Hyperbolic Evolution Equations

Tanabe, Hiroki

doi:10.1201/9780203755419-7

Cited by 5 publications

(5 citation statements)

References 12 publications

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“…Furthermore, the resolvent set of U(t, s) is assumed to be nonempty. Following the standard theory of abstract evolution equations, [17][18][19] the semigroup property…”

Section: Mathematical Settingsmentioning

confidence: 99%

Unbounded generalization of logarithmic representation of infinitesimal generators

Iwata

2023

Math Methods in App Sciences

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The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion, a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms, for example, transforms appearing in the theory of integrable systems.

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“…Furthermore, the resolvent set of U(t, s) is assumed to be nonempty. Following the standard theory of abstract evolution equations, [17][18][19] the semigroup property…”

Section: Mathematical Settingsmentioning

confidence: 99%

Unbounded generalization of logarithmic representation of infinitesimal generators

Iwata

2023

Math Methods in App Sciences

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“…On the other hand, we recall the following resolvent estimate (see e.g. Tanabe [32]): there exist constants z 0 > 0 and C > 0 such that…”

Section: Preliminariesmentioning

confidence: 99%

Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior

Liu¹,

Yamamoto²

2022

Inverse Problems

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As the most significant difference from parabolic equations, long-time or short-time behavior of solutions to time-fractional evolution equations is dominated by the fractional orders, whose unique determination has been frequently investigated in literature. Unlike all the existing results, in this article we prove the uniqueness of orders and parameters (up to a multiplier for the latter) only by principal terms of asymptotic expansions of solutions near t=0 at a single point. Moreover, we discover special conditions on unknown initial values or source terms for the coincidence of observation data. As a byproduct, we can even conclude the uniqueness for initial values or source terms by the same data. The proof relies on the asymptotic expansion after taking the Laplace transform and the completeness of generalized eigenfunctions.

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“…Let R be a positive radius smaller than R (0) and R (1) . Then since both series (52) and ( 53) converge there exists a positive number g such that for all ν…”

Section: Free Propagation Of a Wave Packetmentioning

confidence: 99%

“…The solution of the TDSE will then not be timeanalytic, i.e. series of the form as in (52) and (53) do not exist for this initial state. Note that the requirement of infinite convergence radius by itself is not enough, one really needs to satisfy the constraints (57).…”

Section: Free Propagation Of a Wave Packetmentioning

confidence: 99%

“…This specific approach unitarily transforms the equation to a static singular potential and then uses results from Tanabe [52] which in turn rely on Kato [46]. The study in Yajima [53] resumes the perturbative approach of Phillips [45] and combines it with specific properties of the Schrödinger Hamiltonian, e.g.…”

Section: History Of Explicitly Time-dependent Initial-value Problemsmentioning

confidence: 99%

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Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory

Ruggenthaler

Penz

Leeuwen

2015

J. Phys.: Condens. Matter

142

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In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schrödinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.

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Hyperbolic Evolution Equations

Cited by 5 publications

References 12 publications

Unbounded generalization of logarithmic representation of infinitesimal generators

Unbounded generalization of logarithmic representation of infinitesimal generators

Uniqueness of orders and parameters in multi-term time-fractional diffusion equations by short-time behavior

Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory

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