Transient tunnel effect and Sommerfeld problem (Q2785475)

This monograph involves 80 lemmas, 22 theorems, 7 corollaries along with some conjectures. It consists of a revised version of the author's habilitation thesis and is devoted to the study of two particular problems, namely:NEWLINENEWLINENEWLINE1. The Klein-Gordon equation in one space dimension with a step, andNEWLINENEWLINENEWLINE2. The Helmholtz equation in two space dimensions with Dirichlet conditions imposed on a half-axis (i.e. the Sommerfeld half-plane problem).NEWLINENEWLINENEWLINEA common feature of these problems is the so-called `Limiting Absorption Principle' which plays often an extremely important role in solving diffraction problems. The monograph is composed of two main chapters devoted, respectively, to the above-mentioned problems as well as of a rather long appendix devoted to the study of some properties of the nonnegative selfadjoint operators used in the book. The book involves also a discussion of the physical and philosophical implications of some experimental as well as theoretical results to the first problem (more clearly, to the relativistic tunnel effect). A rather long introductory chapter explains the physical importance of the problems and gives some historical backgrounds. The mathematical results given in Chapter 1, were already published in [Math. Methods Appl. Sci. 17, No. 9, 697-752 (1994; Zbl 0803.35115)].NEWLINENEWLINENEWLINEThe first problem treated in Chapter 1 consists in finding the solution \(u_j(t,x):[0,\infty)\times N_j\to\mathbb{R}\) \((j=1,2)\) satisfying the relations NEWLINE\[NEWLINE\partial^2_tu_j- a_j\partial^2_xu_j+ c_ju_j=0\quad\forall t\in[0,\infty),\quad x\in N_j,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu_j(0,x)= u_{0,j}(x),\;\partial_t u_j(0,x)= v_{0,j}(x)\;\forall x\in N_j,\;u_1(t,0)= u_2(t,0)\;\forall t\in[0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINEd_1\partial_x u_1(t,0^+)+ d_2\partial_xu_2(t, 0^+)= 0\quad\forall t\in [0,\infty),NEWLINE\]NEWLINE where \(a_j\), \(c_j\) and \(d_j\) are given constants while \(u_{0,j}(x)\) and \(v_{0,j}(x)\) are given functions and \(N_j\cong [0,\infty)\). A main objective of this chapter is to show that \(L^\infty\)-type Sobolev norms of the solution and its first time derivative decay at least as \(\text{const }t^{-1/4}\) as the time \(t\) tends to infinity. To this end he uses the explicit knowledge of the ordered spectral representation of the spatial part of the problem and the expansion of the solution in generalized eigenfunctions as well as some Van der Corput-type estimates. The problem treated in Chapter 2 consists of the classical Sommerfeld Problem, namely: To find the functions \(u^{(k)}(x,y)\) satisfying the following relations: NEWLINE\[NEWLINE(\Delta+ k^2)u^{(k)}(x,y)= f(x,y),\;(x,y)\in\mathbb{R}^2\backslash\Sigma,\;u^{(k)}(x,0)=0,\;0\leq x.NEWLINE\]NEWLINE Here \(\Sigma\) stands for the positive \(x\)-axis (Remark that no radiation condition is prescribed. It is known that solutions of this problem for \(\text{Im }k>0\) can easily be obtained via Wiener-Hopf techniques. One of the basic problems connected with these solutions is the study of their limit when \(\text{Im }k\to\pm 0\). The so-called `Limiting Absorption Principle' states that in a convenient topology the above-mentioned solutions converge to limit solutions whenever \(\text{Re }k>0\). The main result of this Chapter is that \(u^{(k_1+ik_2)}\) converges to \(u^{(k_1)}\) for \(k_2\to 0\) in \(L^p_{\text{loc}}(\mathbb{R}^2)\) for all \(1\leq p<\infty\). To this end he starts from a formula due to \textit{F.-O. Speck} [Proc. R. Soc. Edinb., Sect. A, 104, 261-277 (1986; Zbl 0626.35020)] and gives two reformulated versions of it. Then he uses a special method to control rapidly oscillating integrands by conveniently deforming the integration paths in the complex plane. Thus he gets the expressions involving some `generalized eigenfunctions'.