Abstract
We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove
where \(\mathbf {1}_J (\cdot )\) denotes the spectral projection of a self-adjoint operator on a set \(J\in \text {Borel}(\mathbb R)\). The operators A and B are self-adjoint, bounded from below and differ by a rank-one perturbation and \(\xi \) denotes the corresponding spectral shift function. The set I is a union of intervals on the real line such that its boundary lies in the resolvent set of A and B and such that the spectral shift function vanishes there i.e. I contains isolated parts of the spectrum of A and B. We apply this formula to the subspace perturbation problem.
Similar content being viewed by others
References
Albeverio, S., Motovilov, A.K.: Sharpening the norm bound in the subspace perturbation theory. Complex Anal. Oper. Theory 7, 1389–1416 (2013)
Avron, J., Seiler, R., Simon, B.: The index of a pair of projections. J. Funct. Anal. 120, 220–237 (1994)
Bhatia, R.: Matrix Analysis, Graduate Texts in Mathematics, vol. 169. Springer, New York (1997)
Dietlein, A., Gebert, M., Müller, P.: Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function. J. Spectr. Theory. e-print arXiv:1701.02956 (to appear)
Deift, P.A., Its, A.R., Zhou, X.: A Riemann–Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. Math. 2(146), 149–235 (1997)
Frank, R.L., Pushnitski, A.: The spectral density of a product of spectral projections. J. Funct. Anal. 268, 3867–3894 (2015)
Gebert, M.: The asymptotics of an eigenfunction-correlation determinant for Dirac-\(\delta \) perturbations. J. Math. Phys. 56, 072110-1-18 (2015)
Gebert, M., Küttler, H., Müller, P.: Anderson’s orthogonality catastrophe. Commun. Math. Phys. 329, 979–998 (2014)
Gebert, M., Küttler, H., Müller, P., Otte, P.: The exponent in the orthogonality catastrophe for Fermi gases. J. Spectr. Theory 6, 643–683 (2016)
Gesztesy, F., Nichols, R.: An abstract approach to weak convergence of spectral shift functions and applications to multi-dimensional Schrödinger operators. J. Spectr. Theory 2, 225–266 (2012)
Hislop, P.D., Müller, P.: The spectral shift function for compactly supported perturbations of Schrödinger operators on large bounded domains. Proc. Am. Math. Soc. 138, 2141–2150 (2010)
Kostrykin, V., Makarov, K.A., Motovilov, A.K.: On a subspace perturbation problem. Proc. Am. Math. Soc. 131, 3469–3476 (2003)
Küttler, H., Otte, P., Spitzer, W.: Anderson’s orthogonality catastrophe for one-dimensional systems. Ann. Henri Poincaré 15, 1655–1696 (2014)
Knörr, H.K., Otte, P., Spitzer, W.: Anderson’s orthogonality catastrophe in one dimension induced by a magnetic field. J. Phys. A 48, 325202-1-17 (2015)
Liaw, C., Treil, S.: Rank one perturbations and singular integral operators. J. Funct. Anal. 257, 1947–1975 (2009)
Makarov, K.A., Seelmann, A.: The length metric on the set of orthogonal projections and new estimates in the subspace perturbation problem. J. Reine Angew. Math. 708, 1–15 (2015)
Ohtaka, K., Tanabe, Y.: Theory of the soft-X-ray edge problem in simple metals: historical survey and recent developments. Rev. Mod. Phys. 62, 929–991 (1990)
Otte, P.: An integral formula for section determinants of semi-groups of linear operators. J. Phys. A 32, 3793–3803 (1999)
Otte, P.: An adiabatic theorem for section determinants of spectral projections. Math. Nachr. 278, 470–484 (2005)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press, New York (1978)
Seelmann, A.: Notes on the \(\sin 2\Theta \) theorem. Integral Equ. Oper. Theory 79, 579–597 (2014)
Simon, B.: Spectral averaging and the Krein spectral shift. Proc. Am. Math. Soc. 126, 1409–1413 (1998)
Simon, B.: Trace Ideals and Their Applications, Mathematical Surveys and Monographs, vol. 120, 2nd edn. American Mathematical Society, Providence (2005)
Teschl, G.: Mathematical Methods in Quantum Mechanics, Graduate Studies in Mathematics, vol. 99. American Mathematical Society, Providence (2009)
Weyl, H.: The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton (1939)
Yafaev, D.R.: Mathematical Scattering Theory. General Theory. American Mathematical Society, Providence (1992)
Acknowledgements
M.G. thanks Alexander Pushnitski, Peter Müller and especially Adrian Dietlein for many illuminating discussions on this topic.
Author information
Authors and Affiliations
Corresponding author
Additional information
M.G. was supported by the DFG under Grant GE 2871/1-1.
Rights and permissions
About this article
Cite this article
Gebert, M. On an Integral Formula for Fredholm Determinants Related to Pairs of Spectral Projections. Integr. Equ. Oper. Theory 90, 35 (2018). https://doi.org/10.1007/s00020-018-2461-7
Received:
Revised:
Published:
DOI: https://doi.org/10.1007/s00020-018-2461-7
Keywords
- Differences of spectral projections
- Fredholm determinants
- Spectral shift function
- Subspace perturbation problem