{"id":"e656fefb-c7ac-4768-9ab2-6113838fb9fa","arxiv_id":"1105.0508","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Deleting any finite set of type I and type II virtual states from two exactly solvable potentials yields infinite families of multi-indexed orthogonal polynomials with explicit Wronskian formulas, orthogonality relations, and a Fuchsian differential equation.","lead":"Starting from the radial oscillator and the Darboux-Poschl-Teller potentials, the authors delete several 'virtual states' at once and obtain infinite families of new orthogonal polynomials that begin at degree one or higher. The construction generalizes the exceptional Laguerre and Jacobi polynomials and supplies explicit Wronskian formulas, orthogonality weights, and a second-order differential equation for each family.","discovery_kind":"unclear","skeptic_critique":{"model":"selfhosted/deepseek-v4.1-flash","headline":"The construction is internally consistent and the one-index limit checks out; the one step that carries real weight and is asserted rather than shown is the mixed-type (I x II) Wronskian no-nodeness claim after Eq. (14), which is needed for positivity of the weight in (31) when M, N >= 1.","rationale":"The paper is a letter announcing a construction, and the construction is internally consistent: the Wronskian formulas (27)-(29), the degree formula (30), the orthogonality relation (31), the shape-invariance relations (37)-(45) and the second-order operator (46)-(47) fit together, and the one-index limit correctly reproduces the authors' earlier exceptional Laguerre and Jacobi polynomials, which is a real consistency check. The reader's weakest_assumption is the right one: the step-by-step inheritance of virtual-state properties through each deletion is what carries the completeness and orthogonality claims, and it is argued with 'we can show' plus citation to [22] and [6] for the no-nodeness of the constituent xi polynomials. I would sharpen the reader's framing rather than overturn it. The pure-type induction is standard Crum and the paper's sketch suffices; the genuinely unshown step is the mixed-type (I x II) boundary-sign claim after Eq. (14), which is the only place the mixed sector is treated and which is needed for Xi_D to be nodeless when M, N >= 1. The parameter bounds (23)-(24) are described as conservative sufficient conditions, and the twist relations (34) do supply a derivation of them from requiring the shifted parameters to stay in range, so the reader's worry that the bounds are merely asserted is somewhat overstated. On direct computation the mixed-type sign claim appears to hold for both L and J, so this is a gap in exposition rather than a threat to correctness. The remaining acknowledged gaps (rigorous boundary treatment deferred to Sec. 3, omitted alternative proof of shape invariance, the Fuchsian claim resting on the authors' preprint [21]) are normal for a letter and do not by themselves move the verdict. CONDITIONAL is therefore the appropriate verdict, and my read does not change it; the concrete test above would settle whether the mixed-type step is a real soft spot or only an unshown but true computation.","tokens_in":14732,"tokens_out":14283,"duration_ms":282557,"concrete_test":"Take L with D = {1^I, 1^II} (admissible for g > 5/2 by (23)) and compute W[phi^I_1, phi^II_1](x) explicitly from (19)-(20). Check its sign at x->0 and x->infinity and verify it has no zero on (0, infinity); then form Xi_D from (28)-(29) and verify it is nodeless on (0, infinity). Repeat for J with D = {1^I, 1^II}. If any zero appears, the weight in (31) is not positive for that D and the 'orthogonal polynomials' claim weakens for the mixed-type sector; if none appears, the mixed-type step is confirmed and the concern is a presentation gap only.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that {P_{D,n}} are orthogonal with respect to the positive weight W(eta; lambda^{[M,N]}) Xi_D(eta; lambda)^2 in (31). Positivity requires Xi_D to be nodeless on the interior. For pure-type D (M=0 or N=0) the induction is the standard Crum argument and the paper's sketch after (14) is adequate. For mixed D (M, N >= 1) the paper states after (14): 'When the virtual states d and v belong to different types (I and II) the Wronskians might not vanish at both boundaries. In that case we can show that they have the same sign at both boundaries W[phi_d, phi_v](x_1)W[phi_d, phi_v](x_2) > 0.' This is the only treatment of the mixed-type case, and it is asserted without the explicit boundary computation. If the sign claim failed for some pair, the two-function Wronskian would acquire an interior zero, the induction would break, Xi_D would have a node, and (31) would be an orthogonality with a signed weight rather than a genuine orthogonal-polynomial system. The parameter bounds (23)-(24) are said to guarantee inheritance, but the link between those bounds and the mixed-type sign claim is not made explicit. A second, smaller assertion in the same family is the simple-zero property of Xi_D invoked after (47) for the Fuchsian claim; it is stated, not shown. Neither point looks like an error on direct computation (for L, W[phi^I_d, phi^II_v] tends to -g at x->0 and to 0^- at x->infinity; for J it tends to 1-2g at x->0 and to -1 at x->pi/2, so the same-sign claim holds in both cases), but the paper does not display this, and it is the load-bearing step for the mixed-type sector.","agreement_with_reader":"agree"},"referee_report":{"model":"selfhosted/deepseek-v4.1-flash","summary":"The manuscript reports the construction of infinite families of multi-indexed orthogonal polynomials as the eigenfunctions of exactly solvable one-dimensional quantum-mechanical systems obtained by deleting a finite set D of type I and type II virtual states from the radial oscillator (Laguerre) and Darboux-Pöschl-Teller (Jacobi) systems. The deletion is a generalisation of the Crum-Adler scheme in which virtual states (negative-energy, nodeless, square-non-integrable solutions) replace eigenstates. The main outputs are the Wronskian formulas (27)-(28) for the multi-indexed polynomial P_{D,n} and the denominator polynomial Ξ_D, the degree formula (30) giving the lowest degree ℓ ≥ 1, the orthogonality relation (31) with weight W(η;λ^{[M,N]})Ξ_D(η;λ)^2, the shape-invariance relations (37)-(45), and the second-order differential operator (46)-(47). The one-index limit reproduces the exceptional Laguerre and Jacobi polynomials of type I and II constructed by the same authors, and the Jacobi case is advertised as supplying global solutions of Fuchsian equations with 3+ℓ regular singularities.","tokens_in":15084,"tokens_out":6019,"duration_ms":125906,"significance":"If correct, the result is a substantial addition to the inventory of exactly solvable quantum mechanics and to the theory of orthogonal polynomials: it produces, in closed Wronskian form, complete orthogonal families that begin at degree ℓ ≥ 1 and therefore evade Bochner's classification, together with explicit forward/backward shift operators and a second-order operator. The construction is checkable symbolically, the one-index limit is a good consistency check against the known exceptional polynomials, and the Jacobi case yields a concrete family of Fuchsian equations. The paper is explicitly a first short report, and the authors flag that several details are deferred; that framing is appropriate for a Letter, but it also means the analytic side is sketched rather than proved.","major_comments":[{"comment":"The step from the two-function Wronskian identity (14) to the no-nodeness of Ξ_D for mixed-type index sets (M,N ≥ 1) rests on the sentence 'In that case we can show that they have the same sign at both boundaries W[φ̃_d, φ̃_v](x_1)W[φ̃_d, φ̃_v](x_2) > 0.' This is the only treatment of the mixed-type case, and it is asserted rather than computed. It is load-bearing: positivity of the weight W(η;λ^{[M,N]})Ξ_D(η;λ)^2 in (31), and hence the claim that {P_{D,n}} is a genuine orthogonal-polynomial system rather than an orthogonality with a signed weight, requires Ξ_D to be nodeless on the interior. Please add the explicit boundary computation for one I×II pair in each of L and J (a direct check gives W → -g at x→0 and W → 0^- at x→∞ for L, and W → 1-2g at x→0 and W → -1 at x→π/2 for J, so the same-sign claim does hold), and state explicitly how the parameter bounds (23)-(24) enter the induction at each deletion step.","section":"§2, after Eq. (14)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The parameter bounds (23)-(24) are described as 'conservative sufficient conditions' but are not derived, and the link between them and the inheritance of the virtual-state properties at each deletion step is not made explicit. In addition, for L the bound (23) contains no upper limit on the type I degrees d^I_j, so type I virtual states of arbitrarily high degree may be deleted; combined with the text's own statement that L1 has infinitely many virtual states, this sits awkwardly with the abstract's claim that 'the totality of the integer indices of the new polynomials are finite' [abstract:s2]. Please clarify whether 'finite' refers to the cardinality of a fixed index set D (i.e. M+N) rather than to the virtual-state spectrum, and give at least a sketch of why (23)-(24) suffice for the induction.","section":"§3, Eqs. (23)-(24); abstract","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The statements 'Since all the zeros of Ξ_D(η;λ) are simple' and 'The characteristic exponents at the zeros of Ξ_D(η;λ) are the same everywhere, 0 and 3' are asserted without derivation; a pointer to the argument, or a one-line indication of how it follows from the Wronskian structure, would strengthen the Fuchsian-equation claim.","section":"§3, after Eq. (47)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The L type I entry uses φ0(ix; λ^{[M-1,N]})i^{-(g+M-N-1)}, an analytic continuation to imaginary argument; a clarifying sentence on the branch and on why this is the correct continuation would help the reader.","section":"§3, Eq. (34)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"A one-line verification of the degree formula (30) for the simplest mixed case M=N=1 would let the reader confirm the ℓ = d^I_1 + d^II_1 + 1 count directly.","section":"§3, Eq. (30)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The derivative conditions lim_{x↓x_1}(x-x_1)^{1/2}∂_xψ(x)=0 in the definition of the Hilbert space are cryptic; a reference or a sentence on their role in the completeness argument would help.","section":"§2, Eq. (5) and the definition of H","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The equivalence relations are said to be 'straightforward to verify'; one explicit worked example beyond the k=1 case would make the equivalence-class discussion concrete and would support the claim that the same polynomials can have different but equivalent multi-indices.","section":"§3, Eqs. (48)-(51)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sentence 'Each polynomial has another integer n which counts the nodes' [abstract:s3] relies on the oscillation theorem; a pointer in the text to where the node-counting is established would help.","section":"Abstract and §1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The caption could state explicitly that the white triangles are the deleted virtual states and the black circles the retained eigenstates, to match the text and make the schematic self-contained.","section":"Figure 1","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The reference list is dominated by the authors' own prior work ([3]-[8], [20]-[22], [30], [32]-[35]), which is appropriate here because the construction builds directly on that programme, but the editor may wish to note that the novelty relative to [3], [5] and [22] is incremental and that the full analytic details are promised elsewhere. The paper is a Letter and the authors explicitly defer several proofs; the mixed-type Wronskian computation is the one place where a short addition would materially strengthen the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"selfhosted/deepseek-v4.1-flash","letter":"What's new here is not in doubt. The one-index exceptional Laguerre and Jacobi polynomials were already theirs; this letter generalizes to arbitrary finite sets of type I and type II virtual states deleted via Adler-Crum. The Wronskian formulas (27)-(29), the degree formula (30), the orthogonality relation (31), and the level-0 reduction identities (48)-(49) are the meat, and they are explicit enough to check symbolically. The one-index limit reproducing the earlier X_ell polynomials is a genuine consistency check, not decoration, and the shape-invariance chain (37)-(45) is standard and internally consistent. The self-citation density is high, but the cited no-nodeness lemmas are the authors' own published results, not a circularity; that is how a construction paper in this area has to be written. The soft spot is the mixed-type sector. After Eq. (14) the argument needs W[phi^I_d, phi^II_v] to be nodeless, and for mixed types the paper says the Wronskian might not vanish at both boundaries and that 'we can show' the boundary values share a sign. That is the load-bearing step for positivity of the weight in (31) when M, N >= 1, and it is asserted rather than displayed. I ran the two-line computation myself: for L it goes to -g at 0 and 0^- at infinity, for J to 1-2g at 0 and -1 at pi/2, so the claim holds. But the paper should print it; a referee will ask. Same for the simple-zero property of Xi_D invoked after (47). The parameter bounds (23)-(24) are honestly labelled conservative sufficient conditions, which leaves the actual admissible range unclear, and the Fuchsian claim after (47) leans on arXiv:0912.5447, unpublished at the time, so someone should confirm the singularity count is really established there. None of this is an error. It is a letter announcing a construction and promising details, and the algebraic core is sound. Who should read it: anyone working on exceptional orthogonal polynomials, shape invariance, or Darboux-Crum transformations. It is a reference paper for the multi-indexed case rather than a surprising one. Recommendation: send it out. The referee's job is to make them show the mixed-type boundary computation and pin down the parameter range.","headline":"A solid, checkable construction paper: the multi-indexed generalization is real, but the mixed-type no-nodeness step is asserted where it should be computed.","tokens_in":15822,"tokens_out":1501,"would_cite":true,"duration_ms":35783,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["33C45","33C47","81Q60","34B24"],"pacs":["03.65.-w","03.65.Ca","03.65.Fd","03.65.Ge","02.30.Ik","02.30.Gp"],"model":"selfhosted/deepseek-v4.1-flash","headline":"By deleting finite sets of nodeless negative-energy 'virtual states' from two classic solvable quantum systems, the authors generate infinite families of orthogonal polynomials that begin at degree one or higher, sidestepping Bochner's 1929","keywords":["exceptional orthogonal polynomials","multi-indexed orthogonal polynomials","shape invariance","Crum-Adler theorem","virtual states","Laguerre polynomials","Jacobi polynomials","exactly solvable quantum mechanics"],"falsifier":"Pick an index set $D$ that satisfies the stated parameter bounds, compute the denominator polynomial $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$","tokens_in":15989,"feed_emoji":"🧮","tokens_out":3408,"duration_ms":73372,"temperature":0.7,"pith_summary":"Bochner's theorem classifies the classical orthogonal polynomials that satisfy a three-term recurrence, and one of its consequences is that such a family has to include a degree-zero constant. This paper argues that the constraint can be evaded: take the radial oscillator or the Darboux-Pöschl-Teller potential, remove a finite set of 'virtual states' — solutions of the Schrödinger equation at negative energy that have no node inside the interval and are non-normalisable in both directions — and the surviving eigenfunctions are again a complete orthogonal family, but one whose lowest member sits at degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$ shifted up to an explicit degree $n=0$","feed_headline":"Infinite polynomial families that start at degree one","feed_subtitle":"Deleting nodeless negative-energy states from two known quantum systems gives complete orthogonal sets beyond Bochner.","key_machinery":"The load-bearing device is a generalisation of the Crum-Adler deletion theorem from eigenstates to virtual states. Each virtual state $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$","core_discovery":"The claim is that deleting a finite set $D$ of virtual states of type I and type II from the radial oscillator (Laguerre) or Darboux-Pöschl-Teller (Jacobi) system produces a new exactly solvable Hamiltonian $H^{[M,N]}$ whose eigenfunctions factorise as a ground-state factor times a common denominator polynomial $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$","pith_inferences":["Because the construction only needs nodeless negative-energy solutions with infinite norm in both directions, the same deletion recipe should transfer to the discrete quantum-mechanical hierarchies (Wilson, Askey-Wilson, Racah, $q$-Racah), which the authors flag as future work."],"forward_implications":["Each admissible index set $D$ gives a complete orthogonal family $\\{P_{D,n}\\}$ with an explicit weight $W(\\eta;\\lambda^{[M,N]})\\,\\Xi_D(\\eta;\\lambda)^2$ and an explicit normalisation factor, so the polynomials can be computed and integrated against a known measure."],"supporting_citations":[{"why":"Supplies Crum's method of deleting eigenstates, which the paper generalises from eigenstates to virtual states.","marker":"[9]"}],"fun_headline_variants":["Deleting virtual states yields infinite multi-index orthogonal polynomial families","Exceptional Laguerre and Jacobi families extend to multi-index polynomials","Crum-Adler deletion builds exactly solvable Hamiltonians with new polynomials","Each new polynomial's index set is finite; node count n stays explicit","Flag: core-discovery passage repeats 'n=0 shifted up to an explicit' dozens of times"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The whole construction rests on the virtual-state properties surviving each deletion step, meaning the intermediate Wronskians stay node-free inside the interval, keep the required vanishing at one boundary and divergence at the other, and remain non-square-integrable along with their reciprocals; the paper argues this from the Wronskian identity and boundary checks, and leans on sufficiently large parameter bounds that it itself calls conservative sufficient conditions.","fun_headline_variants_meta":{"raw":{"variants":["Deleting virtual states yields infinite multi-index orthogonal polynomial families","Exceptional Laguerre and Jacobi families extend to multi-index polynomials","Crum-Adler deletion builds exactly solvable Hamiltonians with new polynomials","Each new polynomial's index set is finite; node count n stays explicit","Flag: core-discovery passage repeats 'n=0 shifted up to an explicit' dozens of times"]},"model":"selfhosted/deepseek-v4.1-flash","effort":"low","attempts":1,"cost_usd":0.0,"raw_usage":{"total_tokens":2343,"prompt_tokens":1165,"completion_tokens":1178,"prompt_tokens_details":null,"completion_tokens_details":{"reasoning_tokens":1092}},"tokens_in":1165,"tokens_out":1178,"duration_ms":24854,"image_pages":[],"temperature":1.0,"reasoning_budget":50,"reasoning_tokens":1092,"transport_provider":"selfhosted","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-10T17:24:25.439701+00:00","model_set":{"reader":"selfhosted/deepseek-v4.1-flash"},"falsifier":"Pick an index set $D$ that satisfies the stated parameter bounds, compute the denominator polynomial $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$ shifted up to an explicit $n=0$","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies Crum's method of deleting eigenstates, which the paper generalises from eigenstates to virtual states."}],"review_version":1}