Derivatives of the Evans function and (modified) Fredholm determinants for first order systems

Fredholm determinants, Evans functions and Maslov indices for partial differential equations

We develop a difference equations analogue of recent results by F. Gesztesy, K. A. Makarov, and the second author relating the Evans function and Fredholm determinants of operators with semi-separable kernels.

A new approach to finding the exact Green's function to the time-dependent Schr\"odinger equation is described. The approach is conceptually simple: one seeks an operator which when commuted through the Schr\"odinger operator transforms it into the Schr\"odinger operator of a soluble problem. This approach provides an alternative to both the eigenfunction expansion of the propagator and the path integral. The first example of an exact propagator represented as a ``sum over classical paths'' which is not equal to the Gaussian approximation of the path integral is found.

We consider a class of translationally invariant magnetic fields such that the corresponding potential has a constant direction. Our goal is to study basic spectral properties of the Schr\"odinger operator ${\bf H}$ with such a potential. In particular, we show that the spectrum of ${\bf H}$ is absolutely continuous and we find its location. Then we study the long-time behaviour of solutions $\exp(-i {\bf H} t)f$ of the time dependent Schr\"odinger equation. It turnes out that a quantum particle remains localized in the plane orthogonal to the direction of the potential. Its propagation in this direction is determined by group velocities. It is to a some extent similar to a evolution of a one-dimensional free particle but "exits" to $+\infty$ and $-\infty$ might be essentially different.

In this chapter, the properties of the transfer function and frequency response of first and second order systems are studied on some examples from electrical circuit laws. We show that their properties are governed by the poles (i.e., the zeros of the denominator) of the transfer function which is a rational fraction. A geometric argument based on the location of the poles of the transfer function in the complex plane allows a qualitative interpretation of the behavior of the frequency response with varying frequency. This geometric interpretation is easily generalized to situations with any number of zeros and poles. It proves useful for the understanding of the general behavior of filters. The study begins here with the simplest system, the first order system. Then the second order circuit system is presented thoroughly. The logarithmic Bode representation of the frequency gain is introduced and its advantages demonstrated. The quality factor Q of a resonant circuit is defined.

There are two perspectives on the problem of human rights abuses: the perpetrator perspective and the system perspective. The perpetrator perspective focuses on the shortcomings of persons, blaming them for neglect, cruelty, or moral weakness, and it attempts correction through education, activism, and judicial intervention. The system perspective acknowledges that the social circumstances within which individuals live can surreptitiously co-opt otherwise good people into participating in human rights abuses. Here, the corrective approach needs to focus on the complex task of altering structures and functions of systems in order to overcome structurally propelled abuses of human rights, while at the same time recognizing the moral and legal importance of allocating individual blame for the violation of such rights.

The purpose of the present paper is to develop the inverse scattering transform for the nonlocal semi-discrete nonlinear Schrodinger equation (known as Ablowitz-Ladik equation) with PT-symmetry. This includes: the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data and the fundamental analytic solutions. In addition, the paper studies the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann-Hilbert problem, the 1- and 2-soliton solutions for the nonlocal Ablowitz-Ladik equation are derived. Finally, the completeness relation for the associated Jost solutions is proved. Based on this, the expansion formula over the complete set of Jost solutions is derived. This will allow one to interpret the inverse scattering transform as a generalised Fourier transform.

A representation is obtained for the Green’s function associated with the two point boundary problem $y'' = Ay' + By + f$, $y(0) = y(T) = 0$, where y and f are vector valued, A and B are closed linear operators. Certain results are applied to solving nonhomogeneous partial differential equations.

It is proved that the modified Fredholm determinant F of the three-dimensional Lippmann-Schwinger equation in the theory of scattering by spherically symmetric potentials is related to the Jost functions fl of angular momentum l by F= ∏ l=0∞{fl2i+1 exp[(2l+1) TrKl]},where Kl is the kernel of the lth radial Lippmann-Schwinger equation. The relation between the multiplicity of the zeros of F and the degeneracy is discussed, and a relevant theorem for Hilbert-Schmidt operators is proved.

During the 20th century, the electronics industry has evolved from radio, television, and audio systems to computer systems. The trace of evolution is clearly from linear systems to nonlinear systems. This is by no means accidental, and is due to the fact that nonlinear systems are more versatile than linear systems. As of now, the successful commercial optical instruments are all linear systems. From the entropy point of view, as the knowledge of human beings further expands, we need more powerful computing capabilities such as memory association and self organization. Based on the fundamental differences between the electron and the photon, this paper presents the argument that optical computing will provide memory association and self organization ability (which compliments the current electronic logical computing). Thus, one can infer that the main future thrust of the optical industry will be in nonlinear optical systems and hybrid combinations with nonlinear electronic systems in order to provide both self organization and logical computing (to emulate, more or less, the computing power of the human brain). Currently, the stat of the ar has already demonstrated successful combinations of linear optical system with nonlinear electronic systems (computers). These kinds of hybrid systems are called electro-photonic systems in the following curriculum. Based on the above vision for the future and taking into account the need for immediate employment, the following curriculum is designed to provide a core training to the students who are interested in choosing optics or electro-photonics as their career. This curriculum is being offered in the upper years at the Physics Department of Chung Yuan University. Electronics, Optics, Electro-photonic System Design and Analysis, Optical System Design and Analysis, Introduction to Optical Computing, Holography, Nonlinear Optics, The Experimental Technology for Electro- photonics (I), The Experimental Technology for Electro-photonics (II). The main thrust among all these courses is the system modeling and analysis. In this paper, the design of this curriculum will be elaborated and the topics for each course will be listed in detail.

We establish three results on the 3D Schr\"odinger equation with a potential having no spherical symmetry: the existence of a Wiener-Hopf factorization of the scattering operator, a method to recover the potential from the scattering data, and the existence of the 3D Jost operator.

In this paper we show how a second order scalar uniformly elliptic equation in divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary condition. This first order system involves Hodge-Dirac operators and can be seen as a natural generalization of the Beltrami equation in the plane and we develop a theory for this equation, extending results from the plane to higher dimension. The reduction to a first order system applies both to linear as well as quasilinear second order equations and we believe this to be of independent interest. Using the first order system, we give a new representation formula of the solution of the Dirichlet problem both on simply and finitely connected domains. This representation formula involves only singular integral operators of convolution type and Neumann series there of, for which classical Calderón-Zygmund theory is applicable. Moreover, no use is made of any fundamental solution or Green's function beside fundamental solutions of constant coefficient operators. Remarkably, this representation formula applies also for solutions of the fully non-linear first order system. We hope that the representation formula could be used for numerically solving the equations. Using these tools we give a new short proof of Meyers' higher integrability theorem. Furthermore, we show that the solutions of the first order system are Hölder continuous with the same Hölder coefficient as the solutions of the second order equations. Finally, factorization identities and representation formulas for the higher dimensional Beurling-Ahlfors operator are proven using Clifford algebras, and certain integral estimates for the Cauchy transform is extended to higher dimensions.

The spread of the SARS-CoV-2 virus and the related restrictions on commercial transactions have an economic effect whose scale is difficult assess. At the same time, the need for activities intended to prevent and counter the epidemic results in a significant increase in public spending, which undoubtedly threatens the financial security of the state. For the above reasons, numerous rules that modify the principles of public finance management were introduced to the Polish legal system after the state of epidemic had been declared. The aim of the article is to present and analyse the amendments in the field of management of the general reserve and special purpose reserves that were introduced to the Polish legal system in order to ensure the proper execution of budgetary tasks during the pandemic. This aim is achieved through the presentation and analysis of the so-called Second Anti-Crisis Shield, i.e. the Act of 31 March 2020 amending the Act on Special Solutions Related to Preventing, Counteracting and Combating COVID-19, Other Infectious Diseases and the Resulting Crisis, as well as the provisions of the so-called Budget-Related Act for 2021, i.e. the Act of 19 November 2020 r. on Special Solutions to Implement the Budget Act for 2021. The implementation of the indicated research objective leads to conclusion that the introduced changes significantly broaden the scope of the executive’s decision-making competence in the implementation of the state budget, which may raise doubts in the light of the constitutional principle of exclusivity of the legislative authority in shaping state revenues and expenditures.

The present paper deals with special solutions for linear functional differential equations with small delays. The importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions. Sufficient conditions for the existence of special solutions for nonautonomous equations in $\mathbb R^n$ are given. For autonomous equations in $\mathbb R^n$, results relating special solutions and characteristic values are presented. We also consider linear autonomous functional differential equations in Banach spaces, and use special solutions to study the asymptotic behaviour of their solutions. Several applications are given.

We study the determinant det(I−KPII) of an integrable Fredholm operator KPII acting on the interval (−s, s) whose kernel is constructed out of the Ψ-function associated with the Hastings–McLeod solution of the second Painleve equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann–Hilbert method, we evaluate the large s-asymptotics of det(I−KPII) .