The Cauchy problem for Schr?dinger-type partial differential operators with generalized functions in the principal part and as data

We set-up and solve the Cauchy problem for Schr?dinger-type differential operators with generalized functions as coefficients, in particular, allowing for distributional coefficients in the principal part. Equations involving such kind of operators appeared in models of deep earth seismology. We prove existence and uniqueness of Colombeau generalized solutions and analyze the relations with classical and distributional solutions. Furthermore, we provide a construction of generalized initial values that may serve as square roots of arbitrary probability measures.     

Integral problem for Schrödinger-type equations with the general elliptic part

Theoretical and Mathematical Physics - The existence, uniqueness, and regularity properties and Strichartz-type estimates for the solution of an integral-type initial value problem for linear and...     

The Cauchy problem for non-autonomous nonlinear Schrödinger equations

In this paper we study the Cauchy problem for cubic nonlinear Schrödinger equation with space- and time-dependent coefficients on ∝^m and \(\mathbb{T}^m \). By an approximation argument we prove that for suitable initial values, the Cauchy problem admits unique local solutions. Global existence is discussed in the cases of m = 1, 2.     

On the Cauchy problem for the generalized Benjamin–Ono equation with small initial data

We prove global well-posedness results for small initial data in $\text{H}{}^{\mathrm{s}}\text{(}\text{R}\text{),}\text{s>s}{}_{\mathrm{k}}$, and in , s_k=1/2?1/k, for the generalized Benjamin–Ono equation $\text{?}{}_{\mathrm{t}}\text{u+}\text{H}\text{?}{}^2{}_{\mathrm{x}}\text{u+?}{}_{\mathrm{x}}\text{(u}{}^{\mathrm{k+1}}\text{)=0,}\text{k?4}$. We also consider the cases k=2,3. To cite this article: L. Molinet, F. Ribaud, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential

In this paper, we discuss a generalized Camassa–Holm equation whose solutions are velocity potentials of the classical Camassa–Holm equation. By exploiting the connection between these two equations, we first establish the local well-posedness of the new equation in the Besov spaces and deduce several blow-up criteria and blow-up results. Then, we investigate the existence of global strong solutions and present a class of cuspon weak solutions for the new equation.     

The Morse and Maslov indices for Schrödinger operators

We study the spectrum of Schrödinger operators with matrixvalued potentials, utilizing tools from infinite-dimensional symplectic geometry. Using the spaces of abstract boundary values, we derive relations between the Morse and Maslov indices for a family of operators on a Hilbert space obtained by perturbing a given self-adjoint operator by a smooth family of bounded self-adjoint operators. The abstract results are applied to the Schrödinger operators with θ-periodic, Dirichlet, and Neumann boundary conditions. In particular, we derive an analogue of the Morse-Smale Index Theorem for multi-dimensional Schrödinger operators with periodic potentials. For quasi-convex domains in Rⁿ, we recast the results, connecting the Morse and Maslov indices using the Dirichlet and Neumann traces on the boundary of the domain.     

On spectral estimates for Schrödinger-type operators: The case of small local dimension

The behavior of the discrete spectrum of the Schrödinger operator - Δ -V is determined to a large extent by the behavior of the corresponding heat kernel P(t; x,y) as t → 0 and t→ ∞. If this behavior is power-like, i.e.,

$\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - \delta /2} ),t \to 0,\left\| {P(t; \cdot , \cdot )} \right\|_{L^\infty } = O(t^{ - D/2} ),t \to \infty ,$

then it is natural to call the exponents δ and D the local dimension and the dimension at infinity, respectively. The character of spectral estimates depends on a relation between these dimensions. The case where δ < D, which has been insufficiently studied, is analyzed. Applications to operators on combinatorial and metric graphs are considered.

     

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.