Konstantin A. Makarov

Some Applications of Operator-valued Herglotz Functions

We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for classes of Herglotz operators. Moreover, we study the concrete case of Schrodinger operators on a half-line and provide two illustrations of Livsic’s result [44] on quasi-hermitian extensions in the special case of densely defined symmetric operators with deficiency indices (1,1).

The Threshold Effects for the Two-Particle Hamiltonians on Lattices

For a wide class of two-body energy operators h(k) on the d-dimensional lattice ℤd, d≥3, k being the two-particle quasi-momentum, we prove that if the following two assumptions (i) and (ii) are satisfied, then for all nontrivial values k, k≠0, the discrete spectrum of h(k) below its threshold is non-empty. The assumptions are: (i) the two-particle Hamiltonian h(0) corresponding to the zero value of the quasi-momentum has either an eigenvalue or a virtual level at the bottom of its essential spectrum and (ii) the one-particle free Hamiltonians in the coordinate representation generate positivity preserving semi-groups.

The Ξ operator and its relation to Krein's spectral shift function

We explore connections between Kreins spectral shift function ζ(λ,H0, H) associated with the pair of self-adjoint operators (H0, H),H=H0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K*(H0−λ−i0)−1K) associated with the operator-valued Herglotz functionJ+K*(H0−z)−1K, Im(z)>0 inH, whereV=KJK* andJ=sgn(V). Our principal results include a new representation for ζ(λ,H0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (EJ+A(λ)+tB(λ)(−∞, 0)),EJ((−∞, 0))), ℝ, whereA(λ)=Re(K*(H0−λ−i0−1K),B(λ)=Im(K*(H0−λ-i0)−1K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H0, H) coincides with the trindex associated with the pair (Ξ(J+K*(H0−λ−i0)K), Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ operators and the Fredholm determinant of the abstract scattering matrix.We also provide a generalization of the classical Birman—Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.

The Spectral Shift Operator

We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the Birman-Solomyak spectral averaging formula are discussed.

Perturbation of spectra and spectral subspaces

We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

ON A SUBSPACE PERTURBATION PROBLEM

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let A and V be bounded self-adjoint operators. Assume that the spectrum of A consists of two disjoint parts σ and Σ such that d = dist(σ, Σ) > 0. We show that the norm of the difference of the spectral projections E A (σ) and E A+V ({λ | dist(λ,σ) < d/2}) for A and A + V is less than one whenever either (i) ∥V∥ < 2/2+π d or (ii) ∥V∥ < 1/2 d and certain assumptions on the mutual disposition of the sets a and Σ are satisfied.

Extended Hilbert space approach to few‐body problems

A general formulation of the quantum scattering theory for a system of few particles, which have an internal structure, is given. Due to freezing out the internal degrees of freedom in the external channels, a certain class of energy‐dependent potentials is generated. By means of potential theory, a modified Faddeev equation is derived both in external and internal channels. The Fredholmity of these equations is proven and this is what provides a sound basis for solving the addressed scattering problem.

The Efimov Effect and an Extended Szegö–Kac Limit Theorem

Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit formulae obtained can be considered as a genuine generalization of Szegö–Kacs formula to symbols with real zeros. Connections with the Efimov effect are discussed.

Representation Theorems for Indefinite Quadratic Forms Revisited

The first and second representation theorems for sign-inde finite, not necessarily semi-bounded quadratic forms are revisited. New straightforward proofs of these theorems are given. A number of necessary and sufficient conditions ensur ing the second representation the- orem to hold is proved. A new simple and explicit example of a self-adjoint operator for which the second representation theorem does not hold is also provided.

ON KREIN'S EXAMPLE

In his 1953 paper [Matem. Sbornik 33 (1953), 597-626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval [-1,1] and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.