Theory Of Self-adjoint Operators Research Articles

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.

Resolvent and propagation estimates for Klein–Gordon equations with non-positive energy

We study in this paper an abstract class of Klein-Gordon equations: \[ \p_{t}^{2}\phi(t)- 2\i k \p_{t}\phi(t)+ h \phi(t)=0, \] where $\phi: \rr\to \cH$, $\cH$ is a (complex) Hilbert space, and $h$, $k$ are self-adjoint, resp. symmetric operators on $\cH$. We consider their generators $H$ (resp. $K$) in the two natural spaces of Cauchy data, the energy (resp. charge) spaces. We do not assume that the dynamics generated by $H$ or $K$ has any positive conserved quantity, in particular these operators may have complex spectrum. Assuming conditions on $h$ and $k$ which allow to use the theory of selfadjoint operators on Krein spaces, we prove weighted estimates on the boundary values of the resolvents of $H$, $K$ on the real axis. From these resolvent estimates we obtain corresponding propagation estimates on the behavior of the dynamics for large times. Examples include wave or Klein-Gordon equations on asymptotically euclidean or asymptotically hyperbolic manifolds, minimally coupled with an external electro-magnetic field decaying at infinity.

Decay of solutions of a second order differential equation with non-smooth second member

Consider the following abstract initial value problem(⁎)|u″(t)+μ(t)Au(t)+a(|A−θ2u(t)|2)u(t)+b(|A−ηu′(t)|)Au′(t)=f(t)in (0,∞);u(0)=u0,u′(0)=u1 in a real separable Hilbert space H with norm |u|. Here A is a positive self-adjoint operator of H; μ(t),a(s),b(s) positive functions, f(t) a vectorial non-smooth function and θ, η real numbers. In this paper we study the existence, uniqueness and decay of solutions of problem (⁎). In our approach, we use the Theory of Self-Adjoint Operators in Hilbert spaces, the compactness Aubin–Lions Theorem and a Lyapunov functional.

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 J-self-adjoint operators with stable C-symmetries

The paper is devoted to the development of the theory of self-adjoint operators in Krein spaces (J-self-adjoint operators) involving some additional properties arising from the existence of C-symmetries. We mainly focus on the recent notion of stable C-symmetry for J-self-adjoint extensions of a symmetric operator S. The general results involve boundary value techniques and reproducing kernel space methods, and they include an explicit functional model for the class of stable C-symmetries. Some of the results are specialized further by studying the case where S has defect numbers 〈2,2〉 in detail.

Partitioning technique for open systems

The focus of the present contribution is essentially confined to three research areas carried out during the author's turns as visiting (assistant, associate and full) professor at the University of Florida's Quantum Theory Project, QTP. The first two topics relate to perturbation theory and spectral theory for self-adjoint operators in Hilbert space. The third subject concerns analytic extensions to non-self-adjoint problems, where particular consequences of the occurrence of continuous energy spectra are measured. In these studies general partitioning methods serve as general cover for perturbation-, variational- and general matrix theory. In addition we follow up associated inferences for the time dependent problem as well as recent results and conclusions of a rather general yet surprising character. Although the author spent most of his times at QTP during visits in the 1970s and 1980s, collaborations with department members and shorter stays continued through later decades. Nevertheless the impact must be somewhat fragmentary, yet it is hoped that the present account is sufficiently self-contained to be realistic and constructive.

Double Operator Integrals in a Hilbert Space

Double operator integrals are a convenient tool in many problems arising in the theory of self-adjoint operators, especially in the perturbation theory. They allow to give a precise meaning to operations with functions of two ordered operator-valued non-commuting arguments. In a different language, the theory of double operator integrals turns into the problem of scalarvalued multipliers for operator-valued kernels of integral operators. The paper gives a short survey of the main ideas, technical tools and results of the theory. Proofs are given only in the rare occasions, usually they are replaced by references to the original papers. Various applications are discussed.

Irreversibility, Lax± Phillips Approach to Resonance Scattering and Spectral Analysis of Non-Self-Adjoint Operators in Hilbert Space

Since the publication of the very first paperson quantum mechanics the theory of self-adjointoperators in Hilbert space has been a basic tool ofquantum theories. It turns out that the description of the irreversible dynamics of complex systemsrequires the development of the spectral theory ofnon-self-adjoint operators as well. In this paper weconsider the Hilbert space version of the theory ofdissipative operators, which appear as generators of theevolution reduced to a properly selected observationsubspace. The spectral analysis of these operators isbased on ideas of the functional model and dilation theory rather than on traditional resolventanalysis and Riesz integrals. The role of the parameterof the functional model is played by an analyticfunction — the characteristic function —which is interpreted and calculated as a scattering matrix for therelevant scattering problem. Thus the most importantobject of the spectral analysis of dissipative operatorsappears as an element of spectral analysis of a self-adjoint spectral problem. This paper isintended both as an introduction and a sort of bilingualtext for specialists in harmonic analysis and operatortheory who are interested in mathematical problems of the description of irreversible dynamics.The last part describes original results of the authorpublished in different journals during the lastdecade.

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.

Toward the rank-one singular perturbation theory of self-adjoint operators

The perturbation theory is developed for the case where an arbitrary positive self-adjoint operator is perturbed by the projector on a generalized vector. Similar to the well-known problem −Δ+λδ, we obtain, in the general case, explicit representations for singularly perturbed operators and their resolvents, and we find the point spectrum and an explicit form of the corresponding eigenvectors. Our approach differs from the usual ones and is based on the self-adjoint extension theory of semibounded operators.

Multiparameter spectral theory of singular differential operators

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

Strong Resolvent Convergence of Diffusion Operators

It is shown that differential operators arising from boundary value problems with eigenvalue parameter in the boundary condition occur as the limits, in the sense of a generalized notion of strong resolvent convergence, of families of Sturm–Liouville operators modeling heat flow in a rod, where the diffusion coefficient becomes arbitrarily large in half of the rod, thus modeling a mixing-diffusion problem. The generalized notion of strong resolvent convergence is defined, and a development is given in the setting of the abstract theory of self-adjoint operators in a Hilbert space and the theory of semigroups.

Further results on stability boundaries and a weak nonlinear analysis for double diffusive convection with rotation

We review some results for linear stability when a rotating doubly diffusive layer is studied. Some additional stability boundaries for ‘salt-finger’ and diffusive convection are predicted. Using the theory of self adjoint operators, the variation of the critical eigenvalue with physical parameters is examined at steady conditions. A weak non-linear analysis that uses a truncated Fourier series representation provides concentration and temperature profiles, and shows that heat and mass transport increase with Rayleigh number but decrease with Prandtl number diffusivity ratio and Taylor number.

The effects of domain size and fluid conditions at the boundary, on convection inside an annulus heated from below

We use the theory of self adjoint operators to come to some definite conclusions on the behaviour of the critical Rayleigh number with domain size and rigidity of sides in an annulus. The annulus which has two aspect ratios, provides an interesting connection between a cylinder and rectangular channel. The critical Rayleigh number does not monotonically decay with increase in lateral domain size but increases monotonically as the sides become more rigid and as we proceed from total slip to the no slip condition. The analysis also provides a useful way of obtaining first order effects in an annulus as the domain changes or rigidity changes. Our analysis provides monotonicity results only because of the ability to formulate the equations in the form of operator which are self adjoint in a certain inner product.

Some differential inequalities for thermohaline convection in bounded containers of arbitrary shape

Abstract We have used the theory of self adjoint operators to derive a number of useful differential inequalities for thermohaline convection in containers. Variation of critical eigenvalues with boundary conditions, domain size and fluid properties is shown. Exchange of stabilities is assumed in general but is shown to be always true for the case of fluids, which are salted above and heated below. The differential inequalities are useful to obtain first order effects of the various parameters on the critical eigenvalues.

Theory of self-adjoint operators generated by strongly singular, second-order expression of divergence type

Theory of self-adjoint operators generated by strongly singular, second-order expression of divergence type

The theory of self-adjoint operators in banach spaces with a Hermitian form

The theory of self-adjoint operators in banach spaces with a Hermitian form

An elementary inequality in function theory

In the theory of self-adjoint operators in Hilbert space and of formally self-adjoint linear differential equations there are many situations involving analytic functions on the complex plane whose singularities are confined to the real axis and where the growth of the function at such singular points is strictly limited.