Search

The aim of this chapter is to illustrate how useful the functional calculus can be. In particular, we prove a version of the Lieb’s Variational Principle (to estimate traces of operators by means of density matrices). Then, we prove Stone’s formula that relates the spectral projections to the resolvent.

Transition and contraction operators, operator ergodic theorem, maximum inequalities, ratio ergodic theorem, filling operators and functionals, ratio limit theorem, space-time invariance, strong and weak ergodicity and mixing, tail and invariant triviality, Harris recurrence and ergodicity, potential and resolvent operators, convergence dichotomy, recurrence dichotomy, invariant measures, distributional and pathwise limits.

We consider a class of Lax operators L related to BD.I type symmetric spaces. They allow one to solve a special class of vector NLS equations which model Bose-Einstein condensates with spin. We construct the kernel of the resolvent for L and prove that the fundamental analytic solutions of L satisfy completeness relation on a subspace of the vector space \(\mathscr {M}\).

We investigate the low-energy behavior of the resolvent of Schrodinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions.

In this chapter, we apply the concept of resolvent that we developed in Sect. 1.4.1 for vector Volterra integral dynamic equations and show the boundedness of solutions. The resolvent is an abstract term which makes it difficult, if not impossible, to make efficient use of it. However, by the help of Lyapunov functionals and variation of parameters, we will be able to verify all the conditions that are related to the resolvent. In Sect. 6.1.1 we make use of the resolvent along with fixed point theory and show the existence and uniqueness of solutions of nonlinear Volterra dynamic equations. Later in the chapter, we consider nonlinear Volterra integral dynamic equations and construct suitable Lyapunov functional to obtain results regarding boundedness and stability of the zero solution. Contents of this chapter are totally new and not published anywhere else except those of Sect. 6.3 that can be found in Adivar and Raffoul (Appl Math Comput 273:258–266, 2016).

The regularized trace of the heat kernel of a one-dimensional Schrodinger operator with a singular two-particle contact interaction being of Lieb-Liniger type is considered. We derive a complete small-time asymptotic expansion in (fractional) powers of the time, t. Most importantly, we do not invoke standard parametrix constructions for the heat kernel. Instead, we first derive the large-energy expansion of the regularized trace of the resolvent for the considered operator. Then, we exploit that the resolvent may be obtained by a Laplace transformation of the heat semi-group, and an application of a suitable inverse Watson lemma eventually delivers the small-t asymptotic expansion of the heat-kernel trace.

Let S be a symmetric relation with finite and equal defect numbers in the Hilbert space \(\mathfrak H\). If \(\widetilde A\) is a self-adjoint extension of S in some larger Hilbert space \(\widetilde {\mathfrak H}\), the compression of \(\widetilde A\) to \({\mathfrak H}\) is a symmetric extension of S. We study this compression in dependence of the parameter \({\mathcal T}\), which parametrizes the extensions \(\widetilde A\) according to M.G. Krein’s resolvent formula. By means of a fractional transformation, analogous results are proved for the Straus extensions of S at a real point.

Some inverse scattering problems for the three-dimensional biharmonic operator are considered. The operator is perturbed by first and zero order perturbations, which may be complex-valued and singular. We show the existence of the scattering solutions in the Sobolev space \(W^1_{\infty }(R^3)\). One of the main result of this paper is the proof of analogue of Saito’s formula (in different form as known before), which can be used to prove a uniqueness theorem for the inverse scattering problem. Another main result is to obtain the estimates for the kernel of the resolvent of the direct operator in \(W^1_{\infty }\) and to prove the reconstruction formula for the unknown coefficients of this perturbation.

Harnessing the abstract power of the celebrated result due to Borichev and Tomilov (Math. Ann. 347:455–478, 2010, no. 2), we study the energy decay in a one-dimensional coupled wave-heat-wave system. We obtain a sharp estimate for the rate of energy decay of classical solutions by first proving a growth bound for the resolvent of the semigroup generator and then applying the asymptotic theory of \(C_0\)-semigroups. The present article can be naturally thought of as an extension of a recent paper by Batty, Paunonen, and Seifert (J. Evol. Equ. 16:649–664, 2016) which studied a similar wave-heat system via the same theoretical framework.

In this chapter, we provide a brief introduction to time scale calculus and introduce fundamental concepts that we need throughout this book. In Sect. 1.2 based on the work of Peterson and Tisdell (J Differ Equ Appl 10(13–15):1295–1306, 2004), we introduce the concept of Lyapunov functions for ordinary dynamical equations on time scales and prove general theorems in terms of wedges in which we derive specific conditions for stability of the zero solution and the boundedness of all solutions. We proceed to prove new and general theorems in terms of wedges and the existence of Lyapunov functions that satisfy certain conditions and give necessary conditions for stability and instability of the zero solution. In Sect. 1.4 we furnish all the details in proving the existence of the resolvent of Volterra integral dynamic equations by appealing to the results of Adivar and Raffoul (Bull Aust Math Soc 82(1):139–155, 2010). In addition, we will introduce the notion of shift that we make use of in Chaps. 5 and 8, and utilize the notion of resolvent and develop new results concerning Volterra dynamic equations. We end the chapter by introducing the notion of periodicity. In the famous paper (Kuchment, Floquet theory for partial differential equations, Birkhauser Verlag, Basel, 1993) of Raffoul and Kaufmann, the notion of periodicity on time scales was first introduced. Later on, Adivar (Math Slovaca 63(4):817–828, 2013) generalized the definition of periodicity to general time scales by introducing the concept of periodic shift operator. We end the chapter with some interesting and meaningful open problems that should be somewhat challenging.

The control algebraic Riccati equation is studied for a class of systems with unbounded control and observation operators. Using a dichotomy property of the associated Hamiltonian operator matrix, two invariant graph subspaces are constructed which yield a nonnegative and a nonpositive solution of the Riccati equation. The boundedness of the nonnegative solution and the exponential stability of the associated feedback system is proved for the case that the generator of the system has a compact resolvent.

We provide and discuss an integral resolvent criterion for generation of bounded \(C_0\)-semigroups on Banach spaces.

We establish semiclassical resolvent estimates for Schrodinger operators with long-range matrix-valued potentials. As an application we prove resonance free domains both in trapping and non-trapping situations. Our results generalize the well-known results of Burq (Am J Math 124(4):677–735, 2002), Martinez (Ann Henri Poincare 4:736–75, 2002) in the case of scalar Schrodinger operators.

Let A and A1 be unbounded selfadjoint operators in a Hilbert space \( \mathcal H \). Following [3], we call A1 a singular perturbation of A if A and A1 have different domains \( \mathcal D(A), D(A_1) \) but \( \mathcal D(A) \cap D(A_1) \) is dense in \( \mathcal H \) and A = A1 on \( \mathcal D(A) \cap D(A_1) \). In this note we specify without recourse to the theory of selfadjoint extensions of symmetric operators the conditions under which a given bounded holomorphic operator function in the open upper and lower half-planes is the resolvent of a singular perturbation A1 of a given selfadjoint operator A.

This is a short report on the authors’ recent work on spectral theory of N-body Schrodinger operators (Adachi et al., New Methods in Spectral Theory of N-Body Schrodinger Operators, arXiv:1804.07874 [math-ph]). We present Rellich’s theorem, the limiting absorption principle, microlocal resolvent bounds, and a microlocal Sommerfeld uniqueness result. Our assumptions on pair-potentials are minimal: Each of them consists of a long-range term with first order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type. In addition, our setup also allows hard-core interactions. The proofs depend on a commutator scheme recently proposed by two of the authors (Ito and Skibsted, J. Funct. Anal. 278(9), 108449 (2020)).

This chapter is exclusively devoted to the study of Volterra integro-dynamic equations with or without delay. We will display some exotic Lyapunov functionals to obtain stability and instability of the zero solutions and boundedness of solutions. The delay is in the form of shift operators so that all type of general time scales can be considered and without the requirement that they be additive. We prove variant forms of Gronwall’s inequality, so we can determine function bounds for the solutions of the integro-dynamic equations. We will develop in detail the notion of principal matrix solution for Volterra integro-dynamic equations and then build on the same concept to fully develop the existence of resolvent kernel which will be used along with Lyapunov functionals to obtain necessary and sufficient conditions for uniform stability and uniform asymptotic stability of the zero solution. Then we advance to functional delay dynamic equations which we write as Volterra integro-dynamic equations and study their behaviors via Lyapunov functionals. Toward the end of the chapter we classify the positive and negative solutions of nonlinear systems of Volterra integro-dynamic equations by appealing to Schauder’s and Knaster fixed point theorems. As usual, we end the chapter with some interesting and rewarding open problems. Some results are new and the rest can be found in Adivar (Electron J Qual Theory Differ Equ 2010(7):1–22, 2010; Glasg Math J 53(3), 463–480, 2011), Bohner (Far East J Appl Math 18(3):289–304, 2005), Burton (J Math Anal Appl 28:545–552, 1969; Nonlinear Anal 1(4):331–338, 1976/1977), Eloe et al. (Dynam Syst Appl 9(3):331–344, 2000), Kulik and Tisdell (Int J Differ Equ 3(1):103–133, 2008), Miller (J Differ Equ 10, 485–506, 1971), Raffoul (JIPAM J Inequal Pure Appl Math 10(3):70, 9, 2009), and Wang (J Math Anal Appl 186(3):8350–861, 1994; J Math Anal Appl 298(1):33–44, 2004).

In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag–Leffler functions. At the end, some topics are indicated for further study and possible generalizations. Also the aim of the paper is to attract attention and give references to not widely known results on fractional powers of the Bessel differential operator. This class of fractional operators is in close connection with transmutation theory and classic transmutational operators. We also study connections of Bessel fractional operators with different kinds of integral transforms.

This chapter introduces the notion of the spectrum of an operator (possibly unbounded) on a Hilbert space. The theory of the resolvent operator is developed and used to establish basic properties of the spectrum.

In these lecture notes, we study free boundary problems for the Navier–Stokes equations with and without surface tension. The local well-posedness, global well-posedness, and asymptotics of solutions as time goes to infinity are studied in the Lp in time and Lq in space framework. To prove the local well-posedness, we use the tool of maximal Lp–Lq regularity for the Stokes equations with nonhomogeneous free boundary conditions. Our approach to proving maximal Lp–Lq regularity is based on the \({\mathcal R}\)-bounded solution operators of the generalized resolvent problem for the Stokes equations with non-homogeneous free boundary conditions and the Weis operator-valued Fourier multiplier.