Local Spectral Deformation
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 767-804.

We develop an analytic perturbation theory for eigenvalues with finite multiplicities, embedded into the essential spectrum of a self-adjoint operator H. We assume the existence of another self-adjoint operator A for which the family H θ =e iθA He -iθA extends analytically from the real line to a strip in the complex plane. Assuming a Mourre estimate holds for i[H,A] in the vicinity of the eigenvalue, we prove that the essential spectrum is locally deformed away from the eigenvalue, leaving it isolated and thus permitting an application of Kato’s analytic perturbation theory.

Nous construisons dans cet article une théorie de perturbation analytique pour des valeurs propres avec multiplicités finies, plongées dans le spectre essentiel d’un opérateur auto-adjoint H. Pour pouvoir faire ça on suppose l’existence d’un autre opérateur auto-adjoint A pour lequel la famille H θ =e iθA He -iθA a une extension analytique de la ligne réelle à une bande dans le plan complexe. En supposant que l’estimation de Mourre soit vraie pour i[H,A] au voisinage de la valeur propre, on montre que le spectre essentiel est localement déformé afin qu’il ne contienne plus la valeur propre permettant ainsi l’application de la théorie de la perturbation analytique de Kato.

Published online:
DOI: 10.5802/aif.3177
Classification: 81Q10, 47A55, 81Q12
Keywords: Analytic perturbation theory, spectral deformation, Mourre theory
Mot clés : Théorie de la perturbation analytique, Déformation spectrale, Théorie de Mourre
Engelmann, Matthias 1; Møller, Jacob Schach 2; Rasmussen, Morten Grud 3

1 IADM University of Stuttgart Pfaffenwaldring 57 70569 Stuttgart (Germany)
2 Department of Mathematics Aarhus University Ny Munkegade 118, bldg. 1530 DK-8000 Aarhus C (Denmark)
3 Department of Mathematical Sciences Aalborg University Skjernvej 4A 9220 Aalborg Ø (Denmark)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Local {Spectral} {Deformation}},
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Engelmann, Matthias; Møller, Jacob Schach; Rasmussen, Morten Grud. Local Spectral Deformation. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 767-804. doi : 10.5802/aif.3177. https://aif.centre-mersenne.org/articles/10.5802/aif.3177/

[1] Aguiliar, J.; Combes, Jean-Michel A class of analytic perturbations for one-body Schrödinger Hamiltonians, Commun. Math. Phys., Volume 22 (1971), pp. 269-279 | DOI | Zbl

[2] Amrein, Werner O.; Boutet de Monvel, Anne; Georgescu, Vladimir C 0 -groups, commutator methods and spectral theory of N-body Hamiltonians, Progress in Mathematics, Birkhäuser, 1996, xiv+460 pages | Zbl

[3] Balslev, Erik; Combes, Jean-Michel Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Commun. Math. Phys., Volume 22 (1971), pp. 280-294 | DOI | Zbl

[4] Bierstone, Edward; Milman, Pierre D. Semianalytic and subanalytic sets, Publ. Math., Inst. Hautes Étud. Sci., Volume 67 (1988), pp. 5-42 | DOI | Zbl

[5] Delort, Jean-Marc F.B.I. transformation. Second microlocalization and semilinear caustics, Lecture Notes in Mathematics, 1522, Springer, 1992, vi+101 pages | Zbl

[6] Dereziński, Jan; Jakšić, Vojkan Spectral theory of Pauli-Fierz operators, J. Funct. Anal., Volume 180 (2001) no. 2, pp. 243-327 | DOI | Zbl

[7] Derks, Gianne; Sasane, Sara Maad; Sandstede, Björn Perturbations of embedded eigenvalues for the planar bilaplacian, J. Funct. Anal., Volume 260 (2011) no. 2, pp. 340-398 | DOI | Zbl

[8] Engelmann, Matthias; Rasmussen, Morten Grud Spectral deformation for two-body dispersive systems with e.g. the Yukawa potential, Math. Phys. Anal. Geom., Volume 19 (2016) no. 4, 24 pages | DOI

[9] Georgescu, Vladimir; Gérard, Christain On the virial theorem in quantum mechanics, Commun. Math. Phys., Volume 208 (1999) no. 2, pp. 275-281 | DOI | Zbl

[10] Gérard, Christain; Nier, Francis The Mourre theory for analytically fibered operators, J. Funct. Anal., Volume 152 (1998) no. 1, pp. 20-219 | DOI | Zbl

[11] Griesemer, Marcel; Hasler, David G. On the smooth Feshbach-Schur map, J. Funct. Anal., Volume 254 (2008), pp. 2329-2335 | DOI | Zbl

[12] Griesemer, Marcel; Hasler, David G. Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation, Ann. Henri Poincaré, Volume 10 (2009) no. 3, pp. 577-621 | DOI | Zbl

[13] Herbst, Ira W.; Skibsted, Erik Decay of eigenfunctions of elliptic PDE’s, Adv. Math., Volume 270 (2015), pp. 138-180 | DOI | Zbl

[14] Hunziker, W. Distortion analyticity and molecular resonance curves, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 45 (1986), pp. 339-358 | Zbl

[15] Hunziker, W.; Sigal, Israel Michael The quantum N-body problem, J. Math. Phys., Volume 41 (2000) no. 6, pp. 3448-3510 | DOI | Zbl

[16] Jensen, Arne; Mourre, Éric; Perry, Peter Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 41 (1984) no. 2, p. 270-225 | Zbl

[17] Kato, Tosio Perturbation theory of linear operators, Classics in Mathematics, Springer, 1995, xxi+619 pages (Reprint of the 2nd edition) | Zbl

[18] Lieb, Elliott H.; Loss, Michael Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001, xxi+346 pages | Zbl

[19] Møller, Jacob S. Fully coupled Pauli-Fierz systems at zero and positive temperature, J. Math. Phys., Volume 55 (2014) (art. ID 075203) | DOI | Zbl

[20] Møller, Jacob S.; Westrich, Matthias Regularity of eigenstates in regular Mourre theory, J. Funct. Anal., Volume 260 (2011) no. 3, pp. 852-878 | DOI | Zbl

[21] Boutet de Monvel, Anne; Georgescu, Vladimir; Sahbani, Jaouad Boundary values of regular resolvent families, Helv. Phys. Acta, Volume 71 (1998) no. 5, pp. 518-553 | Zbl

[22] Mourre, Éric Absence of singular continuous spectrum for certain self adjoint operators, Commun. Math. Phys., Volume 78 (1981), pp. 391-408 | DOI | Zbl

[23] Nakamura, Shu Distortion analyticity for two-body Schrödinger operators, Ann. Inst. Henri Poincaré, Phys. Théor., Volume 53 (1990) no. 2, pp. 149-157 | Zbl

[24] Rao, Murali; Stetkær, Henrik; Fournais, Søren; Schach Møller, Jacob S. Complex analysis. An invitation, World Scientific Publishing, 2015, x+414 pages | Zbl

[25] Rasmussen, Morten Grud Spectral and scattering theory for translation invariant models in quantum field theory, Aarhus University (Denmark) (2010) (Ph. D. Thesis)

[26] Reed, Michael; Simon, Barry Methods of modern mathematical physics IV, Academic Press, 1978, xv+396 pages | Zbl

[27] Saint Raymond, Xavier Elementary introduction to the theory of pseudodifferential operators, Studies in Advanced Mathematics, Boca Raton, 1991, viii+108 pages | Zbl

[28] Simon, Barry The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Phys., Volume 97 (1976), pp. 279-288 | DOI | Zbl

[29] Simon, Barry The definition of molecular resonance curves by the method of exterior complex scaling, Phys. Lett, t., Volume 71A (1979), pp. 211-214 | DOI

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