By Volodymyr Koshmanenko, Mykola Dudkin, Nataliia Koshmanenko
This monograph provides the newly built approach to rigged Hilbert areas as a latest technique in singular perturbation conception. A key thought of this strategy is the Lax-Berezansky triple of Hilbert areas embedded one into one other, which specifies the well known Gelfand topological triple.
All sorts of singular interactions defined by means of potentials supported on small units (like the Dirac δ-potentials, fractals, singular measures, excessive measure super-singular expressions) admit a rigorous therapy basically by way of the outfitted areas and their scales. the most suggestion of the strategy is to exploit singular perturbations to alter internal items within the beginning rigged area, and the development of the perturbed operator by way of the Berezansky canonical isomorphism (which connects the optimistic and destructive areas from a brand new rigged triplet). The method combines 3 strong instruments of practical research in line with the Birman-Krein-Vishik idea of self-adjoint extensions of symmetric operators, the idea of singular quadratic varieties, and the idea of rigged Hilbert spaces.
The e-book will attract researchers in arithmetic and mathematical physics learning the scales of densely embedded Hilbert areas, the singular perturbations phenomenon, and singular interplay problems.
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Additional resources for The Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators
Example text
The subset of A=0 (A) consisting of positive extensions we denote by A+ (A). One of the creators of the theory of operators and quantum mechanics, J. von Neumann, claimed that there exists at least one self-adjoint extension A˜ such that for each bounded from below symmetric operator A we have mA˜ = mA [161]. Later, Stone [178] and Friedrichs [86] proved this fact, and Kre˘ın [148] described the set of all positive self-adjoint extensions of a positive symmetric operator. If the operator A ≥ 1, then the method of rigged spaces gives an appropriated way for the construction of its self-adjoint extension preserving the lower bound.
If the operator A has eigenvalues, then they form a subset of the purely point spectrum σpp (A) ⊆ σ(A). We recall that λ is an eigenvalue of the operator A if Aϕ = λϕ, where ϕ = 0; then the vector ϕ is called the eigenvector of A. In the next considerations we are interested in unbounded symmetric or selfadjoint operators. 1) that points λ ∈ C with Im(λ) = 0 belong to the resolvent set of any self-adjoint operator. As a corollary, the spectrum of a self-adjoint operator belongs to the real axis.
3. Linear transformations Thus, the mapping 9 (1 − Δ)−k/2 : L2 −→ Wpk originally defined only on functions g ∈ S(Rn ), can be extended to a unitary operator on the closure. The negative Sobolev space W2−k , the space with the index −k < 0, is dual to W2k , k > 0, and is defined as the subset of generalized functions S (Rn ) with a finite norm ω W2−k := F −1 ((1 + |ξ|2 ))−k/2 (F ω)(ξ) L2 < ∞, ω ∈ S (Rn ). In this case, the closure of the mapping (1 − Δ)−k/2 : Wp−k −→ L2 is also a unitary operator. In different applications, the local Sobolev spaces Wpk (Ω) are used, where Ω is a compact or open set in Rn .
