K. A. Makarov

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We revisit the computation of (2-modified) Fredholm determinants for operators with matrix-valued semi-separable integral kernels. The latter occur, for instance, in the form of Green’s functions associated with closed ordinary differential operators on arbitrary intervals on the real line. Our approach determines the (2-modified) Fredholm determinants in(More)
We extend the concept of Lifshits–Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of (admissible) operators that are similar to self-adjoint operators. An operator H is called admissible if: (i) there is a bounded operator V with a bounded inverse such that H = V ĤV for some self-adjoint operator Ĥ; (ii)(More)
Let g(z, x) denote the diagonal Green’s matrix of a self-adjoint m ×m matrix-valued Schrödinger operator H = − d 2 dx2 Im +Q(x) in L2(R)m, m ∈ N. One of the principal results proven in this paper states that for a fixed x0 ∈ R and all z ∈ C+, g(z, x0) and g′(z, x0) uniquely determine the matrixvalued m×m potential Q(x) for a.e. x ∈ R. We also prove the(More)
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.
The principal results of this paper consist of an intrinsic definition of the Evans function in terms of newly introduced generalized matrix-valued Jost solutions for general first-order matrix-valued differential equations on the real line, and a proof of the fact that the Evans function, a finite-dimensional determinant by construction, coincides with a(More)
We provide additional results in connection with Krein’s formula, which describes the resolvent difference of two self-adjoint extensions A1 and A2 of a densely defined closed symmetric linear operator Ȧ with deficiency indices (n, n), n ∈ N ∪ {∞}. In particular, we explicitly derive the linear fractional transformation relating the operator-valued(More)
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(More)
Let A and C be self-adjoint operators such that the spectrum of A lies in a gap of the spectrum of C and let d > 0 be the distance between the spectra of A and C . We prove that under these assumptions the sharp value of the constant c in the condition ‖B‖ < cd implying the solvability of the operator Riccati equation XA−CX+XBX = B∗ is equal to √ 2. We also(More)
Let H0 and V (s) be self-adjoint, V, V ′ continuously differentiable in trace norm with V (s) ≥ 0 for s ∈ (s1, s2), and denote by {EH(s)(λ)}λ∈R the family of spectral projections of H(s) = H0+V (s). Then we prove for given μ ∈ R, that s 7−→ tr ( V ′(s)EH(s)((−∞, μ)) ) is a nonincreasing function with respect to s, extending a result of Birman and Solomyak.(More)