Spectral Theory Of Self-adjoint Operators Research Articles

Large time behavior of weak solutions to the surface growth equation

This paper studies the existence and decay estimates of weak solutions to the surface growth equation. First, the global existence of weak solutions is obtained by the approximation method introduced by Majda and Bertozzi [Vorticity and Incompressible Flow (Cambridge University Press, 2001)]. Then, we derive the L2-decay rates of weak solutions via the Fourier splitting method under the assumption that u0∈L1(R)∩L2(R). For more general cases, i.e., u0∈L2(R), the behavior of weak solutions in L2 is obtained by the spectral theory of self-adjoint operators.

Modelling of derivatives pricing using methods of spectral analysis

In this article expands the method of finding the approximate price for a wide class of derivative financial instruments. Using the spectral theory of self-adjoint operators in Hilbert space and the wave theory of singular and regular perturbations, the analytical formula of the approximate asset price is established. Methods for calculating the approximate price of options using the tools of spectral analysis, singular and regular wave theory in the case of fast and slow factors are developed. Combining methods from the spectral theory of singular and regular perturbations, it is possible to estimate the price of derivative financial instruments as a schedule by eigenfunctions. The approximate value of securities and their rate of return are calculated. Applying the theory of Sturm-Liouville, Fredholm’s alternative and analysis of singular and regular perturbations at different time scales have enabled us to obtain explicit formulas for the approximate value of securities and their yield on the basis of the development of their eigenfunctions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. The theorem of closeness estimates for bond prices approximation is proved. An algorithm for calculating the approximate price of derivatives and the accuracy of estimates has been developed, which allows to analyze and draw precautionary conclusions and suggestions to minimize the risks of pricing derivatives that arise in the stock market. A model for finding the value of derivatives corresponding to the dynamics of the stock market and the size of financial flows has been developed. This model allows you to find the prices of derivatives and their volatility, as well as minimize speculative changes in pricing, analyze the progress of stock market processes and take concrete steps to improve the situation to optimize financial strategies. The used methodology of European options pricing based on the study of volatility behavior and analysis of the yield of financial instruments allows to increase the accuracy of the forecast and make sound management strategic decisions by stock market participants.

Standing Waves for Discrete Nonlinear Schrödinger Equations with Nonperiodic Bounded Potentials

In this paper, we investigate standing waves in discrete nonlinear Schrodinger equations with nonperiodic bounded potentials. By using the critical point theory and the spectral theory of self-adjoint operators, we prove the existence and infinitely many sign-changing solutions of the equation. The results on the exponential decay of standing waves are also provided.

Spectral study of options based on CEV model with multidimensional volatility

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.

On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis

We study conditions of discreteness of spectrum of the operator defined by , . The operator has two singularities at the ends of the interval . The second question is positivity of solutions of the equation under boundary conditions , . The used abstract scheme is close to the well-known MS Birman’s method in the spectral theory of self-adjoint operators. Conditions for discreteness of spectrum and positivity of the Green’s operator are obtained. The result relates to the MS Birman’s result on the necessary and sufficient condition for discreteness of spectrum of a polar-differential operation. The results may be interesting for researchers in qualitative theory of functional-differential equations and spectral theory of self-adjoint operators. MSC: Primary 34K08; 34K10; secondary 34K12.

On the Extension of a Certain Class of Carleman Operators

The integral operators of Carleman play an important role in the spectral theory of selfadjoint operators and made the object of several works such those of G. I. Targonski [Proc. Amer. Math. Soc. 18 (1967)(3)], V. B. Korotkov [Sib. Math. J. 11 (1970)(1)], and J. Weidmann [Manuscripta Math. (1970)(2)]. In the present paper, we study a certain class of these operators in the Hilbert space L2 (X,μ). Precisely, we give necessary and sufficient conditions so that they possess equal defeciency indices. Such operators find their applications in the theory of random variable approximation.

Lower bound for the spectrum and the presence of pure point spectrum of a singular discrete Hamiltonian system

In this paper, criteria for limit-point ( n) case of a singular discrete Hamiltonian system are established. Furthermore, the lower bound of the essential spectrum is obtained and the present of pure point spectrum is discussed for such system by using the spectral theory of self-adjoint operators in a Hilbert space.

Natural oscillations of mechanical systems and the spectral theory of self-adjoint operators

In this paper, the following assertion is illustrated with some examples. In practically all cases, the problem on natural oscillations for linear holonomic nondissipative mechanical systems reduces to the problem of determining eigenvalues and eigenelements of the corresponding self-adjoint operator acting in the Hilbert space generated by the quadratic form of the kinetic energy. Bibliography: 1 title.

Variable hilbert scales and their interpolation inequalities with applications to tikhonov regularization

Variable Hilbert scales are constructed using the spectral theory of self-adjoint operators in Hilbert spaces. An embedding and an interpolation theorem (based on Jenssen's inequality) are proved. They generalize known results about “ordinary” Hilbert scales derived by Natterer [Applic. Anal., 18 (1984), pp.29-37]. Bounds on best possible and actual errors for regularization methods are obtained by applying the interpolation inequality. These bounds extend the standard ones, and, in particular, include exponential and logarithmic error laws. Similar results were established earlier by Hegland [SIAM J. Numer. Anal., 29 (1992), pp. 1446-14611 for compact operators only. Here, they are generalized to include unbounded operators. A detailed discussion of Tikhonov regularization by Nair et al. [Tech. Rep. MR8-94, CMA, Aust. Nat. Uni., 19941 indicates that parameter choice strategies, which were thought to be suboptimal, can give substantially higher convergence rates than the so-called optimal choices! This improvement depends on how the regularizor is chosen. In some particularly difficult situations, however, the choice of regularizor cannot improve the convergence.

Mathematical analysis of the propagation of elastic guided waves in heterogeneous media

In this article, we are concerned with the propagation of elastic waves in isotropic heterogeneous media, invariant under translation in one direction. We give a theoretical analysis of the existence of guided waves and of their properties. In particular the thresholds, or cut-off frequencies, are studied in detail. The main mathematical tool is the spectral theory of selfadjoint operators, and more specifically the Max-Min principle.

On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators

The study of boundary value problems involving linear differentiai equations with real-valued coefficients is by now a well-established area of analysis. On the other hand, much less is known about the solvability of such problems when the coefficients (or boundary conditions) are known to be complex. Examples of this latter type arise naturally, for example, in nuclear physics (e.g., the so-called optical mode1 for low energy scattering [ 1, p. IjO]), electromagnetic field theory (dielectric waveguides with heat loss (c.f. 19 I), or the propagation of radio waves through inhomogeneous media [S]), and elsewhere. In cases like these, the relevant differential expressions are no longer formally symmetric, and hence the powerful methods associated with the spectral theory of selfadjoint operators are not available. To facilitate the study of such problems, Glazman introduced in [4] the concept of a J-symmetric operator: In a complex Hilbert space R, let J be a given conjugation operator on 2’ (i.e., J is a conjugate-linear involution with (Jx, 3~) = (u, x) for all x and y in 2). A closed, densely defined linear operator T in 3’ is said to be J-symmetric if