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$L^p$ boundedness of wave operators for higher order schr\"odinger operators with threshold eigenvalues

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Pith's one-line read For higher-order Schrödinger operators with a zero-energy eigenvalue, the paper proves $L^p$ boundedness of wave operators up to dimension-dependent thresholds, with extra eigenspace orthogonality raising the range and a new $L^\infty$…

desk verdict Solid technical continuation that closes the threshold-eigenvalue case for 2m<n≤4m; the main fix needed is adding the no-resonance hypothesis to Theorem 1.1. read the letter →

arxiv 2506.16378 v2 pith:PCYO26XA submitted 2025-06-19 math.AP math.SP

M. Burak Erdogan , William R. Green , Kevin LaMaster This is my paper · ORCID
classification math.APmath.SP MSC 35P2547A4081U05
keywords waveoperatorsL^pboundednesshigher-orderSchrödingeroperatorthresholdeigenvaluezero-energyresonanceresolventexpansiondispersiveestimatescatteringtheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves bounds on the $L^p$ operator norm of wave operators for $H=(-\Delta)^m+V$ in dimensions $2m3$ that had not been established before. Because wave-operator boundedness transfers mapping properties of $f((-\Delta)^m)$ to $f(H)P_{ac}(H)$, these ranges feed directly into dispersive and Strichartz-type estimates for the perturbed evolution.

What carries the argument

The load-bearing object is the low-energy stationary representation $W_{low,\kappa}$, written through the symmetric resolvent identity as an integral of $R_0^+(\lambda^{2m})v\,\Gamma_\kappa(\lambda)\,v[R_0^+(\lambda^{2m})-R_0^-(\lambda^{2m})]$. The paper expands $\Gamma_\kappa(\lambda)$ into a smooth part plus a singular finite-rank part $\lambda^{-2m}\Gamma_s(\lambda)$ using a threshold resolvent inversion identity and the threshold classification spaces $S_1,S_2,\dots$; the singular part is a combination of operators of type $(k_1,k_2,\alpha)$ acting on free resolvent differences. The crucial quantitative control is Proposition 2.2, which bounds the resulting integral kernels via cancellation lemmas that convert orthogonality of $\phi\in S_{k+1}L^2$ against $v x^\alpha$ into extra powers of $\lambda$ or of $|x-y|^{-1}$, together with oscillatory integral estimates for the remaining $\lambda$-integrals. The classification premise that at zero energy $S_1=S_{k+1}$ with $T_{k+1}$ invertible is what turns the threshold eigenvalue into this tractable finite-rank singularity.

What would settle it

Exhibit a real-valued potential $V$ obeying $|V(x)|\lesssim\langle x\rangle^{-\beta}$ with $\beta>\max(8m-n,n_\star)+2$, $2m<n\le 4m$, such that $H=(-\Delta)^m+V$ has a zero-energy eigenvalue, no resonance, and no positive eigenvalue, yet $W_{low,\kappa}$ fails to be bounded at some endpoint such as $p=n/(2m-k_0)$ for a $k_0$ satisfying the orthogonality hypotheses. Equivalently, compute the kernel of the singular part in equation (8) for a rank-one model with an explicit zero-energy eigenfunction and test whether the oscillatory integral bounds saturate at the claimed endpoint.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.1 and Theorem 1.2 and lifted to the full wave operators by Corollary 1.4, is that the only low-energy obstruction to $L^p$ boundedness is the zero-energy eigenspace, and its effect is controlled by the smallest $k_0$ such that the eigenspace is orthogonal to $x^\alpha V$ for all $|\alpha|<k_0$. In dimensions $2m<n\le 4m$ the generic bound is $p<2n/(n-1)$ for odd $n$ and $p<2n/(n-2)$ for even $n$; with $k_0$ degrees of orthogonality the bound improves to $p<n/(2m-k_0)$, to all finite $p$ when $k_0=2m$, and to $p=\infty$ when $k_0=2m+1$. For $m=1$, $n>3$, the $L^\infty$ case is new: previous results stopped at finite $p$ under two orthogonality conditions, aside from the known three-dimensional case. The proof works uniformly in even and odd dimensions by deriving an explicit singular decomposition of $\Gamma_\kappa(\lambda)$ into a smooth operator plus a finite-rank singular term, then controlling every resulting kernel by the same orthogonality-based cancellation mechanism.

Load-bearing premise

The proof assumes that at zero energy the only obstruction is a genuine square-integrable bound state, never a resonance; if a resonance sits at zero energy instead of or alongside the eigenvalue, the resolvent expansion and the claimed $p$-ranges would need to be replaced.

Editorial extensions

If this is right

  • For any potential satisfying Assumption 1.3 and the theorem's hypotheses, the full wave operators $W_\pm$ are bounded on the stated $L^p$ ranges, so $L^p\to L^{p'}$ dispersive bounds for $e^{-itH}P_{ac}(H)$ follow with rate $|t|^{-(n/m)(1/2-1/p)}$.
  • In dimensions $2m<n<4m$ the low-energy part is bounded for every $\kappa\ge 0$, so a single smooth cutoff suffices and the $p$-ranges are not an artifact of high-energy bookkeeping.
  • For the classical Schrödinger operator ($m=1$) the theorem subsumes and streamlines the eigenvalue-only threshold results, and in $n>3$ it produces the first $L^\infty$ endpoint for wave operators under the orthogonality $k_0=3$.
  • The decomposition into a smooth operator plus a finite-rank singular term reduces threshold eigenvalue effects to explicit kernels, and the paper notes that the same tools were developed with enough flexibility to handle resonances in lower dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The stated upper limits are likely optimal: the paper calls the ranges natural, and known low-dimensional resonance cases exhibit unboundedness at analogous endpoints; one would expect the same saturation here even at eigenvalue-only thresholds.
  • Proposition 2.2 appears applicable beyond the eigenvalue-only case, since its kernel bounds use only orthogonality and decay; a resonance version would likely produce different $p$-ranges and the paper's setup already points toward that extension.
  • The orthogonality conditions $\langle \psi, x^\alpha V\rangle=0$ are conditions that can be checked or engineered for concrete potentials; testing them numerically for radial or compactly supported potentials in low dimensions would give direct evidence on whether the ranges saturate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper proves Lp boundedness of the wave operators for H=(-Δ)^m+V in dimensions n>2m when the perturbed operator has a zero-energy eigenvalue. The main new contribution is the low-energy regime 2m<n≤4m, where threshold resonances are possible: under a no-resonance assumption, the authors obtain the range 1≤p<2n/(n-1) for n odd and 1≤p<2n/(n-2) for n even. With additional orthogonality of the eigenspace to {x^α V: |α|≤k0-1}, the range is extended to 1≤p<n/(2m-k0), with endpoint ranges [1,∞) and [1,∞] when k0=2m and k0=2m+1 respectively. The proof uses the stationary representation of the wave operator, the Jensen-Nenciu resolvent inversion scheme, and a finite-rank decomposition of the singular part of M(λ)^{-1}; the remaining singular terms are controlled by a new Proposition 2.2. The authors state that the arguments apply uniformly to the classical m=1 case and yield a new L∞ endpoint for n>3.

Significance. If the results are correct as intended, this is a substantial extension of the higher-order wave-operator theory: it covers the low-dimensional range 2m<n≤4m where threshold resonances and logarithmic resolvent terms appear, and it does so with a unified even/odd-dimensional treatment. The paper contains genuine technical work: Lemma 4.3 provides the requisite resolvent expansion with logarithmic terms, Proposition 2.2 is proved in-text via the cancellation Lemmas 3.2–3.4, and the dimensional bookkeeping correctly reproduces the claimed endpoint ranges from k0=2m-⌈n/2⌉+1. The claimed new L∞ boundedness in the classical m=1, n>3 eigenvalue-only case is clearly identified and is a concrete advance over the existing p<∞ results. There are no fitted parameters or post hoc exclusions in the derivation. The main limitation is that the theorem statements, as displayed, do not fully state the no-resonance structural hypothesis on which the low-energy inversion argument depends.

major comments (2)
  1. [Section 1, Theorem 1.1] The displayed statement of Theorem 1.1 assumes only a zero-energy eigenvalue and no positive eigenvalues, but the proof requires the additional no-resonance classification stated in Section 4: for 2m<n≤4m, 'having only an eigenvalue at zero (no resonances)' means S1=Sk+1 and T_{k+1}:=S1 v G_{1,L} v S1 is invertible on S1L2. This invertibility enters directly in the Jensen-Nenciu inversion (26)-(27), in the expansion of M(λ)^{-1} in Proposition 4.2, and ultimately in the singular structure (8) used in Section 5. Since zero-energy resonances can coexist with a zero-energy eigenvalue in this dimension range, the stated Lp ranges are not established under the hypotheses as written. The abstract and the paragraph preceding Theorem 1.1 do say 'no resonances,' so this is a statement-level gap rather than a proof error, but it is load-bearing and should be fixed by adding the no-resonance hypothesis to the theorem statement.
  2. [Section 1, Theorem 1.2 and Corollary 1.4] The same gap affects Theorem 1.2 in dimensions 2m<n≤4m. The orthogonality condition on the eigenspace, {x^α V: |α|≤k0-1}, does not by itself rule out zero-energy resonances; the proof requires the stronger structural assumption S1=S_{k0+1}, which is used in Lemma 4.5 and Proposition 4.2. Consequently, the statement of Theorem 1.2 should include the no-resonance (or S1=S_{k+1}) hypothesis for 2m<n≤4m, and Corollary 1.4, which invokes Theorem 1.2, should be revised accordingly. This is a fixable statement-level issue, but it is central because the entire low-energy argument rests on the eigenvalue-only classification.
minor comments (3)
  1. [Section 1, p. 3] The sentence 'We note that this has a slightly less stringent assumption than [3], which requires β>n+4k+5=8m−n+11 for n odd' appears to have an arithmetic slip: with the paper's k=2m−⌈n/2⌉+1, n+4k+5 = 8m−n+7 for n odd. Please verify the formula for the assumption in [3] or clarify the definition of k used there.
  2. [Section 1, p. 2] The sentence 'We note that the classical case and dimensions 4m≥n>2m, (n=3,4), the existence of threshold eigenvalues limits...' has a grammatical issue; 'and dimensions' should likely be 'in dimensions'.
  3. [Section 4, p. 18] The statement 'In the case 2m<n≤4m, with k=2m−⌈n/2⌉+1, having only an eigenvalue at zero (no resonances) means S1=Sk+1 and T_{k+1}:=S1 v G_{1,L} v S1 is invertible on S1L2' is an external structural classification from [9]. Since it is load-bearing, it would help to label it explicitly as an assumption taken from [9] rather than presenting it as a consequence of the immediately preceding definitions.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained, the self-citations are published proofs with independent content, and the only flagged issue is a statement-level omitted hypothesis rather than a circular step.

full rationale

I walked the paper's derivation chain: the stationary representation (1), the symmetric resolvent identity (2), the Jensen-Nenciu inversion scheme (26)-(31), the low-energy decomposition (8), and the kernel bounds in Proposition 2.2 (10). At no point is a conclusion identified with an assumption by construction, and there are no fitted parameters, no post hoc exclusions, and no renaming of an empirical pattern as a derivation. The main external inputs are: the high-energy/born-series bounds and Proposition 2.1 from the authors' earlier papers [6,7], the pointwise resolvent bounds (42) from [7], and the zero-energy classification S1 = S_{k+1} with T_{k+1} invertible from [9]. These are stated, published results with proofs and with hypotheses that do not include the target Lp-boundedness claims, so under the review rules they count as independent evidence rather than circularity. The orthogonality index k0 appears both in the hypothesis and in the final range p < n/(2m-k0), but this is derived: the singular part of M^{-1} (Proposition 4.2) is explicitly computed to be a combination of operators of type (k0,k0,0), (0,k0,k0), (k0,0,k0), and Proposition 2.2 then yields the range from k2 + alpha = k0. This is an algebraic consequence of the resolvent expansion, not an assumption of the desired bound. The one legitimate concern is that the displayed Theorem 1.1 omits the 'no resonances' hypothesis that the proof in Section 4 requires; the abstract and Section 2 state it, but the theorem statement as printed is incomplete in dimensions 2m < n <= 4m. That is a statement-level correctness issue, not circularity, because no equation or conclusion is being defined in terms of itself. I therefore find no significant circularity and assign score 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the assumptions listed above. There are no fitted numerical parameters: β, k0, and κ are hypotheses or technical choices, not values tuned to data. The heavy machinery (Jensen-Nenciu inversion, resolvent expansions) is imported from the literature, while the new work is the extraction of the singular structure (8) and the cancellation arguments in Section 3. No new physical or mathematical entities are postulated.

assumptions (7)
  • domain assumption Threshold obstruction is exactly an eigenvalue: S1 = S_{k+1} and T_{k+1} := S1 v G_{1,L} v S1 is invertible on S1 L² (no resonances).
    This classification, stated in Section 4, isolates the eigenvalue-only case; if a resonance is present the identification fails, the expansion of M(λ)^{-1} changes, and the p-ranges differ (see [3, 19, 20]). It is the load-bearing premise separating the claimed ranges from the resonance-case ranges.
  • domain assumption No positive eigenvalues for H = (−∆)^m + V.
    The paper states this is vital: higher-order operators can have positive eigenvalues even for smooth compactly supported potentials [9]. The high-energy bounds from [6] and the intertwining identity require it.
  • domain assumption Decay and smoothness of V: |V(x)| ≲ ⟨x⟩^{−β} with β > max(8m−n, n⋆)+2, plus Assumption 1.3 for the high-energy part.
    β enters the resolvent expansion Lemma 4.3 and the AE bounds for vR0v; Assumption 1.3(iii) imposes a Fourier-space L^q condition on ⟨x⟩^σ V when n ≥ 4m−1 and is required for the high-energy contribution imported from [6].
  • standard math Jensen-Nenciu inversion formula (26) for M(λ)^{-1} and the invertibility of B(λ) on S1 L².
    Imported from [16]; used in Section 4 to separate the singular part λ^{-2m} S1 B(λ)^{-1} S1 from the admissible error.
  • standard math Stone's formula identity in Lemma 3.2: replacing vφ by λ^{k2−ε} |D|^{−k2+ε} vφ inside the resolvent-difference integral.
    This is the spectral projection identity that powers the cancellation lemmas; it is standard and also used by Galtbayar-Yajima [1, 2], but its proof is only sketched in the paper.
  • standard math Free resolvent kernel bounds of Lemma 2.3 and (39), including pointwise derivative bounds with exponents (n+1)/2 − 2m and (1−n)/2.
    These bounds, rooted in [6, 7], are used throughout Sections 3 and 5 to control the λ-integrals and the iterated resolvents.
  • domain assumption The zero-energy eigenspace is orthogonal to {x^α V(x) : |α| ≤ k0−1} for the stated k0.
    This is the hypothesis in Theorem 1.2 that extends the p-range; it is an assumption the theorem is conditional on, not a derived fact.

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Pith. "Pith review of $L^p$ boundedness of wave operators for higher order schr\"odinger operators with threshold eigenvalues." pith.science (2026). https://pith.science/paper/PCYO26XA

@misc{pith2026250616378,
  author       = {Pith},
  title        = {Pith review of: $L^p$ boundedness of wave operators for higher order schr\"odinger operators with threshold eigenvalues},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PCYO26XA}},
  note         = {Machine review of arXiv:2506.16378}
}
abstract

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$ when $H$ has a threshold eigenvalue. We adapt our recent results for $m\geq 1$ when $n>4m$ to lower dimensions $2m<n\leq 4m$ to show that when $H$ has a threshold eigenvalue and no resonances, the wave operators are bounded on $L^p(\mathbb R^n)$ for the natural range $1\leq p<\frac{2n}{n-1}$ when $n$ is odd and $1\leq p<\frac{2n}{n-2}$ when $n$ is even. We further show that if the zero energy eigenfunctions are orthogonal to $x^\alpha V(x)$ for all $|\alpha|<k_0$, then the wave operators are bounded on $1\leq p<\frac{n}{2m-k_0}$ when $k_0<2m$ in all dimensions $n>2m$. The range is $p\in [1,\infty)$ and $p\in[1,\infty]$ when $k_0=2m$ and $k_0>2m$ respectively. The proofs apply in the classical $m=1$ case as well and streamlines existing arguments in the eigenvalue only case, in particular the $L^\infty(\mathbb R^n)$ boundedness is new when $n>3$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The $L^p$-continuity of wave operators for fractional order Schr\"odinger operators

    math.AP 2025-09 conditional novelty 6.0 of 10

    Wave operators for fractional Schrödinger operators are proven bounded on Lp for all 1≤p≤∞ under decay or smallness conditions on the potential, yielding dispersive and Strichartz estimates for the perturbed flow.

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Works this paper leans on

31 extracted references · 29 canonical work pages · cited by 1 Pith paper

  1. [3]

    Cheng, A

    H. Cheng, A. Soffer, Z. Wu, and X. Yao, The Lp-boundedness of wave operators for higher order Schr¨ odinger operator with zero singularities in low odd dimensions preprint 2025, arXiv:2505.07009

    arXiv 2025

  2. [1]

    The Lp-boundedness of wave operators for 4th order Schr¨ odinger operators on R2, I, preprint 2025, arXiv:2504.11753

    Galtbayar, A., and Yajima, K. The Lp-boundedness of wave operators for 4th order Schr¨ odinger operators on R2, I, preprint 2025, arXiv:2504.11753

    work page arXiv 2025

  3. [2]

    The Lp-boundedness of wave operators for fourth order Schr¨ odinger operators on R4

    Galtbayar, A., and Yajima, K. The Lp-boundedness of wave operators for fourth order Schr¨ odinger operators on R4. J. Spectr. Theory 14 (2024), no. 1, pp. 271–354

    work page 2024

  4. [4]

    Agmon, Spectral properties of Schr¨ odinger operators and scattering theory.Ann

    S. Agmon, Spectral properties of Schr¨ odinger operators and scattering theory.Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218

    work page 1975

  5. [5]

    M. B. Erdo˘ gan, M. Goldberg, and W. R. Green,Counterexamples to Lp boundedness of wave operators for classical and higher order Schr¨ odinger operators,J. Funct. Anal. 285 (2023), no. 5, Paper No. 110008, 18 pp. 28 M. B. ERDO ˘GAN, W. R. GREEN, K. LAMASTER

    work page 2023

  6. [6]

    M. B. Erdo˘ gan, and W. R. Green, The Lp-continuity of wave operators for higher order Schr¨ odinger operators, Adv. Math. 404 (2022), Paper No. 108450

    work page 2022

  7. [7]

    M. B. Erdo˘ gan, and W. R. Green, A note on endpoint Lp-continuity of wave operators for classical and higher order Schr¨ odinger operatorsJ. Differential Equations 355 (2023), 144–161

    work page 2023

  8. [8]

    M. B. Erdo˘ gan, W. R. Green, and K. LaMaster, Lp-continuity of wave operators for higher order Schrodinger operators with threshold eigenvalues in high dimensions , Discrete Contin. Dyn. Syst. 45 (2025), Issue 9: 3258–3274

    work page 2025

Show all 31 references
  1. [9]

    H. Feng, A. Soffer, Z. Wu, and X. Yao, Decay estimates for higher order elliptic operators , Trans. Amer. Math. Soc. 373 (2020), no. 4, 2805—2859

    2020

  2. [10]

    Finco, and K

    D. Finco, and K. Yajima, The Lp boundedness of wave operators for Schr¨ odinger operators with threshold singu- larities II. Even dimensional case . J. Math. Sci. Univ. Tokyo 13 (2006), no. 3, 277–346

    2006

  3. [11]

    Galtbayar, and K

    A. Galtbayar, and K. Yajima. The Lp-boundedness of wave operators for fourth order Schr¨ odinger operators on R4. J. Spectr. Theory 14 (2024), 271–354

    2024

  4. [12]

    Goldberg and W

    M. Goldberg and W. Green, The Lp boundedness of wave operators for Schr¨ odinger operators with threshold singularities. Adv. Math. 303 (2016), 360–389

    2016

  5. [13]

    Goldberg and W

    M. Goldberg and W. Green. On the Lp Boundedness of Wave Operators for Four-Dimensional Schr¨ odinger Op- erators with a Threshold Eigenvalue . Ann. Henri Poincar´ e18 (2017), no. 4, 1269–1288

    2017

  6. [14]

    Goldberg, and W

    M. Goldberg, and W. Green, On theLp boundedness of the Wave Operators for fourth order Schr¨ odinger operators,. Trans. Amer. Math. Soc. 374 (2021), 4075–4092

    2021

  7. [15]

    H¨ ormander,The existence of wave operators in scattering theory

    L. H¨ ormander,The existence of wave operators in scattering theory. Math. Z. 146 (1976), no. 1, 69–91

    1976

  8. [16]

    Jensen, and G

    A. Jensen, and G. Nenciu. A unified approach to resolvent expansions at thresholds . Rev. Mat. Phys. vol. 13, no. 6 (2001), 717–754

    2001

  9. [17]

    Jensen and K

    A. Jensen and K. Yajima. On Lp boundedness of wave operators for 4-dimensional Schr¨ odinger operators with threshold singularities. Proceedings of the London Mathematical Society 96.1 (2008), pp. 136–162

    2008

  10. [18]

    Mizutani, Z

    H. Mizutani, Z. Wan, and X. Yao, Lp-boundedness of wave operators for fourth-order Schr¨ odinger operators on the line, Adv. Math. 451 (2024), Paper No. 109806, 68 pp

    2024

  11. [19]

    Mizutani, Z

    H. Mizutani, Z. Wan, and X. Yao, Lp-boundedness of wave operators for fourth order Schr¨ odinger operators with resonances on R3, J. Funct. Anal. 289 (2025), no. 8, Paper No. 111013, 68 pp

    2025

  12. [20]

    Mizutani, Z

    H. Mizutani, Z. Wan, and X. Yao, Counterexamples and weak (1,1) estimates of wave operators for fourth-order Schr¨dinger operators in dimension three , J. Spectr. Theory 14 (2024), no. 4, 1409–1450

    2024

  13. [21]

    Reed and B

    M. Reed and B. Simon. Methods of Modern Mathematical Physics III: Scattering Theory , Academic Press, New York, NY, 1972

    1972

  14. [22]

    Schechter, Scattering theory for pseudodifferential operators, Quart

    M. Schechter, Scattering theory for pseudodifferential operators, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 111–121

    1976

  15. [23]

    Schechter, Scattering theory for elliptic operators of arbitrary order

    M. Schechter, Scattering theory for elliptic operators of arbitrary order. Comment. Math. Helv. 49 (1974), 84–113

    1974

  16. [24]

    Yajima, The Wk,p-continuity of wave operators for Schr¨ odinger operators.J

    K. Yajima, The Wk,p-continuity of wave operators for Schr¨ odinger operators.J. Math. Soc. Japan 47 (1995), no. 3, 551–581

    1995

  17. [25]

    Yajima, The Wk,p-continuity of wave operators for Schr¨ odinger operators

    K. Yajima, The Wk,p-continuity of wave operators for Schr¨ odinger operators. II. Positive potentials in even dimensions m≥ 4. Spectral and scattering theory (Sanda, 1992), 287–300, Lecture Notes in Pure and Appl. Math., 161, Dekker, New York, 1994

    1992

  18. [26]

    Yajima, The Wk,p-continuity of wave operators for Schr¨ odinger operators

    K. Yajima, The Wk,p-continuity of wave operators for Schr¨ odinger operators. III. Even-dimensional casesm≥ 4. J. Math. Sci. Univ. Tokyo 2 (1995), no. 2, 311–346. WAVE OPERATORS FOR HIGHER ORDER SCHR ¨ODINGER OPERATORS 29

    1995

  19. [27]

    Yajima, The Lp boundedness of wave operators for Schr¨ odinger operators with threshold singularities I

    K. Yajima, The Lp boundedness of wave operators for Schr¨ odinger operators with threshold singularities I. The odd dimensional case. J. Math. Sci. Univ. Tokyo 13 (2006), 43–94

    2006

  20. [28]

    Yajima, Wave Operators for Schr¨ odinger Operators with Threshold Singularities, Revisited

    K. Yajima, Wave Operators for Schr¨ odinger Operators with Threshold Singularities, Revisited . Preprint, arXiv:1508.05738

    arXiv

  21. [29]

    Yajima, Remark on theLp-boundedness of wave operators for Schr¨ odinger operators with threshold singularities, Documenta Mathematica 21 (2016), 391–443

    K. Yajima, Remark on theLp-boundedness of wave operators for Schr¨ odinger operators with threshold singularities, Documenta Mathematica 21 (2016), 391–443

    2016

  22. [30]

    L1 and L∞-boundedness of wave operators for three dimensional Schr¨ odinger operators with threshold singularities, Tokyo J

    Yajima, K. L1 and L∞-boundedness of wave operators for three dimensional Schr¨ odinger operators with threshold singularities, Tokyo J. Math. 41 (2018), no. 2, 385–406

    2018

  23. [31]

    Yajima, K., The Lp-boundedness of wave operators for four-dimensional Schr¨ odinger operators , in The physics and mathematics of Elliott Lieb—the 90th anniversary. Vol. II , 517–563, EMS Press, Berlin. Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A...

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