonlinear superposition principle that the critical points of the reduced functional $J$ are in one-to-one correspondence with critical points of $I_0$."},{"cited_title":"Berkolaiko, P","cited_arxiv_id":null,"evidence_quote":"It supplies the definition of a metric graph and quantum-graph conventions (edges, vertices, compact core, half-lines) used throughout."},{"cited_title":"Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, pp","cited_arxiv_id":null,"evidence_quote":"It is used to pass from weak solutions to $C^1$ spinors on each edge, a step needed in the non-existence arguments and in Lemma 5.2."},{"cited_title":"Sabirov, D.B","cited_arxiv_id":null,"evidence_quote":"It provides the physical motivation of stationary Dirac solitons on networks, the starting point for considering nonlinear Dirac equations on graphs."}],"review_version":1}