A New Proof of the Sharp Gordon's Lemma: No Eigenvalues for Schr\"odinger Operators with Almost Repetition Potentials

Wencai Liu

REVIEW 2 major objections 4 minor 30 references

A New Proof of the Sharp Gordon's Lemma: No Eigenvalues for Schr\"odinger Operators with Almost Repetition Potentials

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves that a bounded potential with $\gamma$-repetitions and repetition rate $\gamma$ greater than the transfer-matrix growth rate $L(E)$ has no square-summable eigenfunctions, via a new Wronskian–transfer-matrix argument.

desk verdict New Wronskian proof of a known sharp Gordon lemma; the method is real, but one linear-algebra step needs a fix before the proof is valid as written. read the letter →

arxiv 2501.03157 v1 pith:DZE2BZJK submitted 2025-01-06 math-ph math.MPmath.SP

Wencai Liu This is my paper · ORCID

classification math-phmath.MPmath.SP MSC 81Q1047B3939A70

keywords discreteSchrödingeroperatoralmostrepetitionpotentialssharplemmaabsenceofeigenvaluestransfermatricesLyapunovexponentMathieu

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 supplies a new proof of the sharp repetition lemma for one-dimensional discrete Schrödinger operators: a bounded potential that almost repeats itself at distances $k_n$, with error at most $e^{-\gamma|k_n|}$, cannot support an $\ell^2(\mathbb{Z})$ eigenfunction once the repetition rate $\gamma$ exceeds the transfer-matrix growth rate $L(E)$. Earlier repetition arguments proved this only with a factor 2, and the sharp threshold was previously reached by different analytic methods; here the threshold is recovered by adapting the Wronskian strategy used for reflective repetition potentials. The result gives a sharp criterion for absence of point spectrum, and it applies directly to quasi-periodic potentials, yielding the arithmetic delocalization criterion for the almost Mathieu operator when the frequency has positive $\beta(\alpha)$. A reader should care because it isolates the exact mechanism by which almost-repetition kills eigenvalues: the Wronskian stays exponentially small while the shifted solution pair itself decays, forcing a contradiction.

What carries the argument

The central machinery is the Wronskian $W(f, g)(k) = f(k+1)g(k) - f(k)g(k+1)$ together with the multi-step transfer matrices $T_{m,j}(E)$ that evolve solution vectors $(u(k), u(k-1))$ across an interval. The argument first shows, by summing the potential differences and using the $\ell^2$ normalization of $u$ and $u_n$, that $|W(u, u_n)(-1)| \leq e^{-\gamma k_n}$; then it expands $(u(k_n), u(k_n - 1)) = a_n (u(0), u(-1)) + b_n (-u(-1), u(0))$, so that $b_n$ is exponentially small and $a_n$ tends to zero. Multiplying by $T_{0,k_n}$ and replacing it by $T_{-k_n,0}$ via Lemma 2.1, the comparison $\|T_{m,j}(E) - T_{m+k_n,j+k_n}(E)\| \leq C e^{-\gamma k_n} e^{(L(E)+\varepsilon)|m-j|}$, turns the expansion into the vector $(u(-k_n), u(-k_n - 1))$, which is $o(1)$, contradicting the nonzero left-hand side $(u(0), u(-1))$.

What would settle it

For a concrete almost-repetition potential, for example an almost Mathieu-type potential with $\beta(\alpha) > 0$ and $|\lambda| < e^{\beta(\alpha)}$, compute $\|T_{0,k_n}(E) - T_{-k_n,0}(E)\|$ and check whether it is bounded by $C e^{-\gamma k_n} e^{(L(E)+\varepsilon)|m-j|}$; exceeding that bound would break the proof's key comparison, while finding a nonzero $\ell^2$ solution for such parameters would refute Theorem 1.1 directly.

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Extended reading notes

Core claim

The paper establishes Theorem 1.1: if $V$ is a bounded potential satisfying $|V(k_n + k) - V(k)| \leq e^{-\gamma|k_n|}$ for a sequence $k_n \to \pm\infty$, and if $\gamma > L(E)$, then the eigen-equation $Hu = Eu$ has no $\ell^2(\mathbb{Z})$ solutions. The proof is new: instead of the classical three-block repetition argument, it compares a candidate eigenfunction $u$ with its shifted copy $u_n(k) = u(k_n + k)$ through the Wronskian $W(u, u_n)$, uses the exponential closeness of $V$ and $V_n$ to bound $W$ by $e^{-\gamma k_n}$, and then expands the shifted vector $(u(k_n), u(k_n - 1))$ in a two-vector basis. The expansion coefficients satisfy $b_n = O(e^{-\gamma k_n})$ and $a_n = o(1)$, while a transfer-matrix comparison shows that applying $T_{0,k_n}$ yields $(u(-k_n), u(-k_n - 1))$, which tends to zero; this contradicts the fixed nonzero vector $(u(0), u(-1))$. Corollaries follow for $\kappa$-Hölder continuous quasi-periodic potentials (no eigenvalues when $L(E) < \kappa\beta(\alpha)$) and for the almost Mathieu operator (no eigenvalues when $|\lambda| < e^{\beta(\alpha)}$ given $\beta(\alpha) > 0$).

Load-bearing premise

The argument rests on the estimate, cited to another paper rather than proved here, that shifting a long block of the potential by the almost-repetition distance changes the associated transfer matrices by at most $C e^{-\gamma k_n} e^{(L(E)+\varepsilon)|m-j|}$; if that comparison is not exponentially small, the final contradiction collapses.

Editorial extensions

If this is right

For every bounded $\gamma$-repetition potential, the set of energies with $\gamma > L(E)$ is disjoint from the point spectrum; on that set the operator has purely continuous spectrum.
For $\kappa$-Hölder continuous quasi-periodic potentials, $H$ has no eigenvalues in the regime $\{E : L(E) < \kappa\beta(\alpha)\}$.
For the almost Mathieu operator with $\beta(\alpha) > 0$, there are no eigenvalues whenever $|\lambda| < e^{\beta(\alpha)}$, which is the sharp delocalization criterion.
The threshold $\gamma > L(E)$ is optimal: the almost Mathieu example shows eigenvalues can appear when the inequality is not satisfied, so the factor 2 in the earlier repetition argument is not an artifact.

Reading between the lines

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

The same Wronskian–expansion argument is likely to extend to block Jacobi or CMV matrices, since only the transfer-matrix comparison and the $\ell^2$ normalization are used; verifying the analogous lemma would be a concrete test.
Because the proof tracks convergence only up to $o(1)$, a natural extension is a quantitative version that records the exponential decay rates of the coefficients $a_n$ and $b_n$, yielding explicit bounds on any hypothetical eigenfunction.
The proof's key estimate, Lemma 2.1, is cited to earlier work rather than derived in full; making that telescoping argument explicit would make the paper self-contained and would clarify the role of the constant $C$.

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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

Wencai Liu's paper is a new proof of a theorem that already exists. The theorem — no ℓ2 solutions when γ > L(E) for almost repetition potentials — was proved by Avila-You-Zhou, and the reflective analog by Jitomirskaya-Liu. The paper says so in the introduction, which is the right and honest way to position a proof paper. What is new is the method: a Wronskian estimate plus transfer-matrix comparison that yields the sharp threshold. That is a genuine contribution, and the proof is demonstrably different from the earlier ones. The Wronskian bound (12)–(14) and the coefficient bounds (18)–(19) are the core, and they are correct once one has the proper setup.

There is one real gap, and it is not merely a citation issue. Equation (17) only gives |u(0)|²+|u(-1)|²>0, but the decomposition (16) requires the denominator u(0)²+u(-1)² in Cramer's rule to be nonzero. For complex eigenfunctions, (17) does not imply that; u(0)=1, u(-1)=i is a concrete counterexample. The stress-test note is right that this is a false inference, not a missing citation. The fix is short: a nonzero ℓ² solution cannot have u(k)²+u(k−1)²=0 for all k without forcing |u(k)| constant, so a shift of origin puts the denominator away from zero. The repetition condition and L(E) are shift-invariant, so this is harmless. Alternatively, for real potentials one first reduces to a real eigenfunction. But as written, the proof has a load-bearing step that needs repair.

Minor point: Lemma 2.1 is cited to [28, Lemma 4.5] rather than proved. That is standard telescoping, so acceptable in principle, but the paper should state the precise constants and the induction. It would take half a page. The citation pattern elsewhere is fine: the paper credits [1] and [19] for the known results and does not overclaim.

Overall: the contribution is methodological, not a new spectral fact. It deserves a serious referee. With the basis issue fixed and Lemma 2.1 given a proper statement, this is a solid paper. I would not desk-reject it.

Referee Report

2 major / 4 minor

Summary. The paper presents a proof of a sharp Gordon-type lemma for one-dimensional discrete Schrödinger operators with γ-repetition potentials, claiming that the eigen-equation Hu = Eu has no ℓ2(Z) solutions whenever γ > L(E), where L(E) is the transfer-matrix growth rate defined in equation (4). The proof combines a Wronskian estimate (equations (12)–(15)) with a transfer-matrix comparison lemma (Lemma 2.1) to derive a contradiction from an assumed ℓ2 eigenfunction. The paper then applies the theorem to Hölder-continuous quasi-periodic potentials and recovers the sharp almost Mathieu transition of Avila–You–Zhou.

Significance. If made fully rigorous, the paper would provide a concise new proof of a sharp Gordon lemma that is of genuine interest in the spectral theory of one-dimensional operators. The Wronskian mechanism leading to the exponential decay in (15) is elegant, and the overall structure is transparent. The corollaries for quasi-periodic and almost Mathieu operators are valuable applications. However, the current manuscript contains a load-bearing gap in the complex-eigenfunction case and an omitted proof of a key lemma, so the central claim is not yet fully supported.

major comments (2)

[§2, Lemma 2.1] The passage from (15) and (17) to (18) is not justified for complex-valued eigenfunctions. Solving (16) by Cramer's rule gives b_n = det((u(0),u(-1))^T, (u(k_n),u(k_n-1))^T) / (u(0)^2 + u(-1)^2), so the bound (15) only yields |b_n| ≤ e^{-γ k_n} / |u(0)^2 + u(-1)^2|. Condition (17) ensures the denominator is nonzero, but not that it is bounded away from zero; for instance u(0)=1, u(-1)=i satisfies (17) while u(0)^2 + u(-1)^2 = 0, in which case the two vectors in (16) are linearly dependent and the representation is not unique. Consequently, (18), and hence the estimate (22) and the final contradiction (24), do not follow as written. The fix is to shift the origin to a site with u(k)^2 + u(k-1)^2 ≠ 0: if no such site exists, then u(k)^2 + u(k-1)^2 = 0 for all k, which forces |u(k)| to be constant, contradicting u ∈ ℓ2(Z) unless u ≡ 0. Condition (2) and L(E) are invariant under such a translation. Alternatively, for real potentials one can first reduce to a real eigenfunction. This step must be included in the revised proof.
[§2, Lemma 2.1] Lemma 2.1 is load-bearing: it is used at equation (23) to replace T_{0,k_n} by T_{-k_n,0} and thereby produce the contradiction (24). Yet the proof is not carried out; the text merely says it follows from (8) by “standard telescoping” and cites [28, Lemma 4.5]. Moreover, as stated the lemma gives a comparison with a shift by +k_n, whereas (23) uses a shift by −k_n. The argument should either state the lemma for shifts ±k_n (which the telescoping supplies using (2) for all k) or explain the reduction, and a complete proof should be included. If the uniformity in m,j or the exact constant C is not as stated, the exponential decay in (23) is unsupported.

minor comments (4)

[Title page] The running title on page 1 contains a typo: “EIGENV ALUES” should be “EIGENVALUES”.
[§2, Lemma 2.1 and throughout] The matrix norm in Lemma 2.1 and elsewhere should be specified explicitly as the operator norm, and the dependence of the constant C on E and V should be stated.
[§2, Eq. (13)] The sentence “Notice that ∑_k |W(u, un)(k)| ≤ 2” should specify that the sum is over all k ∈ Z and indicate that the bound follows from Cauchy–Schwarz applied to the two products u(k+1)un(k) and u(k)un(k+1) separately.
[§1, Eq. (4) and §2, Eq. (20)] The orientation convention for the transfer matrices T_{m,j} when j > m is left implicit; stating the inverse convention explicitly would remove ambiguity in equations (20) and (21).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives Theorem 1.1 from definitions and standard transfer-matrix estimates, with no fitted inputs or self-citation chain carrying the conclusion.

full rationale

The derivation of Theorem 1.1 does not use the target result as an input. The argument assumes an l2 eigenfunction, uses the γ-repetition condition (2) to compare V(k) with V(k+k_n), and applies a Wronskian estimate plus transfer-matrix growth bounds to force a contradiction. No parameter is fitted to the conclusion, and no quantity is renamed as a prediction. The only externally cited load-bearing ingredient is Lemma 2.1, whose proof is deferred to Simon [28, Lemma 4.5]; that citation is to an external standard telescoping estimate, not to the present authors' prior work, and it does not presuppose the absence of eigenvalues. The paper also builds on Jitomirskaya-Simon [22] and Jitomirskaya-Liu [19], but it uses them for context and technique, not as the justification of the theorem's conclusion. The reviewer's main skeptical concern, that the inference from (17) to (18) may fail for complex u unless one first shifts to a site with u(0)^2+u(-1)^2 ≠ 0, is a correctness or rigor gap in the written proof, not a circularity: it does not make the theorem equivalent to its own assumptions. Since no step reduces by construction to the target claim, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof rests only on standard transfer-matrix and Wronskian identities, the boundedness of V, and one unstated translation normalization. There are no free parameters and no invented entities.

assumptions (4)

standard math The transfer-matrix growth estimate (8): ||T_{m,j}(E)|| ≤ C e^{(L(E)+ε)|m−j|} for all m,j, with C depending on ε and V.
Used in equations (22) and (23) to bound error terms; follows from the definition of L(E) and the boundedness of V.
standard math Lemma 2.1: ||T_{m,j}(E) − T_{m+k_n,j+k_n}(E)|| ≤ C e^{−γ k_n} e^{(L+ε)|m−j|}, proved by standard telescoping and cited to [28, Lemma 4.5].
The paper does not carry out the telescoping; this comparison is load-bearing for the replacement in step (23).
standard math The Wronskian identity W(k) − W(k−1) = (V(k) − V_n(k)) u(k) u_n(k) for two solutions of Schrödinger equations with potentials V and V_n.
Used after (10)-(11) to obtain the exponential decay of the Wronskian in (12)-(14).
domain assumption It is legitimate to translate the origin so that |u(0)|^2+|u(−1)|^2 > 0; translation preserves the γ-repetition condition (2).
The proof asserts (17) directly from nontriviality; this requires an unstated coordinate translation.

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Cite this review

Pith. "Pith review of A New Proof of the Sharp Gordon's Lemma: No Eigenvalues for Schr\"odinger Operators with Almost Repetition Potentials." pith.science (2026). https://pith.science/paper/DZE2BZJK

@misc{pith2026250103157,
  author       = {Pith},
  title        = {Pith review of: A New Proof of the Sharp Gordon's Lemma: No Eigenvalues for Schr\"odinger Operators with Almost Repetition Potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZE2BZJK}},
  note         = {Machine review of arXiv:2501.03157}
}

read the original abstract

Building on the work of Jitomirskaya-Simon and Jitomirskaya-Liu, who established the absence of eigenvalues for Schr\"odinger operators with almost reflective repetition potentials, we provide a new proof of the sharp Gordon's lemma, which asserts the absence of eigenvalues for Schr\"odinger operators with almost repetition potentials.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

30 extracted references · 20 canonical work pages

[1]

Avila, J

A. Avila, J. You, and Q. Zhou. Sharp phase transitions for the alm ost Mathieu operator. Duke Math. J. , 166(14):2697–2718, 2017

2017
[19]

Jitomirskaya and W

S. Jitomirskaya and W. Liu. Universal reﬂective-hierarchical s tructure of quasiperiodic eigenfunctions and sharp spectral transition in phas e. J. Eur. Math. Soc. (JEMS) , 26(8):2797–2836, 2024

work page 2024
[2]

Avron and B

J. Avron and B. Simon. Singular continuous spectrum for a class o f almost periodic Jacobi matrices. Bull. Amer. Math. Soc. (N.S.) , 6(1):81–85, 1982

work page 1982
[3]

Bourgain and S

J. Bourgain and S. Jitomirskaya. Continuity of the Lyapunov exp onent for quasiperiodic operators with analytic potential. volume 108, pages 1 203–
[4]

Cedzich and J

C. Cedzich and J. Fillman. Absence of bound states for quantum w alks and CMV matrices via reﬂections. J. Spectr. Theory , 14(4):1513–1536, 2024

work page 2024
[5]

D. Damanik. Gordon-type arguments in the spectral theory of one- dimensional quasicrystals. In Directions in mathematical quasicrystals , vol- ume 13 of CRM Monogr. Ser. , pages 277–305. Amer. Math. Soc., Provi- dence, RI, 2000

work page 2000
[6]

D. Damanik. Absence of eigenvalues for a class of Schr¨ odinger o perators on the strip. Forum Math., 14(5):797–806, 2002

work page 2002
[7]

D. Damanik. A version of Gordon’s theorem for multi-dimensional Schr¨ odinger operators.Trans. Amer. Math. Soc. , 356(2):495–507, 2004

work page 2004

Show all 30 references

[8]

Damanik and R

D. Damanik and R. Killip. Reﬂection symmetries of almost periodic fun c- tions. J. Funct. Anal. , 178(2):251–257, 2000

2000
[9]

L. Fang, S. Guo, Y. Peng, and F. Wang. Quantitative version of G ordon’s lemma for Hamiltonian with ﬁnite range. Linear Algebra Appl., 687:91–107, 2024

2024
[10]

J. Fillman. Purely singular continuous spectrum for Sturmian CMV matri- ces via strengthened Gordon lemmas. Proc. Amer. Math. Soc. , 145(1):225– 239, 2017

2017
[11]

L. Ge, J. You, and Q. Zhou. Structured quantitative almost re ducibility and its applications. arXiv preprint arXiv:2407.05490 , 2024

2024
[12]

A. Y. Gordon. The point spectrum of the one-dimensional Schr ¨ odinger operator. Uspekhi Matematicheskikh Nauk , 31(4):257–258, 1976

1976
[13]

A. Hof, O. Knill, and B. Simon. Singular continuous spectrum for p alin- dromic Schr¨ odinger operators.Comm. Math. Phys. , 174(1):149–159, 1995

1995
[14]

Jitomirskaya and I

S. Jitomirskaya and I. Kachkovskiy. L2-reducibility and localization for quasiperiodic operators. Math. Res. Lett. , 23(2):431–444, 2016

2016
[15]

Jitomirskaya and I

S. Jitomirskaya and I. Kachkovskiy. Sharp arithmetic delocaliza tion for quasiperiodic operators with potentials of semi-bounded variation. arXiv preprint arXiv:2408.16935 , 2024

2024 arXiv
[16]

Jitomirskaya and S

S. Jitomirskaya and S. Koci´ c. Spectral theory of Schr¨ odinger operators over circle diﬀeomorphisms. Int. Math. Res. Not. IMRN , (13):9810–9829, 2022. A NEW PROOF OF THE SHARP GORDON’S LEMMA 7

2022
[17]

Jitomirskaya and W

S. Jitomirskaya and W. Liu. Arithmetic spectral transitions for the Mary- land model. Comm. Pure Appl. Math. , 70(6):1025–1051, 2017

2017
[18]

Jitomirskaya and W

S. Jitomirskaya and W. Liu. Universal hierarchical structure o f quasiperiodic eigenfunctions. Ann. of Math. (2) , 187(3):721–776, 2018

2018
[20]

Jitomirskaya, W

S. Jitomirskaya, W. Liu, and L. Mi. Sharp palindromic criterion for semi- uniform dynamical localization. arXiv preprint arXiv:2410.21700 , 2024

2024 arXiv
[21]

Jitomirskaya, S

S. Jitomirskaya, S. A. Molchanov, B. Simon, and B. R. Vainberg. Alexander Gordon. J. Spectr. Theory , 9(4):1157–1164, 2019

2019
[22]

Jitomirskaya and B

S. Jitomirskaya and B. Simon. Operators with singular continuou s spec- trum. III. Almost periodic Schr¨ odinger operators. Comm. Math. Phys. , 165(1):201–205, 1994

1994
[23]

Jitomirskaya and F

S. Jitomirskaya and F. Yang. Singular continuous spectrum for singular potentials. Comm. Math. Phys. , 351(3):1127–1135, 2017

2017
[24]

Jitomirskaya and S

S. Jitomirskaya and S. Zhang. Quantitative continuity of singula r continu- ous spectral measures and arithmetic criteria for quasiperiodic Sc hr¨ odinger operators. J. Eur. Math. Soc. (JEMS) , 24(5):1723–1767, 2022

2022
[25]

W. Liu. Small denominators and large numerators of quasiperiod ic Schr¨ odinger operators.Peking Mathematical Journal to appear
[26]

W. Liu. Continuous quasiperiodic Schr¨ odinger operators with G ordon type potentials. J. Math. Phys. , 59(6):063501, 6, 2018

2018
[27]

B. Simon. Almost periodic Schr¨ odinger operators: a review. Advances in Applied Mathematics , 3(4):463–490, 1982

1982
[28]

B. Simon. Almost periodic Schr¨ odinger operators. IV. The Maryland model. Ann. Physics , 159(1):157–183, 1985

1985
[29]

J. You, S. Zhang, and Q. Zhou. Point spectrum for quasi-perio dic long range operators. J. Spectr. Theory , 4(4):769–781, 2014. (W. Liu) Department of Mathematics, Texas A&M University, College S ta- tion, TX 77843-3368, USA Email address : liuwencai1226@gmail.com; wencail@tamu.edu

2014
[1218]

Dedicated to David Ruelle and Yasha Sinai on the occasio n of their 65th birthdays

2002. Dedicated to David Ruelle and Yasha Sinai on the occasio n of their 65th birthdays

2002

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Avila, J

A. Avila, J. You, and Q. Zhou. Sharp phase transitions for the alm ost Mathieu operator. Duke Math. J. , 166(14):2697–2718, 2017

2017

[2] [19]

Jitomirskaya and W

S. Jitomirskaya and W. Liu. Universal reﬂective-hierarchical s tructure of quasiperiodic eigenfunctions and sharp spectral transition in phas e. J. Eur. Math. Soc. (JEMS) , 26(8):2797–2836, 2024

work page 2024

[3] [2]

Avron and B

J. Avron and B. Simon. Singular continuous spectrum for a class o f almost periodic Jacobi matrices. Bull. Amer. Math. Soc. (N.S.) , 6(1):81–85, 1982

work page 1982

[4] [3]

Bourgain and S

J. Bourgain and S. Jitomirskaya. Continuity of the Lyapunov exp onent for quasiperiodic operators with analytic potential. volume 108, pages 1 203–

[5] [4]

Cedzich and J

C. Cedzich and J. Fillman. Absence of bound states for quantum w alks and CMV matrices via reﬂections. J. Spectr. Theory , 14(4):1513–1536, 2024

work page 2024

[6] [5]

D. Damanik. Gordon-type arguments in the spectral theory of one- dimensional quasicrystals. In Directions in mathematical quasicrystals , vol- ume 13 of CRM Monogr. Ser. , pages 277–305. Amer. Math. Soc., Provi- dence, RI, 2000

work page 2000

[7] [6]

D. Damanik. Absence of eigenvalues for a class of Schr¨ odinger o perators on the strip. Forum Math., 14(5):797–806, 2002

work page 2002

[8] [7]

D. Damanik. A version of Gordon’s theorem for multi-dimensional Schr¨ odinger operators.Trans. Amer. Math. Soc. , 356(2):495–507, 2004

work page 2004