The essential spectrum of Schr\"{o}dinger operators on lattices

Steffen Roch; Vladimir S. Rabinovich

REVIEW

The essential spectrum of Schr\"{o}dinger operators on lattices

Vladimir S. Rabinovich, Steffen Roch

Mathematical Physics · Submitted 27 Feb 2006 · 28 references

Pith's verdict: Useful limit-operator tour through lattice Schrödinger essential spectra, but Theorem 19 overreaches: its exact inf/sup identities do not follow and fail for single-site potentials.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 5 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDmath-ph/0602061 (version 2)

Pith IDAER6MRQQ, Pith's permanent record of this paper and its review

CitationSteffen Roch, Vladimir S. Rabinovich. "The essential spectrum of Schr\"{o}dinger operators on lattices." arXiv:math-ph/0602061 (2006). https://pith.science/paper/AER6MRQQ

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Thomas—

This one is worth reading, but keep a red pen near Section 3.5. The paper is a clean application of the limit-operators method, developed in the authors' earlier work, to discrete Schrödinger operators. That method is not new, but the applications are: explicit essential-spectrum intervals for slowly oscillating potentials, periodic/semi-periodic potentials, waveguides, and one-dimensional potentials with infinitely many discontinuities. Theorem 16 is correct under the stated interval/gap condition; the enumeration of limit operators in its proof is sound, and the condition is exactly what makes it work.

The soft spot is Theorem 19. The identity (34), sp_ess(H)=sp(H1)∪sp(H2)∪sp(H12), is the honest HWZ-type statement and it is fine. But the exact inf/sup formulas in Theorem 19 do not follow from the inclusions in Proposition 18, and they are false. The inclusions are one-sided: they bound the bottom of each subsystem spectrum from below and the top from above; they do not locate it. Take W1=-V0δ_0 with V0 large, W2=W12=0. The single-site operator h1=(1/2m1)Δ-V0δ_0 has a negative bound state E0 with E0>-V0, because the discrete kinetic energy is nonnegative. The limit operator H1 contributes E0+[0,6/m2] to the essential spectrum, so inf sp_ess(H)=E0>-V0, while min(inf W1,0,0)=-V0. A positive bump breaks the upper equality in the same way. So Theorem 19 should be weakened to inequalities, or restated in terms of the actual bottom/top of the subsystem spectra. The abstract and introduction only promise estimates here, so the overstatement is localized, but a referee should catch it.

Minor typos (the gamma index in Case (d) of Theorem 16, and the swapped 6/m constants in (36)-(37)) are harmless. The self-citation pattern is transparent and legitimate—the core essential-spectrum identity comes from their monograph, and the applications are new.

Bottom line: a serious referee should engage with this. The main body is a useful reference for lattice essential spectra, and the flaw in Theorem 19 is correctable. After revision it would be a solid paper; as it stands, the three-particle section overclaims.

Referee report

Major objections (2)

[Section 3.5, Theorem 19 and the display after Proposition 18] The equalities for inf spess(H) and sup spess(H) are not established by the proof. Proposition 18 only yields sp(Hj) ⊆ [inf Wj, sup Wj + m] for j=1,2 and sp(H12) ⊆ [inf W12, sup W12 + m]; together with [0,m] ⊆ these spectra, this gives only the inequalities min(inf W1, inf W2, inf W12) ≤ inf spess(H) ≤ 0 and 0 ≤ sup spess(H) ≤ max(sup W1+m, sup W2+m, sup W12+m). The text itself says that the inclusions 'yield lower and upper bounds' and that only 'a simple estimate' for H12 is available. The theorem's exact statements therefore overstate the proven result and should be replaced by the corresponding inequalities, or by additional hypotheses that guarantee attainment of the potential extrema.
[Section 3.5, Theorem 19] The claimed equalities are in fact false for admissible potentials. Let W1 = -V0 δ_0 for some V0 > 0 and W2 = W12 = 0; these functions tend to zero at infinity as required. Then inf W1 = -V0, but the bottom of sp(H1) is either 0 (if (1/(2m1))∆ + W1 has no negative eigenvalue) or a negative eigenvalue E0 with -V0 < E0 < 0, because the nonnegative discrete kinetic term raises the bottom above the bare potential minimum. By (34), inf spess(H) = min(inf sp(H1), 0) > -V0, contradicting the theorem's lower equality. An analogous positive-bump example contradicts the upper equality. Thus Theorem 19 is not merely unproven but incorrect; the correct statement is a set of one-sided bounds, e.g., min(inf Wj) ≤ inf spess(H) and sup spess(H) ≤ max(sup Wj + m).

Minor objections (5)

[Section 3.4, proof of Theorem 16, Case (d)] In the fourth case of the enumeration, the text says 'one can assume again that g(k) = γ−_k − h where h is constant'; this should read 'g(k) = γ+_k − h', since the sequence is approaching the right endpoint of the interval. The subsequent computation is otherwise correct.
[Section 3.3, formula for sp(L^g_N)] The displayed formula for the spectrum of the one-dimensional operator L^g_N lists intervals [Φ^g_+, 4N + Φ^g_+] and [Φ^g_-, 4N + Φ^g_-]; these should have length 4 rather than 4N. The next line, which gives the spectrum of the full N-dimensional operator, correctly uses 4N.
[Section 2.1, paragraph before Proposition 1] The word 'essential spectum' is a typo for 'essential spectrum'.
[References] References [5] and [16] contain apparent OCR artifacts ('Rieman-Hilbert', 'b f 5'); these should be cleaned up in the final version.
[Abstract and Introduction, equation (5)] The abstract and introduction present the three-particle result as an exact spectral identity. If Theorem 19 is weakened to the bounds that the proof actually provides, the abstract and the surrounding text should be adjusted accordingly, so that the reader is not led to expect exact bottom and top formulas.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The essential spectrum of a lattice Schrödinger operator is exactly the union of the spectra of its limit operators, and this identity yields explicit spectral intervals for several potential classes.

The reading

This paper proves that the essential spectrum of a discrete Schrödinger operator on the lattice $\mathbb{Z}^N$ is the union of the spectra of its limit operators—the operators that arise when one views the system from farther and farther out along sequences of lattice points. For potentials that are slowly oscillating, periodic or semi-periodic, piecewise constant with infinitely many jumps, of waveguide type, or of three-particle form, the limit operators are simple enough that this union becomes explicit intervals. A sympathetic reader would care because the formulas locate the continuous spectral bands of standard tight-binding and lattice few-body models exactly, and they show, for instance, that slowly oscillating magnetic potentials do not affect the essential spectrum at all. The method works for non-selfadjoint operators on $\ell^p$ spaces, not only for selfadjoint operators on $\ell^2$.

What carries the argument

The central object is the operator spectrum $\operatorname{op}(A)$: the collection of all strong limits $A_g=\operatorname{s-lim}_{j\to\infty}V_{-g(j)}AV_{g(j)}$ along sequences $g(j)\to\infty$, where $V_g$ is the shift by $g$. The load-bearing identity is $\operatorname{sp}_{\mathrm{ess}}(A)=\bigcup_{A_g\in \operatorname{op}(A)}\operatorname{sp}(A_g)$, valid in the Wiener algebra of operators with absolutely summable, bounded coefficients. The limit operators simplify at infinity—slowly oscillating coefficients become constants, periodic coefficients remain periodic, step potentials become half-space characteristic functions—and computing their spectra is a one-variable or finite-matrix problem. Assembling the union yields the paper's explicit spectral intervals.

Extended reading notes

Core claim

On the paper's own terms, the central discovery is the identity $\operatorname{sp}_{\mathrm{ess}}(A)=\bigcup_{A_g\in \operatorname{op}(A)}\operatorname{sp}(A_g)$ for band-dominated operators in the Wiener algebra on $\mathbb{Z}^N$: the essential spectrum is completely determined by the spectra of the limit operators, obtained as strong limits $A_g=\operatorname{s-lim}_{j\to\infty}V_{-g(j)}AV_{g(j)}$ along sequences $g(j)\to\infty$. The paper then computes these limit-operator spectra for the five potential classes. Slowly oscillating coefficients collapse to constants, so the essential spectrum is $[m(\Phi),M(\Phi)+4N]$; periodic operators are unitarily equivalent to multiplication by a matrix function, giving finitely many intervals; step potentials with long intervals and long gaps give $[a,a+4]\cup[b,b+4]$; waveguides split into one-dimensional Jacobi operators; and the three-particle operator decomposes into two two-body Hamiltonians plus the pair-interaction Hamiltonian.

Load-bearing premise

For the central identity to hold, every sequence of shifts of the operator must have a convergent subsequence producing a limit operator, and for the discontinuous-potential theorem the intervals where the potential is $a$ and the gaps between them must both grow without bound.

Editorial extensions

If this is right

For slowly oscillating electric and magnetic potentials, the essential spectrum is the single interval $[\liminf_{x\to\infty}\Phi(x), \limsup_{x\to\infty}\Phi(x)+4N]$, independent of the magnetic potential.
For periodic potentials the essential spectrum coincides with the spectrum and is a finite union of intervals; a slowly oscillating perturbation widens these intervals and can close spectral gaps once its oscillation amplitude is large enough.
For the one-dimensional two-valued potential with intervals and gaps both growing to infinity, the essential spectrum is exactly $[a,a+4]\cup[b,b+4]$.
For the discrete quantum waveguide, the essential spectrum is the union of the two asymptotic half-guide bands plus finitely many discrete branches coming from the transversal one-dimensional operator.
For two particles moving around a heavy nucleus on $\mathbb{Z}^3$, the essential spectrum is the union of the spectra of the two single-particle subsystems and the pair-interaction Hamiltonian, with bounds given by the extrema of the potentials.

Reading between the lines

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

Because the limit-operator identity holds on $\ell^p$ for $p\in(1,\infty)$, the same interval formulas should hold for non-selfadjoint or complex-valued potentials on $\ell^p$, a direct testable extension of the paper's selfadjoint setting.
Relaxing the condition that the gaps between the intervals of a discontinuous potential grow to infinity should produce additional limit operators with moving step profiles, so the essential spectrum would no longer be the simple pair of intervals; classifying those extra limits is a natural next problem.
The three-particle decomposition suggests a general cluster rule for $N$-particle lattice Hamiltonians: the essential spectrum should be the union over all cluster decompositions of the subsystem Hamiltonians, mirroring the continuous multi-particle result.

Original abstract

The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schr\"{o}dinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schr\"{o}dinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schr\"{o}dinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schr\"{o}dinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,864 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.54 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schr\"{o}dinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well. We consider the following classes of the discrete Schr\"{o}dinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schr\"{o}dinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schr\"{o}dinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The paper is devoted to the study of the essential spectrum of discrete Schr\"{o}dinger operators on the lattice $\mathbb{Z}^{N}$ by means of the limit operators method.	contribution 0.94	0.60	0.09	asserted 0.97
s1	This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schr\"{o}dinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity.	background 0.98	0.49	0.04	asserted 1.00
s2	The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well.	contribution 0.89	0.24	0.09	asserted 0.82
s3	We consider the following classes of the discrete Schr\"{o}dinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schr\"{o}dinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schr\"{o}dinger operators which describe the motion of two particles around a heavy nuclei on the lattice $\mathbb{Z}^3$.	contribution 0.99	0.65	0.06	asserted 0.89

What kind of paper the classifier reports this is: theoretical result 0.84 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (28)

[22]

Rabinovich, V. S., Roch, S., Silbermann, B., Limit Operators and Their Applications in Operator Theory . Operator Theory: Advances and Appli- cations 150, Birkh¨ auser Verlag, 2004

View on Pith 2004
[1]

Albeverio, S., Lakaev, S. N., Abdullaev, J. I., On the ﬁniteness of the discrete spectrum of four-particle lattice Schr¨ odinger operators. Reports on Mathematical Physics 51(2003), 1, 43 – 70

View on Pith 2003
[2]

Albeverio, S., Lakaev, S. N., Muminov, Z. I., The structure of the essential spectrum of the three-particle Schr¨ odinger operators on a lattice. ArXiv math-ph/0312050, 2003

View on Pith Pith review arXiv 2003
[3]

Amerin, W., M˘ antoiu, M., Purice, R., Propagation properties for Schr¨ o- dinger operators aﬃliated with certain C∗-algebras. Ann. H. Poincar´ e In-t 3(2002), 6, 1215 – 1232

View on Pith 2002
[4]

Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B., Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry . Springer- Verlag, Berlin, Heidelberg, New York 1987

View on Pith 1987
[5]

Deift, P., Orthogonal Polynomials and Random Matrices: A Rieman- Hilbert Approach. AMS, Courant Lectures Notes in Mathematics 3, 2000

View on Pith 2000
[6]

Georgescu, V., Iftimovici, A., Crossed products of C∗-algebras and spec- tral analysis of quantum Hamiltonians . Communications in Mathematical Physics, 228(2002), 519 – 560

View on Pith 2002
[7]

Georgescu, V., Iftimovici, A., Localization at inﬁnity and essential spectrum of quantum Hamiltonians . ArXiv math-ph/0506051V1, 2005

View on Pith Pith review arXiv 2005
[8]

I. Gohberg, S. Goldberg, M. A. Kaashoek , Classes of Linear Oper- ators. Vol. I. – Birkh¨ auser Verlag, Basel, Boston, Berlin 1990

View on Pith 1990
[9]

Jirari, A., Second Order Sturm-Liouville Diﬀerence Equations and Orth og- onal Polynoms . Memoirs AMS 542(1995)

View on Pith 1995
[10]

Kato, T., Perturbation Theory for Linear Operators . Springer-Verlag, Berlin, Heidelberg, New York 1966

View on Pith 1966
[11]

Lakaev, S. N., Muminov, Z. I., The asymptotics of the number of eigenval- ues of a three-particle lattice Schr¨ odinger operator. Functional Anal. Appl. 37(2003), 3, 228 – 231

View on Pith 2003
[12]

Last, Y., Simon, B., The essential spectrum of Schr¨ odinger, Jacobi, and CMV operators. Preprint 304 at http://www.math.caltech.edu/people/biblio.html 20

View on Pith
[13]

Lax P. D., Functional Analysis. Wiley-Interscience, 2002

View on Pith 2002
[14]

M˘ antoiu, M.,C∗-algebras, dynamical systems at inﬁnity, and the essen- tial spectrum of generalized Schr¨ odinger operators. J. Reine Angew. Math. 550(2002), 211 – 229

View on Pith 2002
[15]

Mattis, D. C., The few-body problem on a lattice . Rev. Modern. Phys. 58(1986), 361 – 379

View on Pith 1986
[16]

Mogilner, A., Hamiltonians in solid state physics as multiparticle discr ete Schr¨ odinger operators: Problems and results . Advances in Soviet Mathe- matics bf 5, AMS, Providence, R. I., 1991

View on Pith 1991
[17]

Rabinovich, V. S., Limiting absorption principle for a class of diﬀer- ence equations . In: Operator Theory: Advances and Applications 147, Birkh¨ auser Verlag, Basel 2004, 403 – 422

View on Pith 2004
[18]

Rabinovich, V. S., Essential spectrum of perturbed pseudodiﬀerential op- erators. Applications to Schr¨ odinger, Klein-Gordon, and Dirac operators. Russian Journal of Math. Physics 12(2005), 1, 62 – 80

View on Pith 2005
[19]

Rabinovich, V. S., Roch, S., Pseudodiﬀerence operators on weighted spaces, and applications to discrete Schr¨ odinger operators. Acta Applicandae Math. 84(2004), 55 – 96

View on Pith 2004
[20]

Rabinovich, V. S., Roch, S., Silbermann, B., Fredholm theory and ﬁnite sec- tion method for band-dominated operators. Integral Equations Oper. Theory 30(1998), 452 – 495

View on Pith 1998
[21]

Rabinovich, V. S., Roch, S., Silbermann, B., Band-dominated operators with operator-valued coeﬃcients, their Fredholm properti es and ﬁnite sec- tions. Integral Equations Oper. Theory 40(2001), 3, 342 – 381

View on Pith 2001
[23]

Reed, M., Simon, B., Methods of Modern Mathematical Physics III: Scat- tering Theory. Academic Press, 1979

View on Pith 1979
[24]

Shubin, M. A., Discrete magnetic Laplacian . Comm. Math. Phys. 164 (1994), 2, 259 – 275

View on Pith 1994
[25]

Teschl, G., Jacobi Operators and Completely Integrable Nonlinear Latt ices. AMS, Mathematical Surveys and Monographs 72, 1999

View on Pith 1999
[26]

Weder, R., Spectral and Scattering Theory for Wave Propagation in Per- turbed Stratiﬁed Media . Appl. Math. Sci. 87, Springer-Verlag, 1991

View on Pith 1991
[27]

Wilcox, C. H., Sound Propagation in Stratiﬁed Fluids . Applied Mathemat- ical Science 50, Springer, New York, Berlin Heidelberg, Tokyo 1984. 21

View on Pith 1984
[28]

Yafaev, D. R., Scattering Theory: Some Old and New Problems . Lecture Notes Math. 1735, Springer, Berlin 2000. Authors’ addresses: Vladimir S. Rabinovich, Instituto Polit´ ecnico Nacional, ESIME-Zacatenco, Av. IPN, edif. 1, M´ exico D.F., 07738, M ´EXICO. e-mail: vladimir − rabinovich@hotmail.com Steﬀen Roch, Technische Universit¨ at Darmstadt, Schlossgar...

View on Pith 2000

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [22]

Rabinovich, V. S., Roch, S., Silbermann, B., Limit Operators and Their Applications in Operator Theory . Operator Theory: Advances and Appli- cations 150, Birkh¨ auser Verlag, 2004

View on Pith 2004

[2] [1]

Albeverio, S., Lakaev, S. N., Abdullaev, J. I., On the ﬁniteness of the discrete spectrum of four-particle lattice Schr¨ odinger operators. Reports on Mathematical Physics 51(2003), 1, 43 – 70

View on Pith 2003

[3] [2]

Albeverio, S., Lakaev, S. N., Muminov, Z. I., The structure of the essential spectrum of the three-particle Schr¨ odinger operators on a lattice. ArXiv math-ph/0312050, 2003

View on Pith Pith review arXiv 2003

[4] [3]

Amerin, W., M˘ antoiu, M., Purice, R., Propagation properties for Schr¨ o- dinger operators aﬃliated with certain C∗-algebras. Ann. H. Poincar´ e In-t 3(2002), 6, 1215 – 1232

View on Pith 2002

[5] [4]

Cycon, H. L., Froese, R. G., Kirsch, W., Simon, B., Schr¨ odinger Operators with Applications to Quantum Mechanics and Global Geometry . Springer- Verlag, Berlin, Heidelberg, New York 1987

View on Pith 1987

[6] [5]

Deift, P., Orthogonal Polynomials and Random Matrices: A Rieman- Hilbert Approach. AMS, Courant Lectures Notes in Mathematics 3, 2000

View on Pith 2000

[7] [6]

Georgescu, V., Iftimovici, A., Crossed products of C∗-algebras and spec- tral analysis of quantum Hamiltonians . Communications in Mathematical Physics, 228(2002), 519 – 560

View on Pith 2002

[8] [7]

Georgescu, V., Iftimovici, A., Localization at inﬁnity and essential spectrum of quantum Hamiltonians . ArXiv math-ph/0506051V1, 2005

View on Pith Pith review arXiv 2005

[9] [8]

I. Gohberg, S. Goldberg, M. A. Kaashoek , Classes of Linear Oper- ators. Vol. I. – Birkh¨ auser Verlag, Basel, Boston, Berlin 1990

View on Pith 1990

[10] [9]

Jirari, A., Second Order Sturm-Liouville Diﬀerence Equations and Orth og- onal Polynoms . Memoirs AMS 542(1995)

View on Pith 1995

[11] [10]

Kato, T., Perturbation Theory for Linear Operators . Springer-Verlag, Berlin, Heidelberg, New York 1966

View on Pith 1966

[12] [11]

Lakaev, S. N., Muminov, Z. I., The asymptotics of the number of eigenval- ues of a three-particle lattice Schr¨ odinger operator. Functional Anal. Appl. 37(2003), 3, 228 – 231

View on Pith 2003

[13] [12]

Last, Y., Simon, B., The essential spectrum of Schr¨ odinger, Jacobi, and CMV operators. Preprint 304 at http://www.math.caltech.edu/people/biblio.html 20

View on Pith

[14] [13]

Lax P. D., Functional Analysis. Wiley-Interscience, 2002

View on Pith 2002

[15] [14]

M˘ antoiu, M.,C∗-algebras, dynamical systems at inﬁnity, and the essen- tial spectrum of generalized Schr¨ odinger operators. J. Reine Angew. Math. 550(2002), 211 – 229

View on Pith 2002

[16] [15]

Mattis, D. C., The few-body problem on a lattice . Rev. Modern. Phys. 58(1986), 361 – 379

View on Pith 1986

[17] [16]

Mogilner, A., Hamiltonians in solid state physics as multiparticle discr ete Schr¨ odinger operators: Problems and results . Advances in Soviet Mathe- matics bf 5, AMS, Providence, R. I., 1991

View on Pith 1991

[18] [17]

Rabinovich, V. S., Limiting absorption principle for a class of diﬀer- ence equations . In: Operator Theory: Advances and Applications 147, Birkh¨ auser Verlag, Basel 2004, 403 – 422

View on Pith 2004

[19] [18]

Rabinovich, V. S., Essential spectrum of perturbed pseudodiﬀerential op- erators. Applications to Schr¨ odinger, Klein-Gordon, and Dirac operators. Russian Journal of Math. Physics 12(2005), 1, 62 – 80

View on Pith 2005

[20] [19]

Rabinovich, V. S., Roch, S., Pseudodiﬀerence operators on weighted spaces, and applications to discrete Schr¨ odinger operators. Acta Applicandae Math. 84(2004), 55 – 96

View on Pith 2004

[21] [20]

Rabinovich, V. S., Roch, S., Silbermann, B., Fredholm theory and ﬁnite sec- tion method for band-dominated operators. Integral Equations Oper. Theory 30(1998), 452 – 495

View on Pith 1998

[22] [21]

Rabinovich, V. S., Roch, S., Silbermann, B., Band-dominated operators with operator-valued coeﬃcients, their Fredholm properti es and ﬁnite sec- tions. Integral Equations Oper. Theory 40(2001), 3, 342 – 381

View on Pith 2001

[23] [23]

Reed, M., Simon, B., Methods of Modern Mathematical Physics III: Scat- tering Theory. Academic Press, 1979

View on Pith 1979

[24] [24]

Shubin, M. A., Discrete magnetic Laplacian . Comm. Math. Phys. 164 (1994), 2, 259 – 275

View on Pith 1994

[25] [25]

Teschl, G., Jacobi Operators and Completely Integrable Nonlinear Latt ices. AMS, Mathematical Surveys and Monographs 72, 1999

View on Pith 1999

[26] [26]

Weder, R., Spectral and Scattering Theory for Wave Propagation in Per- turbed Stratiﬁed Media . Appl. Math. Sci. 87, Springer-Verlag, 1991

View on Pith 1991

[27] [27]

Wilcox, C. H., Sound Propagation in Stratiﬁed Fluids . Applied Mathemat- ical Science 50, Springer, New York, Berlin Heidelberg, Tokyo 1984. 21

View on Pith 1984

[28] [28]

Yafaev, D. R., Scattering Theory: Some Old and New Problems . Lecture Notes Math. 1735, Springer, Berlin 2000. Authors’ addresses: Vladimir S. Rabinovich, Instituto Polit´ ecnico Nacional, ESIME-Zacatenco, Av. IPN, edif. 1, M´ exico D.F., 07738, M ´EXICO. e-mail: vladimir − rabinovich@hotmail.com Steﬀen Roch, Technische Universit¨ at Darmstadt, Schlossgar...

View on Pith 2000