Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients
Noboru Okazawa Tomomi Yokota
The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on $\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and $b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$, while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$. Over twenty years ago late Professor Kato proved that the minimal realization $T_{min}$ is essentially quasi-$m$-accretive in $L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is shown that under some additional conditions the same conclusion remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.
keywords: Schrödinger operators quasi-$m$-accretive operators.
Schrödinger type evolution equations with monotone nonlinearity of non-power type
Yoshiki Maeda Noboru Okazawa
Existence of unique strong solutions is established for Schrödinger type evolution equations with monotone nonlinearity. The proof is based on a perturbation theorem for $m$-accretive operators in a complex Hilbert space.
keywords: $m$-accretive operators. Schrödinger type evolution equations (nondiffusive complex Ginzburg-Landau equations) monotone nonlinearity
Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains
Noboru Okazawa Tomomi Yokota
Please refer to Full Text.
keywords: subdifferential operators Smoothing effect semigroups of nonlinear operators accretive operators the complex Ginzburg-Landau equation.
Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation
Noboru Okazawa Tomomi Yokota
Two theorems concerning strong wellposedness are established for the complex Ginzburg-Landau equation. One of them is concerned with strong $L^{2}$-wellposedness, that is, strong wellposedness for $L^{2}$-initial data. The other deals with $H_{0}^{1}$-initial data as a partial extension. By a technical innovation it becomes possible to prove the convergence of approximate solutions without compactness. This type of convergence is known with accretivity methods when the argument of the complex coefficient is small. The new device yields the generation of a class of non-contraction semigroups even when the argument is large. The results are both obtained as application of abstract theory of semilinear evolution equations with subdifferential operators.
keywords: subdifferential operators strong wellposedness smoothing effect accretive operators semigroups of nonlinear operators. The complex Ginzburg-Landau equation
Semigroup-theoretic approach to identification of linear diffusion coefficients
Gianluca Mola Noboru Okazawa Jan Prüss Tomomi Yokota
Let $X$ be a complex Banach space and $A:\,D(A) \to X$ a quasi-$m$-sectorial operator in $X$. This paper is concerned with the identification of diffusion coefficients $\nu > 0$ in the initial-value problem: \[ (d/dt)u(t) + {\nu}Au(t) = 0, \quad t \in (0,T), \quad u(0) = x \in X, \] with additional condition $\|u(T)\| = \rho$, where $\rho >0$ is known. Except for the additional condition, the solution to the initial-value problem is given by $u(t) := e^{-t\,{\nu}A} x \in C([0,T];X) \cap C^{1}((0,T];X)$. Therefore, the identification of $\nu$ is reduced to solving the equation $\|e^{-{\nu}TA}x\| = \rho$. It will be shown that the unique root $\nu = \nu(x,\rho)$ depends on $(x,\rho)$ locally Lipschitz continuously if the datum $(x,\rho)$ fulfills the restriction $\|x\|> \rho$. This extends those results in Mola [6](2011).
keywords: linear evolution equations in Banach spaces linear parabolic equations Identification problems unknown constants well-posedness results.
Linear evolution equations with strongly measurable families and application to the Dirac equation
Noboru Okazawa Kentarou Yoshii
A new existence and uniqueness theorem is established for linear evolution equations of hyperbolic type with strongly measurable coefficients in a separable Hilbert space. The result is applied to the Dirac equation with time-dependent potential.
keywords: hyperbolic type the Dirac equation. Linear evolution equations measurable coefficients
Energy methods for abstract nonlinear Schrödinger equations
Noboru Okazawa Toshiyuki Suzuki Tomomi Yokota
So far there seems to be no abstract formulations for nonlinear Schrödinger equations (NLS). In some sense Cazenave[2, Chapter 3] has given a guiding principle to replace the free Schrödinger group with the approximate identity of resolvents. In fact, he succeeded in separating the existence theory from the Strichartz estimates. This paper is a proposal to extend his guiding principle by using the square root of the resolvent. More precisely, the abstract theory here unifies the local existence of weak solutions to (NLS) with not only typical nonlinearities but also some critical cases. Moreover, the theory yields the improvement of [21].
keywords: abstract nonlinear Schrödinger equation inverse-square potentials. Energy methods nonnegative selfadjoint operators

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