On the approximation by regular potentials of Schr\"odinger operators with point interactions

Artbazar Galtbayar; Kenji Yajima

REVIEW 3 major objections 4 minor 14 references

On the approximation by regular potentials of Schr\"odinger operators with point interactions

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that the wave operators of multi-center point-interaction Schrödinger operators are the strong scaling limits of the wave operators of regular potentials, with the point-coupling constants determined by the…

desk verdict Worth refereeing: the multi-center wave-operator convergence is new and the proof is mostly sound, but Lemma 4.9 states the resonance classification for N where M is needed, so the exceptional expansion needs a correction before publication. read the letter →

arxiv 1908.02936 v1 pith:PLTY2ZYY submitted 2019-08-08 math-ph math.MP

Artbazar Galtbayar , Kenji Yajima This is my paper · ORCID

classification math-phmath.MP MSC 47A1081Q1081Uxx

keywords pointinteractionswaveoperatorsSchrödingerzero-energyresonancescalinglimitsstrongresolventconvergencescatteringtheoryself-adjointextensions

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 that the scattering theory of Schrödinger operators with point interactions at finitely many sites is a genuine scaling limit of scattering by regular potentials. For potentials that are scaled as $\varepsilon^{-2}V_j((x-y_j)/\varepsilon)$ and tuned by couplings $\lambda_j(\varepsilon)$ with $\lambda_j(0)=1$ and $\lambda'_j(0)\ne 0$, the resolvents and wave operators of the scaled Hamiltonians converge strongly to those of the local point-interaction Hamiltonian $H_{\alpha,Y}$. The interaction parameter at site $y_j$ is forced to be $\alpha_j = -\lambda'_j(0)/|(a_j,\varphi_{j1})|^2$, where $\varphi_{j1}$ is the threshold-resonance state of the single-center Hamiltonian $-\Delta+V_j$. The result extends the known one-center case to any finite number of centers and provides a substantially simpler proof of the resolvent convergence previously treated in the standard monograph on point interactions. In short, point interactions are not an ad hoc model but the zero-range limit of ordinary quantum mechanics, including its time-dependent scattering.

What carries the argument

The central mechanism is the stationary representation of wave operators combined with a finite-rank perturbation formula for the scaled resolvent. After the unitary scaling (11), the relevant operator is $M_\varepsilon(\varepsilon k)=\Lambda(\varepsilon)B\tau_\varepsilon G_0(\varepsilon k)\tau_\varepsilon^*A$ on $L^2(\mathbb{R}^3)^N$, whose diagonal part $D_\varepsilon(\varepsilon k)$ collects single-center contributions and whose off-diagonal part $\varepsilon E_\varepsilon(\varepsilon k)$ describes propagation between centers. Here a threshold resonance is a bounded solution of $(-\Delta+V)u=0$ with the asymptotic $u(x)=L(\varphi)/|x|+O(|x|^{-2})$, i.e. a zero-energy resonance rather than a bound state. The key limit is Proposition 4.11: in the exceptional (threshold-resonance) case, $\varepsilon(1+D_\varepsilon(\varepsilon k))^{-1}$ converges to an explicit finite-rank operator $L$ built from the resonance state $\varphi_1$ and the dual state $\psi_1$, with scalar factor $-(\lambda'(0)+ik|(a,\varphi_1)|^2/4\pi)^{-1}$. The off-diagonal part tends to $|B\rangle\hat G(k)\langle A|$, a rank-$N$ operator with kernel $G_k(y_i-y_j)$, and Lemma 4.14's matrix reduction turns the inversion of $1+\langle A|L|B\rangle\hat G(k)$ into the inverse of the $N\times N$ matrix $\tilde\Gamma(k)$ whose entries are $(\alpha_j-ik/4\pi)\delta_{j\ell}-G_k(y_j-y_\ell)(1-\delta_{j\ell})$. That matrix is exactly the $\Gamma_{\alpha,Y}(k)$ of the point-interaction resolvent, which is why the same $\alpha$ and $Y$ appear in the limit.

What would settle it

For two identical resonant potential wells separated by $d>0$, set $\lambda_j(\varepsilon)=1+\varepsilon c_j$ and compute the two-center wave operator of $H_Y(\varepsilon)$ for small $\varepsilon$; the theorem predicts that the limit is $W^+_{\alpha,Y}$ with matrix $\Gamma_{\alpha,Y}(k)$ whose off-diagonal entries are $G_k(y_j-y_\ell)=e^{ikd}/(4\pi d)$. A direct numerical check of this identity for a fixed $u,v\in\mathcal D^*$ would settle the claim, since any deviation in the off-diagonal phase or in the predicted $\alpha_j=-c_j/|(a_j,\varphi_{j1})|^2$ falsifies Theorem 1.1.

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Extended reading notes

Core claim

Let $Y=\{y_1,\dots,y_N\}\subset\mathbb{R}^3$ and let $H_{\alpha,Y}$ be the self-adjoint local point-interaction operator with parameter $\alpha_j$ at $y_j$, defined by the resolvent formula (1)-(2). The paper's central claim, Theorem 1.1, is that for real potentials $V_j$ with $\langle x\rangle^2V_j\in L^p\cap L^q$ ($p<3/2$, $q>3$) and real $C^2$ couplings $\lambda_j(\varepsilon)$ with $\lambda_j(0)=1$ and $\lambda'_j(0)\ne 0$, whenever every $H_j=-\Delta+V_j$ has a threshold resonance at $0$, the operators $H_Y(\varepsilon)=-\Delta+\sum_{j=1}^N \lambda_j(\varepsilon)\varepsilon^{-2}V_j((x-y_j)/\varepsilon)$ converge in the strong resolvent sense to $H_{\alpha,Y}$, and the wave operators $W^\pm_{Y,\varepsilon}$ for the pair $(H_Y(\varepsilon),H_0)$ converge strongly to $W^\pm_{\alpha,Y}$. The proof derives an explicit formula for the limiting coupling $\alpha_j$ in terms of the resonance state $\varphi_{j1}$ and the derivative $\lambda'_j(0)$, and shows that the multi-center matrix $\Gamma_{\alpha,Y}(k)$ emerges from a finite-rank reduction of the scaled resolvent. Thus the scattering amplitudes of the point-interaction model are exactly the $\varepsilon\to 0$ limits of the scattering amplitudes of regular potentials concentrated near the points $y_j$.

Load-bearing premise

The load-bearing premise is that every single-center Hamiltonian $H_j=-\Delta+V_j$ has a threshold resonance at zero energy and that $\lambda'_j(0)\ne 0$; without the resonance a center is removed from the limiting point-interaction operator, and without the nonzero slope the coupling constant $\alpha_j$ would not be finite.

Editorial extensions

If this is right

The point-interaction Hamiltonian $H_{\alpha,Y}$ is not an independent model: its resolvent and wave operators are the strong limits of the corresponding objects for regular potentials, with the coupling constants $\alpha_j$ fixed by the resonance data of the single-center potentials.
Because the proof covers any finite $N$, multi-center point-interaction scattering can be computed or simulated by solving a regular potential problem with scaled potentials and then taking $\varepsilon\to 0$, rather than by solving the singular boundary-value problem directly.
When $\lambda_j(\varepsilon)=1$ for all $j$, the $\varepsilon$-independent $L^p$ bounds for the regular wave operators imply, by the convergence proved here, that the $\alpha=0$ point-interaction wave operators are bounded on $L^p(\mathbb{R}^3)$ for $1<p<3$ (Remark 1.2(i)).
If a center's potential lacks a threshold resonance, it simply drops out of the limiting operator (Remark 1.2(ii)); only the resonant centers contribute to the effective point-interaction model.
The stationary formulas converge uniformly on compact $k$-sets, so the wave-operator convergence holds strongly in $L^2(\mathbb{R}^3)$, not merely in a distributional or weighted sense.

Reading between the lines

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

The same finite-rank reduction suggests that changing the scaling exponent from $\varepsilon^{-2}$ to a different power would generically produce an energy-dependent coupling $\alpha_j(k)$, or a different class of boundary conditions; this is testable by repeating the threshold expansion with $\varepsilon^{-\beta}$.
The formula $\alpha_j=-\lambda'_j(0)/|(a_j,\varphi_{j1})|^2$ gives a practical tuning knob: by choosing $\lambda_j(\varepsilon)$ with a controlled slope at $\varepsilon=0$ one can dial the effective interaction strength from zero to arbitrarily large values, which may be useful in effective few-body models for ultracold atoms near a Feshbach resonance.
Since only centers with zero-energy resonances survive the limit, the construction offers a way to realize a desired set of interaction sites by positioning potential wells that are individually tuned to resonance; non-resonant wells act as inert background.
A natural extension would be to check whether the same wave-operator convergence holds for complex or non-self-adjoint couplings $\lambda_j(\varepsilon)$, or for time-dependent scalings, which would connect the result to the theory of open quantum systems.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper deserves a serious referee. The new part is Theorem 1.1(b), strong convergence of multi-center wave operators for general couplings alpha, which is not in [5] (only N=1, alpha=0) and not in [2] (resolvent only). If correct, it makes the scattering theory of point interactions literally the eps->0 limit of regular-potential scattering. The paper also gives a genuinely shorter resolvent proof.

What is genuinely good: the off-diagonal analysis in Lemma 4.2, the finite-rank matrix reduction via Deift's formula, and the explicit alpha_j = -lambda'_j(0)/|(a_j,phi_{j1})|^2. The proof is concrete and mostly careful, and the use of Jensen-Nenciu is appropriate. I see no circularity problem: the alpha's are read off from the potentials, not fit to the limit. The citation pattern is fine too; [5] is used as a black box for the point-interaction wave-operator formula, but that result is proved independently there.

Two corrections are needed. First, the more substantive one: Lemma 4.9(2) says that for phi in N, u = D0(a phi) satisfies (-Delta+V)u=0. The computation gives (-Delta+V)u = a(1+bD0a)phi, so the equation holds for phi in M, not phi in N. The proof itself assumes (1+bD0a)phi=0, which is exactly M. The case classification, the choice of phi_1, and the alpha formula all refer back to Lemma 4.9, so read literally the resonance direction is not established. The fix is to replace N by M throughout Lemma 4.9 and in Proposition 4.11's reference to it. The dual basis from Lemma 4.7 is the right one, so this looks like a genuine misstatement rather than a failed argument.

Second, Lemma 4.1(3) claims L2 convergence uniformly on compact subsets of the closed upper half-plane, but the Plancherel estimate in the proof has poles on the real axis and the statement is false there. The resolvent application only needs Im k > 0, and the wave-operator part uses (1), which is fine for real k. So this is repairable too.

One honest limitation: the threshold-resonance assumption on every center is load-bearing. If some center is regular, Remark 1.2(ii) removes it from the limiting operator, so Theorem 1.1 does not deliver all N centers. That is a limitation, but it is stated clearly and is not a hidden flaw.

Bottom line: this is a useful paper for people working on point interactions and scaling limits, and the wave-operator convergence is a real new result. I would not desk-reject it; I would send it to peer review and recommend acceptance after the N/M correction and the Lemma 4.1(3) range fix.

Referee Report

3 major / 4 minor

Summary. The paper studies N-center Schrödinger operators with local point interactions H_{\alpha,Y} in R^3. Under the assumption that each single-center potential V_j obeys condition (7) and has a threshold resonance at zero, and that \lambda_j(0)=1, \lambda_j'(0)\neq0, the authors prove that the scaled regular Hamiltonians H_Y(\varepsilon) converge to H_{\alpha,Y} in the strong resolvent sense, with \alpha_j=-\lambda_j'(0)/|(a_j,\phi_{j1})|^2, and that the corresponding wave operators W^\pm_{Y,\varepsilon} converge strongly to W^\pm_{\alpha,Y}. The proof combines the stationary representation from [5] with a resolvent identity in which the scaled potential is factored as AB, followed by threshold expansions of \varepsilon(1+M_\varepsilon(\varepsilon k))^{-1} via the Jensen\u2013Nenciu lemma. A simplified proof of the resolvent convergence is also presented.

Significance. If the proof is made fully rigorous, the result is a substantial generalization of the single-center case in [5] to N centers with arbitrary real coupling parameters, and it yields L^p-boundedness of the limiting wave operators. The paper is clearly organized, the factorization strategy is natural, and the reduction to the matrix \tilde\Gamma(k) is elegant. The use of the previously proved stationary representation is not circular. The main problems are local: one lemma is stated with the wrong kernel space, one scaling-back formula is misprinted, and one convergence statement is formulated for an overly large set of k. I regard all of these as fixable in revision.

major comments (3)

[4.1, Lemma 4.9 and Proposition 4.11] Lemma 4.9 is stated for N=Ker(1+aD0b), with u=D0(a\phi) claimed to satisfy (-\Delta+V)u=0. However, (-\Delta+V)D0(a\phi)=a(1+bD0a)\phi, so the equation holds for \phi\in M=Ker(1+bD0a), not for a general \phi\in N; the proof of part (2a) itself uses (1+bD0a)\phi=0. The case analysis and the choice of \phi_1 with L(\phi_1)>0 therefore apply to M, and Proposition 4.11's computation SQ1S=-\lambda'(0)S-(ik/4\pi)|(a,\phi_1)|^2\phi_1\otimes\psi_1 relies on \phi_1\in M. Since the formula for \alpha_j in Lemma 4.14, Eq. (69), depends on this \phi_1, the central threshold expansion is not justified as written. The intended repair is to replace N by M throughout Lemma 4.9 and the case discussion after it; the existing proofs then go through unchanged.
[4, Lemma 4.1(3), Eq. (25)] The statement claims L^2 convergence to |G_k\rangle\langle a,u| uniformly for k in compact subsets of C_+, and Lemma 3.4 defines C_+={k:\Im k\ge0}. For real k, G_k is not in L^2, so the claimed limit cannot hold; the Plancherel estimate in the proof, sup_{k\in\tilde K}|(|\xi|^2-k^2)^{-1}|\le C\langle\xi\rangle^{-2}, is false as \Im k\to0. The resolvent application only needs compact subsets of {\Im k>0}, and the proof is valid there. Please restrict the statement to \Im k>0 and correct the definition of C_+ in Lemma 3.4.
[4.2, Eq. (71)-(72)] The displayed formula (71) contains U_\varepsilon u and U_\varepsilon v, but Lemma 3.5(1) and Eq. (21) contain U_\varepsilon^*\tau u and U_\varepsilon^*\tau v. Lemma 4.1(1) concerns U_\varepsilon^*, so the convergence claimed from (23) and (70) is not justified for (71) as printed. The later sentence 'replacing u and v respectively by \tau u and \tau v' appears to compensate for the missing \tau, but the displayed vector should be U_\varepsilon^*\tau u; as written, the wave-operator convergence proof contains a gap in the scaling-back step.

minor comments (4)

[4.2, Lemma 4.14] In the displayed matrix after Eq. (69), the diagonal entries are written as \alpha_{n_2+1},\dots,\alpha_{n_1+n_2}; since the resonant centers are n_1+1,\dots,N=n_1+n_2, the first index should be n_1+1.
[4.1, Proposition 4.11] The compact set \Omega is not specified in the statement of Proposition 4.11; please state explicitly whether it is a compact subset of {\Im k>0} or of R\setminus\{0\}, and whether the uniformity is over real or complex k.
[4.1, Lemma 4.9(2a)] In the proof of part (2a), the phrase 'If (1+bD0a)\phi=0' introduces an assumption that is not part of the lemma statement; this is part of the M/N confusion and should be resolved by the correction described in the first major comment.
[1, Remark 1.2(i)] The sentence 'W_{Y,\varepsilon} converges to W_{\alpha=0,Y} weakly in L^p' should say 'weakly' in the sense of pairings with L^{p'} functions; the wording is slightly terse but the intended meaning is clear.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: coupling parameters α_j are derived from explicit ε→0 asymptotics of the regular Hamiltonians, not fitted to the point-interaction target.

full rationale

The paper's central claim (Theorem 1.1(b)) is a scaling-limit theorem: for regular potentials satisfying the stated resonance hypothesis, the wave operators W^±_{Y,ε} are shown to converge to the wave operators W^±_{α,Y} of the point-interaction Hamiltonian, with the coupling constants α_j determined by the explicit formula α_j = -λ'_j(0)/|(a_j,φ_{j1})|^2 (Lemma 4.14, Eq. (69)). This α_j is computed from the single-center Schrödinger operator -Δ+V_j and the scaling parameter λ_j; it is not obtained by matching the point-interaction resolvent or wave operator, so there is no fitted-input-called-prediction pattern. The only material citation to the authors' own prior work is Lemma 3.2, the stationary representation of W^+_{α,Y} from [5]; that formula is used as a representation of the independently defined wave operator (4), it is stated in full, and it does not assert the convergence being proved. The proof's core expansion (Proposition 4.11) is derived from the Jensen–Nenciu operator lemma and the explicit expansion of bG0(kε)a, not from the target result. The acknowledged limitation in Remark 1.2(ii) — points whose potentials lack a threshold resonance are removed from the limiting operator — is an honest statement of the hypothesis's role, not a circular step. The apparent N/M mismatch in Lemma 4.9 is a possible correctness defect, but it is not a reduction of the conclusion to the hypothesis, and therefore does not constitute circularity.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

No parameters are fitted: V_j and λ_j are arbitrary inputs satisfying stated assumptions, and α_j are derived. The proof rests on standard scattering theory results and on the stated threshold-resonance assumption. No invented entities.

assumptions (9)

standard math Limiting absorption principle for the free resolvent
Invoked in Lemma 3.4 to extend boundary values of A τ_ε G0(k+iδ) u to real k; cited to Agmon [1] and Kuroda [11].
standard math Absence of positive eigenvalues for H(ε) and H_j(ε)
Used in Lemma 3.4(3) and in Section 4 to ensure invertibility of 1+λ_j(ε)b_jG0(kε)a_j for k≠0; cited to Koch-Tataru [10].
standard math Jensen-Nenciu threshold resolvent expansion
Lemma 4.10 from [8] is the basis for the expansion of ε(1+D_ε(εk))^{-1} in Proposition 4.11.
standard math Deift's commutation formula
Lemma 4.12 from [4] is used to reduce the inverse of 1+L|B⟩Ĝ(k)⟨A| on H^{(N)} to a matrix inverse.
standard math Stationary representation of point-interaction wave operators
Lemma 3.2 from [5] provides the target expression for W^+_{α,Y} against which the limit is compared; this is an external published result.
standard math Resolvent formula for point interactions
Equation (1) from Albeverio et al. [2] defines H_{α,Y} and is the benchmark for resolvent convergence in part (a).
standard math Existence and completeness of wave operators for short-range potentials
Used in Section 3 to justify W^±_{Y,ε}; cited to Kuroda [12] and Reed-Simon [13].
domain assumption Each single-center potential V_j has a threshold resonance at 0 (Theorem 1.1 assumption (3))
Needed for the exceptional-type expansion of ε(1+D_ε(εk))^{-1}; if absent, the corresponding point interaction is removed from the limit (Remark 1.2(ii)).
domain assumption λ_j are C^2 with λ_j(0)=1 and λ'_j(0)≠0 (assumption (2))
Ensures the leading ε-order term in (40) is nonzero and produces finite α_j in (69).

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Pith. "Pith review of On the approximation by regular potentials of Schr\"odinger operators with point interactions." pith.science (2026). https://pith.science/paper/PLTY2ZYY

@misc{pith2026190802936,
  author       = {Pith},
  title        = {Pith review of: On the approximation by regular potentials of Schr\"odinger operators with point interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PLTY2ZYY}},
  note         = {Machine review of arXiv:1908.02936}
}

read the original abstract

We prove that the wave operators for Schr\"odinger operators with multi-center local point interactions are the scaling limits of the ones for Schr\"odinger operators with regular potentials. We simultaneously present a proof of the corresponding well-known result for the resolvent which substantially simplifies the one by Albeverio et al.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[5]

Dell’Antonio, A

G. Dell’Antonio, A. Michelangeli, R. Scandone and K. Yajima, The Lp- boundedness of wave operators for the three dimensional mul ti-centre point interaction, Ann. Inst. H. Poincar´ e 19 (2018), 283–322. 27

work page 2018
[2]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics , AMS Chelsea Publishing, (2004),

work page 2004
[1]

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), 151–218

work page 1975
[3]

Cornean, A

H. Cornean, A. Michelangeli and K. Yajima, Two dimensional Schr¨ odinger operators with point interactions: Threshol d expansions, zero modes and Lp-boundedness of wave operators . Reviews in Math- ematical Physics Vol. 31, No. 04, 1950012 (2019)

work page 2019
[4]

P. A. Deift, Applications of a commutation formula , Duke Math. J. bf 45 (1978), 267–309

work page 1978
[6]

Holden, Konvergens mot punkt-interaksjoner (In Norwegian)

H. Holden, Konvergens mot punkt-interaksjoner (In Norwegian). Cand. real.thesis, University of Oslo, Norway, 1981. KK

work page 1981
[7]

A. D. Ionescu and W. Schlag, Agmon-Kato-Kuroda theorems for a large class of perturbations , Duke Math. J. 131 No. 3 (2006), 397440

work page 2006
[8]

Jensen and G

A. Jensen and G. Nenciu, A uniﬁed approach to resolvent expansions at thresholds, Reviews in Mathematical Physics, 13, No. 6 (2001) 717-754

work page 2001

Show all 14 references

[9]

Heidelberg- New-York-Tokyo, 1966

T.Kato, Perturbation of Linear Operators , Springer Verlag. Heidelberg- New-York-Tokyo, 1966

1966
[10]

Koch and D

H. Koch and D. Tataru. Carleman estimates and absence of embedded eigenvalues. Comm. Math. Phys., 267 (2006), 419-449

2006
[11]

S. T. Kuroda, An introduction to scattering theory , vol. 51. Lecture Notes Series, Aarhus Universitet, Matematisk Institut, Aarhus, 1978

1978
[12]

S. T. Kuroda, On the existence and the unitary property of the scattering operator, Nuovo Cimento, 12 (1959)

1959
[13]

Reed and B

M. Reed and B. Simon, Method of modern mathematical physics IV, Scattering theory, New York, Academic Press, 1979

1979
[14]

Yajima, L1 and L∞-boundedness of wave operators for three di- mensional Schr¨ odinger operators with Threshold Singularities, Tokyo J

K. Yajima, L1 and L∞-boundedness of wave operators for three di- mensional Schr¨ odinger operators with Threshold Singularities, Tokyo J. Math., 41 (2018), 385-406. Artbazar Galtbayar Center of Mathematics for Applications and Department of Applied Mathematics National Univers...

2018

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [5]

Dell’Antonio, A

G. Dell’Antonio, A. Michelangeli, R. Scandone and K. Yajima, The Lp- boundedness of wave operators for the three dimensional mul ti-centre point interaction, Ann. Inst. H. Poincar´ e 19 (2018), 283–322. 27

work page 2018

[2] [2]

Albeverio, F

S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, H. Holden, Solvable Models in Quantum Mechanics , AMS Chelsea Publishing, (2004),

work page 2004

[3] [1]

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), 151–218

work page 1975

[4] [3]

Cornean, A

H. Cornean, A. Michelangeli and K. Yajima, Two dimensional Schr¨ odinger operators with point interactions: Threshol d expansions, zero modes and Lp-boundedness of wave operators . Reviews in Math- ematical Physics Vol. 31, No. 04, 1950012 (2019)

work page 2019

[5] [4]

P. A. Deift, Applications of a commutation formula , Duke Math. J. bf 45 (1978), 267–309

work page 1978

[6] [6]

Holden, Konvergens mot punkt-interaksjoner (In Norwegian)

H. Holden, Konvergens mot punkt-interaksjoner (In Norwegian). Cand. real.thesis, University of Oslo, Norway, 1981. KK

work page 1981

[7] [7]

A. D. Ionescu and W. Schlag, Agmon-Kato-Kuroda theorems for a large class of perturbations , Duke Math. J. 131 No. 3 (2006), 397440

work page 2006

[8] [8]

Jensen and G

A. Jensen and G. Nenciu, A uniﬁed approach to resolvent expansions at thresholds, Reviews in Mathematical Physics, 13, No. 6 (2001) 717-754

work page 2001