distributions; singular perturbations; scattering theory; spectral theory of operators; indefinite metric; Pontryagin and Krein spaces; linear relations; operator representation of holomorphic functions.
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.
Biography
Graduated from Faculty of Physics of M. V. Lomonosov Moscow State University (MSU) in 1975 (department of quantum theory). Ph.D. thesis was prepared and defended in 1979 at the V. A. Steklov Mathematical Institute. Senior Scientist since 1988. A list of my works contains more than 30 titles.
In the period 1978–1991 I worked at the Institute of Applied Physics of Moldovian Academy of Science (Department of Statistical Physics). Since 1991 I have been working at Nizhny Novgorod Pedagogical University (Department of Theoretical Physics). Chief of this department since 2001.
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.
Yu. G. Shondin, “Singular point perturbations of an odd operator in a $\mathbb Z_2$-graded space”, Mat. Zametki, 66:6 (1999), 924–940; Math. Notes, 66:6 (1999), 764–776
Yu. G. Shondin, G. F. Sarafanov, “Спектральная задача для двумерных операторов Шредингера и Дирака при наличии сингулярных магнитных вихрей”, Matem. Mod., 9:10 (1997), 23
1996
3.
Yu. G. Shondin, “Semibounded local hamiltonians for perturbations of the laplacian supported by curves with angle points in $\mathbb R^4$”, TMF, 106:2 (1996), 179–199; Theoret. and Math. Phys., 106:2 (1996), 151–166
Yu. G. Shondin, “On the semi-boundedness for point interactions with the support on curves with corner points”, Matem. Mod., 7:5 (1995), 69
5.
Yu. G. Shondin, “About semiboundness of $\delta$-perturbations of the Laplacian supported by curves with angle points”, TMF, 105:1 (1995), 3–17; Theoret. and Math. Phys., 105:1 (1995), 1189–1200
Yu. G. Shondin, “Perturbations of elliptic operators on high codimension subsets and the extension theory on an indefinite metric space”, Zap. Nauchn. Sem. POMI, 222 (1995), 246–292; J. Math. Sci. (New York), 87:5 (1997), 3941–3970
Yu. G. Shondin, “Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space”, TMF, 92:3 (1992), 466–472; Theoret. and Math. Phys., 92:3 (1992), 1032–1037
Yu. G. Shondin, “Quantum-mechanical models in $R_n$ associated with extensions of the energy operator in a Pontryagin space”, TMF, 74:3 (1988), 331–344; Theoret. and Math. Phys., 74:3 (1988), 220–230
Yu. G. Shondin, “Generalized pointlike interactions in $R_3$ and related models with rational $S$ matrix II. $l=1$”, TMF, 65:1 (1985), 24–34; Theoret. and Math. Phys., 65:1 (1985), 985–992
Yu. G. Shondin, “Generalized pointlike interactions in $R_3$ and related models with rational $S$-matrix”, TMF, 64:3 (1985), 432–441; Theoret. and Math. Phys., 64:3 (1985), 937–944
Yu. G. Shondin, “Asymptotic expansion of the commutator of heisenberg fields with respect to finite-dimensional irreducible representations of the Lorentz group”, TMF, 37:1 (1978), 58–65; Theoret. and Math. Phys., 37:1 (1978), 874–879
13.
Yu. G. Shondin, “Fourier expansion associated with the Lorentz group in the space of functions with support outisde the light cone”, TMF, 36:3 (1978), 303–312; Theoret. and Math. Phys., 36:3 (1978), 752–759
Yu. G. Shondin, “Fourier expansion associated with the Lorentz group in the space of generalized functions with support in the light cone”, TMF, 34:1 (1978), 23–33; Theoret. and Math. Phys., 34:1 (1978), 14–21
Yu. M. Shirokov, Yu. G. Shondin, “Expansion of generalized functions with support in the light cone with respect to a continuous set of irreducible representations of the Lorentz group and comparison with Wilson expansions”, TMF, 31:2 (1977), 147–155; Theoret. and Math. Phys., 31:2 (1977), 375–380
Yu. G. Shondin, “Spectral properties of bilocal operators in the expansion of a product of local operators on the light cone”, TMF, 26:3 (1976), 309–315; Theoret. and Math. Phys., 26:3 (1976), 208–212
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