Digest for "Square Root Operators and the Well-Posedness of Pseudodifferential Parabolic Models of Wave Phenomena"

Abstract

Pseudodifferential parabolic equations with an operator square root arise in wave propagation problems as a one-way counterpart of the Helmholtz equation. The expression under the square root usually involves a differential operator and a known function. We discuss a rigorous definition of such operator square roots and show well-posedness of the pseudodifferential parabolic equation by using the theory of strongly continuous semigroups. This provides a justification for a family of widely-used numerical methods for wavefield simulations in various areas of physics.

Proposition 1 (Uniqueness of square roots)

Let \( H\) be a complex Hilbert space and let \( A\colon H \supseteq D(A) \to H\) be a closed linear operator such that \( (-\infty,0] \subseteq \rho(A)\) . There exists at most one closed linear operator \( B\colon H \supseteq D(B) \to H\) with the following properties:

\( B^2 = A\) .
The spectrum \( \sigma(B)\) is contained in the right half plane \( \{\lambda \in \mathbb{C} \mid \rm{Re} \lambda \ge 0\}\) .

Definition 1 (The square root of an operator)

Let \( H\) be a complex Hilbert space and let \( A\colon H \supseteq D(A) \to H\) be a closed linear operator such that \( (-\infty,0] \subseteq \rho(A)\) . We say that \( A\) has a square root if there exists a closed linear operator \( B\colon H \supseteq D(B) \to H\) which satisfies \( B^2 = A\) and \( \sigma(B) \subseteq \{\lambda \in \mathbb{C} \mid \rm{Re} \lambda \ge 0\}\) . In this case we call \( B\) the square root of \( A\) and denote it by \( B =: A^{1/2}\) .

Proposition 2 (Existence of square roots)

Let \( H\) be a complex Hilbert space and let \( A\colon H\supseteq D(A)\to H\) be a closed linear operator. Assume that there is a \( \delta > 0\) such that

\[ \sigma(A) \cup \overline{W(A)} \subseteq \mathbb{C}_{\rm{Im} \ge \delta} := \{\lambda \in \mathbb{C} \mid \rm{Im} \lambda \ge \delta \}. \]

Then \( A\) has a square root and the spectrum and numerical range of \( A^{1/2}\) are located in the first quadrant of \( \mathbb{C}\) and satisfy \( \sigma(A^{1/2}) \subseteq \overline{W(A^{1/2})}\) .

Corollary 1 (Generation theorem for \( \mathrm{i}\) times a square root)

Under the assumptions of Proposition 2 the operator \( \mathrm{i} A^{1/2}\) generates a contractive \( C_0\) -semigroup on \( H\) .

Example 1

Let \( H = L^2(\Omega)\) for a domain \( \Omega \subseteq \mathbb{R}^n\) , let \( L\colon H \supseteq D(L) \to H\) be a self-adjoint linear operator and let \( m\colon \Omega \to \mathbb{C}\) be a bounded and continuous (or, more generally, bounded and measurable) function that satisfies \( \rm{Im}(m(\omega)) \ge \delta\) for a \( \delta > 0\) and all \( \omega \in \Omega\) . Then the operator \( A := L+m\) with domain \( D(A) := D(L)\) satisfies \( \sigma(A) \cup \overline{W(A)} \subseteq \mathbb{C}_\rm{{Im} \ge \delta}\) , so Proposition 2 and Corollary 1 are applicable to \( A\) .

Remark 1

Example 1 gives the well-posedness of the PDPE (3) under the assumptions discussed before the example. The argument assumed that \( k^2(x,y)\) does not depend on \( x\) , but it can be directly generalized to the case where \( k^2(x,y)\) is piecewise-constant in \( x\) . More precisely, assume that the interval \( x\in [0,L]\) is divided into a set of \( N\) subintervals \( [x_{j-1},x_{j}]\) , \( j = 1,\dots,N\) , where \( x_0=0\) , \( x_{N}=L\) , and \( k^2(x,y) = k_j^2(y)\) for all \( x\in [x_{j-1},x_{j}]\) . Then the Cauchy problem (3) can be solved piecewise on the subintervals \( [x_{j-1},x_{j}]\) .

Table of contents

Digest for "Square Root Operators and the Well-Posedness of Pseudodifferential Parabolic Models of Wave Phenomena"

Abstract