Unique Continuation and Absence of Positive Eigenvalues for Schrodinger Operators

Localization for random perturbations of periodic Schrödinger operators

“…Since P has finite dimensional range, there is an accumulation point g = P g of the g k with (Wg\g) = 0. Since W is strictly positive on some open set, this would contradict the unique continuation result from [12], since g is an eigenfunction of #e(o) with eigenvalue E n (ff). (See [28] for more recent developments concerning unique continuation.…”

Asymptotically linear problems driven by fractional Laplacian operators

“…Due to the generality of our non-local framework, it is an open problem whether or not all the eigenfunctions of −L K + q satisfy this property. In the case of the Laplacian all the eigenfunctions verify this condition on the nodal set and this is a direct consequence of a Unique Continuation Principle (see, for instance, [6,10]). As far as we know, there is not a non-local counterpart of this continuation property.…”

“…We also would like to note that this condition is compatible with the classical case of the Laplace operator (which corresponds to the choice s = 1), since in this context it is satisfied by every eigenvalue (see e.g. [6,10]). …”

“…In fact, in this standard situation it is well know that all the eigenfunctions are almost everywhere different from zero. This is a consequence of a Unique Continuation Principle (see [6,10]), which, as far as we know, it is not available in the non-local framework. We think it could be interesting to investigate this problem.…”

Strong unique continuation of eigenfunctions for p‐Laplacian operator

“…We refer to the work of Jerison-Kenig on the unique continuation for Shrödinger operators (cf. [3]). The same work is done by Gossez and Figueiredo, but for linear elliptic operator in the case V ∈ L N/2 , where N > 2, (cf.…”