Speciality:
01.04.02 (Theoretical physics)
Birth date:
30.01.1952
Phone: +7 (8312) 47 04 73
E-mail: Keywords: distributions; singular perturbations; scattering theory; spectral theory of operators; indefinite metric; Pontryagin and Krein spaces; linear relations; operator representation of holomorphic functions.
UDC: 517, 517.5, 517.9, 517.43, 530.145
MSC: 47B50, 47B25, 47A70, 81Q10, 81Q15
Subject:
The self-adjoint realization problem in an indefinite metric space for high-order singular perturbations of a differential operator was investigated. The point perturbation of the Laplacian in ${\bf R}^n$ with $n\ge 4$ as well as point-like perturbations acting in subspaces with nonzero angular momentum are good examples. In the case of perturbations supported by finite number points a solution of this problem was given in terms of canonical self-adjoint extensions of a symmetric operators in appropriate Pontryagin spaces $\Pi_\kappa$ with a negative index $\kappa$, which is determined by the order and rank of singular perturbation. This symmetric operator is completely characterized by a generalized Nevanlinna function of the form $Q(z)=(z^2+1)^{\kappa}Q_0(z)+P_{2kappa-1}(z)$, where $Q_0(z)$ is a matrix Nevanlinna function and $P_{2kappa-1}(z)$ is a self-adjoint matrix polynomial of degree at most $2\kapa-1$. The generalized Nevanlinna functions of this form and their operator representations play the key role in our method. More detail theory was developed (jointly with A. Dijksma, H. Langer, A. Luther and C. Zeinstra) for singular rank one perturbation. In particular an algorithm was described which solves separation of the nonpositive type spectrum from the rest of the spectrum. As by-product a new factorization of generalized Nevanlinna functions was derived. It was shown also how factorization lead to Hilbert space operators which serve as suitable Hamiltonians. The last result was generalized and the factorization of arbitrary generalized Nevanlinna function from the class $N_\kappa$ in product of a rational Blaschke factor and a Nevanlinna function from $N_0$ was proven.
Main publications:
Dijksma A., Langer H., Luger A., Shondin Yu. A factorization result for generalized Nevanlinna functions of the class $N_\kappa$ // Integral Equations and Operator Theory. 2000. V. 36. P. 121–125.
Dijksma A., Langer H., Shondin Yu., Zeinstra C. Self-adjoint operators with inner singularities and Pontryagin spaces // Operator Theory: Adv. Appl. V. 118. Birkhauser Verlag, Basel, 2000. P. 105–175.
