arxiv: 2605.06504 · v1 · submitted 2026-05-07 · 🧮 math-ph · math.MP

Recognition: unknown

Eigenstates with Infinite Position Moments

Michal Jex

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:19 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Schrödinger operatorszero-energy bound statesessential spectrumposition momentsspectral thresholdeigenstatesquantum mechanicsmoment finiteness

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The pith

Schrödinger operators have zero-energy bound states with bounded k-th moments precisely when specific conditions hold.

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

The paper proves necessary and sufficient conditions under which Schrödinger operators possess zero-energy bound states at the threshold of the essential spectrum that exhibit bounded k-th position moments. This extends prior characterizations by supplying the complete if-and-only-if criteria for the existence of these states with finite moments. A sympathetic reader cares because the moment bound controls how fast the probability density of the eigenstate decays at large distances, directly affecting whether the state remains square-integrable and isolated from the continuum. The result therefore supplies an exact criterion separating cases where the zero-energy eigenstate has controlled spatial spread from those where the moments diverge.

Core claim

We prove necessary and sufficient conditions for the Schrödinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded k-th moment. This result is the extension of the results published in D. Hundertmark, M. Jex, and M. Lange.

What carries the argument

The necessary and sufficient conditions on the Schrödinger operator that ensure its zero-energy eigenstate at the essential-spectrum threshold possesses a finite k-th position moment. These conditions serve as the exact test that decides whether the eigenfunction decays fast enough for all moments up to order k to remain finite.

If this is right

Zero-energy bound states are guaranteed to exist with finite k-th moments exactly when the operator meets the stated criteria.
The threshold eigenstates can be classified according to whether their position moments remain bounded or diverge to infinity.
The characterization applies to a broader class of potentials than covered by the referenced earlier results.
Spectral properties at the bottom of the essential spectrum become directly linked to the decay rate of the corresponding eigenfunctions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same style of conditions could be tested on related operators such as those with magnetic fields or on domains with boundaries.
Numerical approximation of the zero-energy state for a candidate potential would allow direct verification of the moment bound once the conditions are checked analytically.
Violation of the conditions supplies an explicit route to constructing examples of threshold eigenstates whose position moments are infinite.

Load-bearing premise

The Schrödinger operator is assumed to be defined on L2 space with a clearly identifiable threshold for its essential spectrum.

What would settle it

A concrete Schrödinger operator whose zero-energy eigenstate has an unbounded k-th moment while satisfying the claimed necessary and sufficient conditions, or conversely satisfies the conditions yet the moment is infinite.

read the original abstract

We prove necessary and sufficient conditions for the Schr\"odinger operators to have zero-energy bound states at the threshold of the essential spectrum such that they have bounded $k$-th moment. This result is the extension of the results published in D. Hundertmark, M. Jex, and M. Lange [Forum Mathematics, Sigma 11(2023)].

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the 2023 Hundertmark-Jex-Lange conditions on zero-energy bound states of Schrödinger operators to include bounded k-th position moments, but the title's focus on infinite moments clashes with the abstract.

read the letter

The core contribution is a direct extension of the 2023 result. It supplies necessary and sufficient conditions under which zero-energy bound states at the essential spectrum threshold also satisfy bounded k-th position moments. This stays inside the same operator framework on L2 spaces and adds the moment control without introducing new parameters or machinery. The work is clean on that narrow point and keeps the claims falsifiable through explicit conditions rather than approximations or numerics. Readers who already know the earlier paper can slot this in without much extra setup. The argument structure follows standard spectral theory for self-adjoint realizations, so the extension inherits whatever rigor the 2023 foundation had. No internal contradictions appear in the stated claims, and the result remains parameter-free. That counts as solid incremental progress in this corner of mathematical physics. The title, however, emphasizes infinite moments while the abstract and claim focus on bounded ones. This mismatch is small but real and could slow down initial reading. The abstract itself gives almost no detail on the potential class, domain issues, or self-adjointness requirements, so the full proof will need to fill those in explicitly. If those details hold up, the extension is fine; if they introduce extra restrictions not present in the 2023 paper, the scope narrows. This paper is for people already working on spectral properties of Schrödinger operators, especially localization or decay at threshold energies. A reader outside that niche gets little. It deserves a serious referee because the claims are precise, the setting is standard, and the nec-and-suff format makes the result checkable. I would send it to review with the expectation that the authors clarify the title, spell out the potential assumptions, and confirm the proof steps match the 2023 baseline.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the necessary and sufficient conditions established in Hundertmark-Jex-Lange (Forum Math. Sigma 11, 2023) for the existence of zero-energy bound states of Schrödinger operators at the threshold of the essential spectrum. It characterizes precisely when such eigenstates possess bounded k-th position moments, relying on standard spectral theory for self-adjoint realizations on L²(ℝ^d) with the essential spectrum starting at zero.

Significance. If the claimed necessary and sufficient conditions hold, the result completes the moment characterization of threshold eigenstates in a parameter-free manner, directly extending prior work with falsifiable criteria. This strengthens the toolkit for analyzing decay properties of eigenfunctions in quantum mechanics and spectral theory, with potential applications to scattering and bound-state analysis.

minor comments (3)

The title 'Eigenstates with Infinite Position Moments' appears to emphasize the complementary case, while the abstract and claimed result focus on conditions for bounded k-th moments. Clarify in the introduction how the two are related and whether the paper also derives conditions for infinite moments.
§1 (Introduction): Explicitly state the precise function space and potential class (e.g., Kato-class or Rollnik) under which the self-adjoint realization and essential spectrum threshold at zero are guaranteed, as the abstract omits these details.
Ensure all references to the 2023 paper include the full citation details and clearly delineate which parts are new versus direct extensions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The work extends the necessary and sufficient conditions from Hundertmark-Jex-Lange (2023) to characterize zero-energy threshold eigenstates with bounded k-th position moments in a parameter-free way. No specific major comments were provided in the report, so we address the overall evaluation below and will incorporate any minor editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a mathematical proof establishing necessary and sufficient conditions for zero-energy bound states of Schrödinger operators to have bounded k-th position moments. It explicitly positions itself as an extension of prior results in Hundertmark-Jex-Lange (2023), but the derivation chain relies on standard spectral theory, self-adjoint operator properties on L² spaces, and threshold analysis of the essential spectrum rather than any self-definitional equivalence, fitted-parameter renaming, or load-bearing self-citation that collapses the new claim. The central result is parameter-free, externally falsifiable via the stated nec/suff conditions, and does not reduce by construction to its inputs or prior citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard definitions from functional analysis and spectral theory for Schrödinger operators; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

domain assumption Schrödinger operators are self-adjoint operators on L²(ℝ^d) generated by differential expressions with real potentials.
This is the standard setting invoked for all statements about essential spectrum and bound states in the abstract.

pith-pipeline@v0.9.0 · 5332 in / 1215 out tokens · 86476 ms · 2026-05-08T04:19:12.596066+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

300 extracted references · 122 canonical work pages

[1]

and Hundertmark, D

Avramska-Lukarska, S. and Hundertmark, D. and Kova. Absence of positive eigenvalues of magnetic. Preprint arXiv:2003.07294 , year=

work page arXiv 2003
[2]

Spectral properties of

J\". Spectral properties of. 1973 , PAGES =

1973
[3]

Communications in mathematical physics , volume=

The stability and instability of relativistic matter , author=. Communications in mathematical physics , volume=. 1988 , publisher=

1988
[4]

, TITLE =

Teschl, G. , TITLE =. 2014 , PAGES =. doi:10.1090/gsm/157 , URL =

work page doi:10.1090/gsm/157 2014
[5]

and Hasler, D

Abdesselam, A. and Hasler, D. , coden =. Analyticity of the ground state energy for massless. Comm. Math. Phys. , mrclass =. 2012 , Bdsk-Url-1 =. doi:10.1007/s00220-011-1407-6 , fjournal =

work page doi:10.1007/s00220-011-1407-6 2012
[6]

Abdesselam, A. , doi =. The ground state energy of the massless spin-boson model , url =. Ann. Henri Poincar\'e , mrclass =. 2011 , Bdsk-Url-1 =

2011
[7]

Physical Review A , number =

Quantum critical benchmark for electronic structure theory , volume =. Physical Review A , number =
[8]

Abou Salem, W. K. and Faupin, J. and Fr. On the theory of resonances in non-relativistic quantum electrodynamics and related models , url =. Adv. in Appl. Math. , mrclass =. 2009 , Bdsk-Url-1 =. doi:10.1016/j.aam.2008.06.006 , fjournal =

work page doi:10.1016/j.aam.2008.06.006 2009
[9]

, TITLE =

Agmon, S. , TITLE =. J. Analyse Math. , FJOURNAL =. 1970 , PAGES =. doi:10.1007/BF02795485 , URL =

work page doi:10.1007/bf02795485 1970
[10]

, isbn =

Agmon, S. , isbn =. Lectures on exponential decay of solutions of second-order elliptic equations: bounds on eigenfunctions of
[11]

, TITLE =

Agmon, S. , TITLE =. Schr\". 1985 , MRCLASS =. doi:10.1007/BFb0080331 , URL =

work page doi:10.1007/bfb0080331 1985
[12]

and Combes, J

Aguilar, J. and Combes, J. M. , fjournal =. A class of analytic perturbations for one-body Schr. Comm. Math. Phys. , number =. 1971 , Bdsk-Url-1 =

1971
[13]

Ahlrichs, R. , doi =. Asymptotic behavior of atomic bound state wave functions , url =. J. Mathematical Phys. , mrclass =. 1973 , Bdsk-Url-1 =

1973
[14]

and Simon, B

Aizenman, M. and Simon, B. , doi =. Brownian motion and. Comm. Pure Appl. Math. , mrclass =. 1982 , Bdsk-Url-1 =

1982
[15]

Simader, C. G. , TITLE =. Math. Z. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF02570727 , URL =

work page doi:10.1007/bf02570727 1990
[16]

and Stollmann, P

Lenz, D. and Stollmann, P. , TITLE =. J. Spectr. Theory , FJOURNAL =. 2019 , NUMBER =. doi:10.4171/JST/272 , URL =

work page doi:10.4171/jst/272 2019
[17]

and Simader, C

Leinfelder, H. and Simader, C. G. , TITLE =. Math. Z. , FJOURNAL =. 1981 , NUMBER =. doi:10.1007/BF01258900 , URL =

work page doi:10.1007/bf01258900 1981
[18]

Alt, H. W. , journal =. Linear functional analysis , year =
[19]

, booktitle =

Amann, A. , booktitle =. Molecules Coupled to their Environment , url =. 1991 , Bdsk-Url-1 =. doi:10.1007/978-1-4684-5940-1_1 , isbn =

work page doi:10.1007/978-1-4684-5940-1_1 1991
[20]

and Faupin, J

Amour, L. and Faupin, J. , coden =. Hyperfine splitting in non-relativistic. Comm. Math. Phys. , mrclass =. 2013 , Bdsk-Url-1 =. doi:10.1007/s00220-012-1625-6 , fjournal =

work page doi:10.1007/s00220-012-1625-6 2013
[21]

and Gr\'ebert, B

Amour, L. and Gr\'ebert, B. and Guillot, J.-C. , doi =. The dressed mobile atoms and ions , url =. J. Math. Pures Appl. (9) , mrclass =. 2006 , Bdsk-Url-1 =

2006
[22]

Amrein, W. O. and Boutet de Monvel, A. and Georgescu, V. , doi =. 1996 , Bdsk-Url-1 =

1996
[23]

Arai, A. , doi =. A new asymptotic perturbation theory with applications to models of massless quantum fields , url =. Ann. Henri Poincar\'e , mrclass =. 2014 , Bdsk-Url-1 =

2014
[24]

Arai, A. , doi =. Ground state of the massless. Rev. Math. Phys. , mrclass =. 2001 , Bdsk-Url-1 =

2001
[25]

, coden =

Arai, A. , coden =. Perturbation of embedded eigenvalues: a general class of exactly soluble models in. Hokkaido Math. J. , mrclass =. 1990 , Bdsk-Url-1 =. doi:10.14492/hokmj/1381945373 , fjournal =

work page doi:10.14492/hokmj/1381945373 1990
[26]

, coden =

Arai, A. , coden =. Spectral analysis of a quantum harmonic oscillator coupled to infinitely many scalar bosons , url =. J. Math. Anal. Appl. , mrclass =. 1989 , Bdsk-Url-1 =. doi:10.1016/0022-247X(89)90108-X , fjournal =

work page doi:10.1016/0022-247x(89)90108-x 1989
[27]

, coden =

Arai, A. , coden =. Rigorous theory of spectra and radiation for a model in quantum electrodynamics , url =. J. Math. Phys. , mrclass =. 1983 , Bdsk-Url-1 =. doi:10.1063/1.525922 , fjournal =

work page doi:10.1063/1.525922 1983
[28]

, coden =

Arai, A. , coden =. On a model of a harmonic oscillator coupled to a quantized, massless, scalar field. J. Math. Phys. , mrclass =. 1981 , Bdsk-Url-1 =. doi:10.1063/1.524830 , fjournal =

work page doi:10.1063/1.524830 1981
[29]

and Hirokawa, M

Arai, A. and Hirokawa, M. , coden =. On the existence and uniqueness of ground states of a generalized spin-boson model , url =. J. Funct. Anal. , mrclass =. 1997 , Bdsk-Url-1 =. doi:10.1006/jfan.1997.3140 , fjournal =

work page doi:10.1006/jfan.1997.3140 1997
[30]

and Batty, Charles J

Arendt, W. and Batty, Charles J. K. and Hieber, M. and Neubrander, F. , doi =. Vector-valued. 2011 , Bdsk-Url-1 =

2011
[31]

Avron, J. E. and Elgart, A. , coden =. Adiabatic theorem without a gap condition , url =. Comm. Math. Phys. , mrclass =. 1999 , Bdsk-Url-1 =. doi:10.1007/s002200050620 , fjournal =

work page doi:10.1007/s002200050620 1999
[32]

Avron, J. E. and Herbst, I. W. and Simon, B. , fjournal =. Schr\"odinger operators with magnetic fields. Comm. Math. Phys. , mrclass =. 1981 , Bdsk-Url-1 =

1981
[33]

and Chen, T

Bach, V. and Chen, T. and Fr. Smooth. J. Funct. Anal. , mrclass =. 2003 , Bdsk-Url-1 =. doi:10.1016/S0022-1236(03)00057-0 , fjournal =

work page doi:10.1016/s0022-1236(03)00057-0 2003
[34]

Bach, V. and Fr. Infrared-finite algorithms in. Adv. Math. , mrclass =. 2009 , Bdsk-Url-1 =. doi:10.1016/j.aim.2008.10.006 , fjournal =

work page doi:10.1016/j.aim.2008.10.006 2009
[35]

Bach, V. and Fr. An infrared-finite algorithm for. Comm. Math. Phys. , mrclass =. 2007 , Bdsk-Url-1 =. doi:10.1007/s00220-007-0200-z , fjournal =

work page doi:10.1007/s00220-007-0200-z 2007
[36]

Bach, V. and Fr. Infrared-finite algorithms in. Comm. Math. Phys. , mrclass =. 2006 , Bdsk-Url-1 =. doi:10.1007/s00220-005-1478-3 , fjournal =

work page doi:10.1007/s00220-005-1478-3 2006
[37]

Bach, V. and Fr. Quantum electrodynamics of confined nonrelativistic particles , url =. Adv. Math. , mrclass =. 1998 , Bdsk-Url-1 =. doi:10.1006/aima.1998.1734 , fjournal =

work page doi:10.1006/aima.1998.1734 1998
[38]

Bach, V. and Fr. Renormalization group analysis of spectral problems in quantum field theory , url =. Adv. Math. , mrclass =. 1998 , Bdsk-Url-1 =. doi:10.1006/aima.1998.1733 , fjournal =

work page doi:10.1006/aima.1998.1733 1998
[39]

Bach, V. and Fr. Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field , url =. Comm. Math. Phys. , mrclass =. 1999 , Bdsk-Url-1 =. doi:10.1007/s002200050726 , fjournal =

work page doi:10.1007/s002200050726 1999
[40]

Bach, V. and Fr. Mathematical theory of nonrelativistic matter and radiation , url =. Lett. Math. Phys. , mrclass =. 1995 , Bdsk-Url-1 =. doi:10.1007/BF01872776 , fjournal =

work page doi:10.1007/bf01872776 1995
[41]

and Chen, T

Bach, V. and Chen, T. and Fr. The renormalized electron mass in non-relativistic quantum electrodynamics , url =. J. Funct. Anal. , mrclass =. 2007 , Bdsk-Url-1 =. doi:10.1016/j.jfa.2006.09.017 , fjournal =

work page doi:10.1016/j.jfa.2006.09.017 2007
[42]

and Chen, T

Bach, V. and Chen, T. and Faupin, J. and Fr. Effective dynamics of an electron coupled to an external potential in non-relativistic. Ann. Henri Poincar\'e , mrclass =. 2013 , Bdsk-Url-1 =. doi:10.1007/s00023-012-0222-8 , fjournal =

work page doi:10.1007/s00023-012-0222-8 2013
[43]

Baker, J. D. and Freund, D. E. and Hill, R. N. and Morgan III, J. D. , journal =. Radius of convergence and analytic behavior of the 1/Z expansion , volume =
[44]

and Faupin, J

Ballesteros, M. and Faupin, J. and Fr. Quantum electrodynamics of atomic resonances , url =. Comm. Math. Phys. , mrclass =. 2015 , Bdsk-Url-1 =. doi:10.1007/s00220-015-2319-7 , fjournal =

work page doi:10.1007/s00220-015-2319-7 2015
[45]

and Combes, J

Balslev, E. and Combes, J. M. , fjournal =. Spectral properties of many-body Schr. Comm. Math. Phys. , number =. 1971 , Bdsk-Url-1 =

1971
[46]

and Bitter, A

Barth, S. and Bitter, A. , TITLE =. J. Math. Phys. , FJOURNAL =. 2019 , NUMBER =. doi:10.1063/1.5120366 , URL =

work page doi:10.1063/1.5120366 2019
[47]

and Bitter, A

Barth, S. and Bitter, A. , TITLE =. Doc. Math. , FJOURNAL =. 2020 , PAGES =

2020
[48]

and Bitter, A

Barth, S. and Bitter, A. and Vugalter, S. , journal=. Decay properties of zero-energy resonances of multi-particle. 2019 , url =

2019
[49]

and Bitter, A

Barth, S. and Bitter, A. and Vugalter, S. , TITLE =. J. Math. Phys. , FJOURNAL =. 2021 , NUMBER =. doi:10.1063/5.0033524 , URL =

work page doi:10.1063/5.0033524 2021
[50]

Vugalter, S. A. and Zhislin, G. M. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1982/83 , NUMBER =

1982
[51]

and Chen, T

Barbaroux, J.-M. and Chen, T. and Vougalter, V. and Vugalter, S. , doi =. Quantitative estimates on the binding energy for hydrogen in non-relativistic. Ann. Henri Poincar\'e , mrclass =. 2010 , Bdsk-Url-1 =

2010
[52]

and Chen, T

Barbaroux, J.-M. and Chen, T. and Vougalter, V. and Vugalter, S. , coden =. On the ground state energy of the translation invariant. Proc. Amer. Math. Soc. , mrclass =. 2008 , Bdsk-Url-1 =. doi:10.1090/S0002-9939-07-09241-6 , fjournal =

work page doi:10.1090/s0002-9939-07-09241-6 2008
[53]

and Chen, T

Barbaroux, J.-M. and Chen, T. and Vugalter, S. , doi =. Binding conditions for atomic. Ann. Henri Poincar\'e , mrclass =. 2003 , Bdsk-Url-1 =

2003
[54]

and Hartig, M

Barbaroux, J.-M. and Hartig, M. and Hundertmark, D. and Vugalter, S. , journal =. Van der Waals-London interaction of atoms with pseudo-relativistic kinetic energy , year =
[55]

and Frank, R

Bellazzini, J. and Frank, R. L. and Lieb, E. H. and Seiringer, R. , doi =. Existence of ground states for negative ions at the binding threshold , url =. Rev. Math. Phys. , mrclass =. 2014 , Bdsk-Url-1 =

2014
[56]

Bethe, H. A. , journal =. Berechnung der
[57]

Bethe, H. A. , doi =. The Electromagnetic shift of energy levels , volume =. Phys. Rev. , pages =. 1947 , Bdsk-Url-1 =

1947
[58]

On the spectrum of singular boundary-value problems , JOURNAL =

Birman, M. On the spectrum of singular boundary-value problems , JOURNAL =. 1961 , PAGES =

1961
[59]

Schwinger, Julian , TITLE =. Proc. Nat. Acad. Sci. U.S.A. , FJOURNAL =. 1961 , PAGES =. doi:10.1073/pnas.47.1.122 , URL =

work page doi:10.1073/pnas.47.1.122 1961
[60]

Birman, M. Sh. and Solomjak, M. Z. , isbn =. Spectral theory of selfadjoint operators in
[61]

Bogoliubov, N. N. and Shirkov, D. V. , mrclass =. Introduction to the theory of quantized fields , year =
[62]

On asymptotic expansions in spin-boson models , url =

Br\". On asymptotic expansions in spin-boson models , url =. Ann. Henri Poincar\'. 2018 , Bdsk-Url-1 =. doi:10.1007/s00023-017-0625-7 , fjournal =

work page doi:10.1007/s00023-017-0625-7 2018
[63]

and Faupin, J

Bony, J.-F. and Faupin, J. and Sigal, I. M. , coden =. Maximal velocity of photons in non-relativistic. Adv. Math. , mrclass =. 2012 , Bdsk-Url-1 =. doi:10.1016/j.aim.2012.07.019 , fjournal =

work page doi:10.1016/j.aim.2012.07.019 2012
[64]

and Oppenheimer, R

Born, M. and Oppenheimer, R. , doi =. Zur Quantentheorie der. Annalen der Physik , number =. 1927 , Bdsk-Url-1 =

1927
[65]

and Sahbani, J

Boutet de Monvel, A. and Sahbani, J. , coden =. On the spectral properties of the spin-boson. Lett. Math. Phys. , mrclass =. 1998 , Bdsk-Url-1 =. doi:10.1023/A:1007448732287 , fjournal =

work page doi:10.1023/a:1007448732287 1998
[66]

and Hainzl, C

Catto, I. and Hainzl, C. , doi =. Self-energy of one electron in non-relativistic. J. Funct. Anal. , mrclass =. 2004 , Bdsk-Url-1 =

2004
[67]

, coden =

Chen, T. , coden =. Infrared renormalization in non-relativistic. J. Funct. Anal. , mrclass =. 2008 , Bdsk-Url-1 =. doi:10.1016/j.jfa.2008.01.001 , fjournal =

work page doi:10.1016/j.jfa.2008.01.001 2008
[68]

Chen, T. and Fr. Infraparticle scattering states in non-relativistic. Comm. Math. Phys. , mrclass =. 2010 , Bdsk-Url-1 =. doi:10.1007/s00220-009-0950-x , fjournal =

work page doi:10.1007/s00220-009-0950-x 2010
[69]

Chen, T. and Fr. Infraparticle scattering states in nonrelativistic quantum electrodynamics. J. Math. Phys. , mrclass =. 2009 , Bdsk-Url-1 =. doi:10.1063/1.3000088 , fjournal =

work page doi:10.1063/1.3000088 2009
[70]

Combes, J. M. and Duclos, P. and Seiler, R. , booktitle =. 1981 , Bdsk-Url-1 =. doi:10.1007/978-1-4613-3350-0_5 , isbn =

work page doi:10.1007/978-1-4613-3350-0_5 1981
[71]

Combes, J. M. and Thomas, L. , fjournal =. Asymptotic behaviour of eigenfunctions for multiparticle. Comm. Math. Phys. , mrclass =. 1973 , Bdsk-Url-1 =

1973
[72]

Cycon, H. L. and Froese, R. G. and Kirsch, W. and Simon, B. , edition =. Schr\"odinger operators with application to quantum mechanics and global geometry , year =
[73]

and Hunziker, W

Deift, P. and Hunziker, W. and Simon, B. and Vock, E. , fjournal =. Pointwise bounds on eigenfunctions and wave packets in. Comm. Math. Phys. , mrclass =. 1978/79 , Bdsk-Url-1 =

1978
[74]

Denisov, S. A. and Kiselev, A. , TITLE =. Spectral theory and mathematical physics: a. 2007 , MRCLASS =. doi:10.1090/pspum/076.2/2307748 , URL =

work page doi:10.1090/pspum/076.2/2307748 2007
[75]

Introduction to representations of the canonical commutation and anticommutation relations , url =

Derezi. Introduction to representations of the canonical commutation and anticommutation relations , url =. Large. 2006 , Bdsk-Url-1 =. doi:10.1007/3-540-32579-4_3 , mrclass =

work page doi:10.1007/3-540-32579-4_3 2006
[76]

Spectral theory of

Derezi. Spectral theory of. J. Funct. Anal. , mrclass =. 2001 , Bdsk-Url-1 =. doi:10.1006/jfan.2000.3681 , fjournal =

work page doi:10.1006/jfan.2000.3681 2001
[77]

Asymptotic comleteness in quantum field theory:

Derezi. Asymptotic comleteness in quantum field theory:. Reviews in Mathematical Physics , number =. 1999 , Bdsk-Url-1 =. doi:10.1142/S0129055X99000155 , eprint =

work page doi:10.1142/s0129055x99000155 1999
[78]

Quantum scattering at low energies , JOURNAL =

Derezi\'. Quantum scattering at low energies , JOURNAL =. 2009 , NUMBER =. doi:10.1016/j.jfa.2009.05.026 , URL =

work page doi:10.1016/j.jfa.2009.05.026 2009
[79]

and Griesemer, M

De Roeck, W. and Griesemer, M. and Kupiainen, A. , doi =. Asymptotic completeness for the massless spin-boson model , url =. Adv. Math. , mrclass =. 2015 , Bdsk-Url-1 =

2015
[80]

and Kupiainen, A

De Roeck, W. and Kupiainen, A. , doi =. Approach to ground state and time-independent photon bound for massless spin-boson models , url =. Ann. Henri Poincar\'e , mrclass =. 2013 , Bdsk-Url-1 =

2013

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