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

Matthias Ehrhardt, Jochen Glück, Pavel Petrov, Stefan Tappe

TL;DR

The paper addresses well-posedness of pseudodifferential parabolic models for wave phenomena by rigorously defining the operator square root $A^{1/2}$ and proving that $\mathrm{i}A^{1/2}$ generates a strongly continuous semigroup, ensuring existence, uniqueness, and continuous dependence for the Cauchy problem $\partial u/\partial x = i\sqrt{A}\,u$. It develops existence and uniqueness theory for operator square roots under accretivity and spectral-range conditions, leveraging Kato's theory and the Lumer–Phillips theorem to show generation of a contractive $C_0$-semigroup. The framework applies to $A = \frac{\partial^2}{\partial y^2} + k^2$ with positive real and imaginary parts of $k$, and generalizes to piecewise-constant in $x$ media. These results provide a solid mathematical justification for widely used PDPE-based numerical methods in wavefield simulations, such as SSP approaches, by guaranteeing well-posed evolution in the range coordinate.

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.

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

TL;DR

The paper addresses well-posedness of pseudodifferential parabolic models for wave phenomena by rigorously defining the operator square root and proving that generates a strongly continuous semigroup, ensuring existence, uniqueness, and continuous dependence for the Cauchy problem . It develops existence and uniqueness theory for operator square roots under accretivity and spectral-range conditions, leveraging Kato's theory and the Lumer–Phillips theorem to show generation of a contractive -semigroup. The framework applies to with positive real and imaginary parts of , and generalizes to piecewise-constant in media. These results provide a solid mathematical justification for widely used PDPE-based numerical methods in wavefield simulations, such as SSP approaches, by guaranteeing well-posed evolution in the range coordinate.

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.
Paper Structure (4 sections, 3 theorems, 9 equations)

This paper contains 4 sections, 3 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The Pseudo-Differential Parabolic Equation
  3. Definition and generator properties of $\mathrm{i}\sqrt{A}$
  4. Conclusion

Key Result

Proposition 3.1

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:

Theorems & Definitions (10)

  • Proposition 3.1: Uniqueness of square roots
  • proof
  • Definition 3.2: The square root of an operator
  • Proposition 3.3: Existence of square roots
  • proof
  • Corollary 3.4: Generation theorem for $\mathrm{i}$ times a square root
  • proof
  • Example 3.5
  • proof
  • Remark 3.6