Solvability of the operator Riccati equation in the Feshbach case. (English. Russian original) Zbl 07083420
Math. Notes 105, No. 4, 485-502 (2019); translation from Mat. Zametki 105, No. 4, 483-505 (2019).
Summary: Let \(L\) be a bounded \(2 \times 2\) block operator matrix whose main-diagonal entries are self-adjoint operators. It is assumed that the spectrum of one of these entries is absolutely continuous, being presented by a single finite band, and the spectrum of the other main-diagonal entry is entirely contained in this band. We establish conditions under which the operator matrix \(L\) admits a complex deformation and, simultaneously, the operator Riccati equations associated with the deformed \(L\) possess bounded solutions. The same conditions also ensure a Markus-Matsaev-type factorization of one of the initial Schur complements analytically continued onto the unphysical sheet(s) of the complex plane of the spectral parameter. We prove that the operator roots of this Schur complement are explicitly expressed through the respective solutions to the deformed Riccati equations.
MSC:
| 47A52 | Linear operators and ill-posed problems, regularization |
Keywords:
operator Riccati equation; Feshbach case; Friedrichs model; graph subspace; Resonance; unphysical sheetReferences:
| [1] | S. Albeverio, K. A. Makarov, and A. K. Motovilov, “Graph subspaces and the spectral shift function,” Canad. J. Math. 55 (3), 449-503 (2003). · Zbl 1074.47007 |
| [2] | K. A. Makarov, S. Schmitz, and A. Seelmann, “On invariant graph subspaces,” Integral Equations Operator Theory 85 (3), 399-425 (2016). · Zbl 1373.47016 |
| [3] | Kostrykin, V.; Makarov, K. A.; Motovilov, A. K., Existence and uniqueness of solutions to the operator Riccati equation. A geometric approach, 181-198 (2003), Providence, RI · Zbl 1066.47014 |
| [4] | S. Albeverio and A. K. Motovilov, “Sharpening the norm bound in the subspace perturbation theory,” Complex Anal. Oper. Theory 7 (4), 1389-1416 (2013). · Zbl 1288.47006 |
| [5] | A. Seelmann, “Notes on the subspace perturbation problem for off-diagonal perturbations,” Proc. Amer. Math. Soc. 144 (9), 3825-3832 (2016). · Zbl 1347.47010 |
| [6] | A. Seelmann, “On an estimate in the subspace perturbation problem,” J. Anal. Math. 135 (1), 313-343 (2018). · Zbl 06919519 |
| [7] | S. Okubo, “Diagonalization of Hamiltonian and Tamm-Dancoff equation,” Progr. Theoret. Phys. 12, 603-622 (1954). · Zbl 0058.43107 |
| [8] | L. L. Foldy and S. A. Wouthuysen, “On the Dirac theory of spin 1/2 particles and its non-relativistic limit,” Phys. Rev. 78, 29-36 (1950). · Zbl 0039.22605 |
| [9] | A. S. Markus and V. I. Matsaev, “On the spectral theory of holomorphic operator-valued functions in Hilbert space,” Funktsional. Anal. Prilozhen. 9 (1), 76-77 (1975) [Functional Anal. Appl. 9 (1), 73-74 (1975)]. · Zbl 0331.47008 |
| [10] | V. Adamjan and H. Langer, “Spectral properties of a class of operator-valued functions,” J. Operator Theory 33 (2), 259-277 (1995). · Zbl 0841.47010 |
| [11] | V. Adamyan, H. Langer, and C. Tretter, “Existence and uniqueness of contractive solutions of some Riccati equations,” J. Funct. Anal. 179 (2), 448-473 (2001). · Zbl 0982.47019 |
| [12] | V. M. Adamyan, H. Langer, R. Mennicken, and J. Saurer, “Spectral components of selfadjoint block operator matrices with unbounded entries,” Math. Nachr. 178, 43-80 (1996). · Zbl 0848.47019 |
| [13] | H. Langer, A. Markus, V. Matsaev, and C. Tretter, “A new concept for block operator matrices: the quadratic numerical range,” Linear Algebra Appl. 330 (1-3), 89-112 (2001). · Zbl 0998.15035 |
| [14] | R. Mennicken and A.A. Shkalikov, “Spectral decomposition of symmetric operator matrices,” Math. Nachr. 179, 259-273 (1996). · Zbl 0874.47009 |
| [15] | A. K. Motovilov, “Removal of the resolvent-like energy dependence from interactions and invariant subspaces of a total Hamiltonian,” J. Math. Phys. 36 (12), 6647-6664 (1995). · Zbl 0884.47056 |
| [16] | Kostrykin, V.; Makarov, K. A.; Motovilov, A. K., A generalization of the tan 2Θ theorem, 349-372 (2004), Basel · Zbl 1084.47002 |
| [17] | V. Kostrykin, K. A. Makarov, and A. K. Motovilov, “On the existence of solutions to the operator Riccati equation and the tan Θ theorem,” Integral Equations Operator Theory 51 (1), 121-140 (2005). · Zbl 1089.47015 |
| [18] | C. Davis and W. M. Kahan, “The rotation of eigenvectors by a perturbation. III,” SIAM J. Numer. Anal. 7, 1-46 (1970). · Zbl 0198.47201 |
| [19] | S. Albeverio and A. K. Motovilov, “The a priori tan Θ theorem for spectral subspaces,” Integral Equations Operator Theory 73 (3), 413-430 (2012). · Zbl 1276.47010 |
| [20] | A. K. Motovilov and A. V. Selin, “Some sharp norm estimates in the subspace perturbation problem,” Integral Equations Operator Theory 56 (4), 511-542 (2006). · Zbl 1121.47007 |
| [21] | Albeverio, S.; Motovilov, A. K., Bounds on variation of the spectrum and spectral subspaces of a few-body Hamiltonian, 98-106 (2016), Khabarovsk |
| [22] | S. Albeverio, A. K. Motovilov, and A. A. Shkalikov, “Bounds on variation of spectral subspaces under <Emphasis Type=”Italic“>J-self-adjoint perturbations,” Integral Equations Operator Theory 64 (4), 455-486 (2009). · Zbl 1197.47024 |
| [23] | S. Albeverio, A. K. Motovilov, and C. Tretter, “Bounds on the spectrum and reducing subspaces of a <Emphasis Type=”Italic“>J-self-adjoint operator,” Indiana Univ. Math. J. 59 (5), 1737-1776 (2010). · Zbl 1236.47032 |
| [24] | K. Veselić, “On spectral properties of a class of <Emphasis Type=”Italic“>J-selfadjoint operators. I,” Glasnik Mat. Ser. III 7, 229-248 (1972). · Zbl 0249.47027 |
| [25] | K. Veselić, “On spectral properties of a class of <Emphasis Type=”Italic“>J-selfadjoint operators. II,” Glasnik Mat. Ser. III 7, 249-254 (1972). · Zbl 0249.47028 |
| [26] | Al’beverio, S.; Motovilov, A. K., Operator Stieltjes integrals with respect to a spectral measure and solutions of some operator equations, 63-103 (2011), Moscow |
| [27] | R. Mennicken and A. K. Motovilov, “Operator interpretation of resonances arising in spectral problems for 2 × 2 operator matrices,” Math. Nachr. 201, 117-181 (1999). · Zbl 0932.47010 |
| [28] | H. Feschbach, “Unified theory of nuclear reactions,” Ann. Phys. 5 (4), 357-390 (1958). · Zbl 0083.44202 |
| [29] | V. Hardt, R. Mennicken, and A. K. Motovilov, “Factorization theorem for the transfer function associated with a 2 × 2 operator matrix having unbounded couplings,” J. Operator Theory 48 (1), 187-226 (2002). · Zbl 1019.47016 |
| [30] | Hardt, V.; Mennicken, R.; Motovilov, A. K., Factorization theorem for the transfer function associated with an unbounded non-self-adjoint 2 × 2 operator matrix, 117-132 (2003), Basel · Zbl 1057.47013 |
| [31] | S. Albeverio and A. K. Motovilov, “On invariant graph subspaces of a <Emphasis Type=”Italic“>J-self-adjoint operator in the Feshbach case,” Math. Notes 100 (6), 761-773 (2016). · Zbl 06701370 |
| [32] | K. O. Friedrichs, “On the perturbation of continuous spectra,” Comm. Pure Appl. Math. 1, 361-406 (1948). · Zbl 0031.31204 |
| [33] | G. Hagen, J. S. Vaagen, and M. Hjorth-Jensen, “The contour deformation method in momentum space, applied to subatomic physics,” J. Phys. A. Math. Gen. 37 (38), 8991-9021 (2004). · Zbl 1055.81077 |
| [34] | E. Balslev and J. M. Combes, “Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions,” Comm. Math. Phys. 22, 280-294 (1971). · Zbl 0219.47005 |
| [35] | C. Lovelace, “Practical theory of three-particle states. I. Nonrealtivistic,” Phys. Rev. B 135, 1225-1249 (1964). |
| [36] | M. Reed and B. Simon, Methods of Modern Mathematical Physics. IY. Analysis of Operators (Academic Press, London, 1978). · Zbl 0401.47001 |
| [37] | V. Hardt, A. Konstantinov, and R. Mennicken, “On the spectrum of the product of closed operators,” Math. Nachr. 215, 91-102 (2000). · Zbl 0965.47002 |
| [38] | M. Sh. Birman and M. Z. Solomyak, Spectrol Theory of Self-Adjoint Operators in Hilbert Space (Lan’, St. Petersburg., 2010) [in Russian]. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
