Local Spectral Deformation
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, p. 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.
Received : 2016-09-03
Revised : 2017-04-13
Accepted : 2017-04-28
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3177
Classification:  81Q10,  47A55,  81Q12
Keywords: Analytic perturbation theory, spectral deformation, Mourre theory
@article{AIF_2018__68_2_767_0,
     author = {Engelmann, Matthias and M\o ller, Jacob Schach and Rasmussen, Morten Grud},
     title = {Local Spectral Deformation},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {2},
     year = {2018},
     pages = {767-804},
     doi = {10.5802/aif.3177},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_2_767_0}
}
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/item/AIF_2018__68_2_767_0/

[1] Aguiliar, J.; Combes, Jean-Michel A class of analytic perturbations for one-body Schrödinger Hamiltonians, Commun. Math. Phys., Tome 22 (1971), pp. 269-279 | Article | Zbl 0219.47011

[2] Amrein, Werner O.; Boutet De Monvel, Anne; Georgescu, Vladimir C 0 -groups, commutator methods and spectral theory of N-body Hamiltonians, Birkhäuser, Progress in Mathematics (1996), xiv+460 pages | Zbl 0962.47500

[3] Balslev, Erik; Combes, Jean-Michel Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Commun. Math. Phys., Tome 22 (1971), pp. 280-294 | Article | Zbl 0219.47005

[4] Bierstone, Edward; Milman, Pierre D. Semianalytic and subanalytic sets, Publ. Math., Inst. Hautes Étud. Sci., Tome 67 (1988), pp. 5-42 | Article | Zbl 0674.32002

[5] Delort, Jean-Marc F.B.I. transformation. Second microlocalization and semilinear caustics, Springer, Lecture Notes in Mathematics, Tome 1522 (1992), vi+101 pages | Zbl 0760.35004

[6] Dereziński, Jan; Jakšić, Vojkan Spectral theory of Pauli-Fierz operators, J. Funct. Anal., Tome 180 (2001) no. 2, pp. 243-327 | Article | Zbl 1034.81016

[7] Derks, Gianne; Sasane, Sara Maad; Sandstede, Björn Perturbations of embedded eigenvalues for the planar bilaplacian, J. Funct. Anal., Tome 260 (2011) no. 2, pp. 340-398 | Article | Zbl 1219.35148

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

[9] Georgescu, Vladimir; Gérard, Christain On the virial theorem in quantum mechanics, Commun. Math. Phys., Tome 208 (1999) no. 2, pp. 275-281 | Article | Zbl 0961.81009

[10] Gérard, Christain; Nier, Francis The Mourre theory for analytically fibered operators, J. Funct. Anal., Tome 152 (1998) no. 1, pp. 20-219 | Article | Zbl 0939.47019

[11] Griesemer, Marcel; Hasler, David G. On the smooth Feshbach-Schur map, J. Funct. Anal., Tome 254 (2008), pp. 2329-2335 | Article | Zbl 1138.81055

[12] Griesemer, Marcel; Hasler, David G. Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation, Ann. Henri Poincaré, Tome 10 (2009) no. 3, pp. 577-621 | Article | Zbl 1207.81208

[13] Herbst, Ira W.; Skibsted, Erik Decay of eigenfunctions of elliptic PDE’s, Adv. Math., Tome 270 (2015), pp. 138-180 | Article | Zbl 1328.81115

[14] Hunziker, W. Distortion analyticity and molecular resonance curves, Ann. Inst. Henri Poincaré, Phys. Théor., Tome 45 (1986), pp. 339-358 | Zbl 0619.46068

[15] Hunziker, W.; Sigal, Israel Michael The quantum N-body problem, J. Math. Phys., Tome 41 (2000) no. 6, pp. 3448-3510 | Article | Zbl 0981.81026

[16] Jensen, Arne; Mourre, Éric; Perry, Peter Multiple commutator estimates and resolvent smoothness in quantum scattering theory, Ann. Inst. Henri Poincaré, Phys. Théor., Tome 41 (1984) no. 2, p. 270-225 | Zbl 0561.47007

[17] Kato, Tosio Perturbation theory of linear operators, Springer, Classics in Mathematics (1995), xxi+619 pages (Reprint of the 2nd edition) | Zbl 0836.47009

[18] Lieb, Elliott H.; Loss, Michael Analysis, American Mathematical Society, Graduate Studies in Mathematics, Tome 14 (2001), xxi+346 pages | Zbl 0966.26002

[19] Møller, Jacob S. Fully coupled Pauli-Fierz systems at zero and positive temperature, J. Math. Phys., Tome 55 (2014) (art. ID 075203) | Article | Zbl 1375.81158

[20] Møller, Jacob S.; Westrich, Matthias Regularity of eigenstates in regular Mourre theory, J. Funct. Anal., Tome 260 (2011) no. 3, pp. 852-878 | Article | Zbl 1205.81087

[21] Boutet De Monvel, Anne; Georgescu, Vladimir; Sahbani, Jaouad Boundary values of regular resolvent families, Helv. Phys. Acta, Tome 71 (1998) no. 5, pp. 518-553 | Zbl 0926.47003

[22] Mourre, Éric Absence of singular continuous spectrum for certain self adjoint operators, Commun. Math. Phys., Tome 78 (1981), pp. 391-408 | Article | Zbl 0489.47010

[23] Nakamura, Shu Distortion analyticity for two-body Schrödinger operators, Ann. Inst. Henri Poincaré, Phys. Théor., Tome 53 (1990) no. 2, pp. 149-157 | Zbl 0736.35026

[24] Rao, Murali; Stetkær, Henrik; Fournais, Søren; Schach Møller, Jacob S. Complex analysis. An invitation, World Scientific Publishing (2015), x+414 pages | Zbl 1317.30001

[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 0401.47001

[27] Saint Raymond, Xavier Elementary introduction to the theory of pseudodifferential operators, Boca Raton, Studies in Advanced Mathematics (1991), viii+108 pages | Zbl 0847.47035

[28] Simon, Barry The bound state of weakly coupled Schrödinger operators in one and two dimensions, Ann. Phys., Tome 97 (1976), pp. 279-288 | Article | Zbl 0325.35029

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