Abstract
Hardy type unique continuation properties for abstract Schrödinger equations and applications
Highlights
The unique continuation properties of the following abstract Schrödinger equation i∂tu + ∆u + A (x) u + V (x, t) u = 0, x ∈ Rn, t ∈ [0, T], (1.1)are studied, where A = A (x) is a linear and V (x, t) is a given potential operator functions in a Hilbert space H; ∆ denotes the Laplace operator in Rn and u = u(x, t) is the H-valued unknown function
Are studied, where A = A (x) is a linear and V (x, t) is a given potential operator functions in a Hilbert space H; ∆ denotes the Laplace operator in Rn and u = u(x, t) is the H-valued unknown function. This linear result was applied to show that two regular solutions u1, u2 of non-linear abstract Schrödinger equations i∂tu + ∆u + A ((x)) u = F (u, u), x ∈ Rn, t ∈ [0, T]
For general non-linearities F must agree in Rn × [0, T], when u1 − u2 and its gradient decay faster than any quadratic exponential at times 0 and T
Summary
Shakhmurov these results we will study the unique continuation properties of abstract Schrödinger equations with the operator potentials. If we choose H to be a concrete Hilbert space, for example H = L2 (Ω), A = L, where Ω is a domain in Rm with sufficiently smooth boundary and L is a regular elliptic operator we obtain the unique continuation properties of the anisotropic Schrödinger equation. From our general results we obtain the unique continuation properties of the Wentzell–Robin type boundary value problem (BVP) for the following Schrödinger equation. Let C (Ω; E) denote the space of E-valued uniformly bounded continuous functions on Ω with norm u C(Ω;E) = sup u (x) E. Let S (Rn; E) denote the space of all continuous linear operators, L : S → E, equipped with topology of bounded convergence. When we want to specify the dependence of such a constant on a parameter, say α, we write Cα
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