Related papers: Wave equations with non-commutative space and time

We consider the Cauchy problem for the weakly dissipative wave equation $$ \square u+\frac\mu{1+t} u_t=0 $$ with parameter $\mu\ge2$. Based on the explicit representations of solutions provided in [Math. Meth. Appl. Sci. 2004; {\bf…

Given a Hamiltonian $H$ on a Hilbert space $\mathcal H$ it is shown that, under the assumption that $\sigma(H)=\sigma_{ac}(H)=R^+$, there exist unique positive operators $T_F$ and $T_B$ registering the Schr\"odinger time evolution generated…

We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form…

We consider Scr\"odinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may be not even differentiable. The well-posedness of the Cauchy problem is proved in the frame of…

We study the generalized fractional linear problem $D^{\nu}_{a+*} f(x) =A(x)f(x)+g(x)$, where $D^{\nu}$ is an arbitrary mixture of Caputo derivatives of order at most one and $A(x)$ a family of operators in a Banach space generating…

Waves on ``commutative'' spacetimes like R^d are elements of the commutative algebra C^0(R^d) of functions on R^d. When C^0(R^d) is deformed to a noncommutative algebra {\cal A}_\theta (R^d) with deformation parameter \theta ({\cal A}_0…

We consider the linear integro-differential operator $L$ defined by \[ Lu(x) =\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(\alpha) 1_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like $|y|^{-d-\alpha}$,…

We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of the special form (Minkowski plane with a handle) containing closed timelike curves (time machines). We prove that the classical solution of the…

Wave-like partial differential equations occur in many engineering applications. Here the engineering setup is embedded into the Hilbert space framework of functional analysis of modern mathematical physics. The notion wave-like is a…

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane…

Traditional boundary integral methods suffer from the singularity of Green's kernels. The paper develops, for a model problem of 2D scattering as an illustrative example, singularity-free boundary difference equations. Instead of converting…

A sketch of an approach towards Lorentzian spectral geometry (based on joint work with Mario Paschke) is described, together with a general way to define abstractly the quantized Dirac field on such Lorentzian spectral geometries.…

The irregular solutions of the stationary Schr\"odinger equation are important for the fundamental formal development of scattering theory. They are also necessary for the analytical properties of the Green function, which in practice can…

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal…

Given an open, bounded and connected set $\Omega\subset\mathbb{R}^{3}$ and its rescaling $\Omega_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider the solutions of the Cauchy problem for the inhomogeneous wave equation $$…

This brief note wants to bring to attention that the formulation of physically reasonable initial-boundary value problems for wave equations in Lorentzian space-times is not unique, i.e., that there are inequivalent such formulations that…

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional…

We present a systematic treatment of scattering processes for quantum systems whose time evolution is discrete. We define and show some general properties of the scattering operator, in particular the conservation of quasi-energy which is…

We study the Cauchy problem for non-linear non-local operators that may be degenerate. Our general framework includes cases where the jump intensity is allowed to depend on the values of the solution itself, e.g. the porous medium equation…

A waveguide G lies in the (n+1)-dimensional Euclidean space for positive integer n, and outside a large ball coincides with the union of finitely many non-overlapping semi-cylinders ("cylindrical ends"). The waveguide is described by the…