Similarity of $$sgn x( - \tfrac{{d^2 }}{{dx^2 }} + c\delta )$$ Type Operators to Normal and Self-adjoint Operators

Mathematical Notes -     

The similarity problem for J-nonnegative Sturm-Liouville operators

Sufficient conditions for the similarity of the operator with an indefinite weight r(x)=(sgnx)|r(x)| are obtained. These conditions are formulated in terms of Titchmarsh-Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm-Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm-Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r(x)=sgnx and , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q≡0, we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward-backward” diffusion equations.     

Feedback controls for stochastic equations with monotone coefficients

We study the behavior of step approximations of solutions of stochastic evolution equations. The results obtained are applied to controlled stochastic equations. Under certain assumptions about the cost functional and the coefficients of the equation the coincidence of optimal costs for the classes of feedback controls and generalized control is proved.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 28–33, 1988.     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in L²(-p, p)\mathcal{L}^2(-\pi, \pi). Our method can be applied to a wide range of evolution problems given by PT-symmetric operators.     

Optimization of Quasi-Normal Eigenvalues for Krein–Nudelman Strings

The paper is devoted to optimization of resonances for Krein strings with total mass and statical moment constraints. The problem is to design for a given ${\alpha \in \mathbb{R}}$ a string that has a resonance on the line ${\alpha + {\rm i}\mathbb{R}}$ with a minimal possible modulus of the imaginary part. We find optimal resonances and strings explicitly.     

On approximations of solutions of stochastic equations with monotone coefficients

An estimate is obtained for the rate of convergence for step function approximations of solutions of stochastic differential equations with monotone coefficients.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 25–28, 1986.