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Bifurcation of double eigenvalues for Aharonov–Bohm operators with a moving pole
Full text linkWe study double eigenvalues of Aharonov-Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential's circulation is a fixed half-integer number and the operator's pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the center, but there exists a whole neighborhood where it bifurcates into two distinct branches
Unique Continuation from Conical Boundary Points for Fractional Equations
No full textWe provide fine asymptotics of solutions of fractional elliptic equations at boundary points where the domain is locally conical; that is, corner type singularities appear. Our method relies on a suitable smoothing of the corner singularity and an approximation scheme, which allow us to provide a Pohozaev-type inequality. Then the asymptotics of solutions at the conical point follow by an Almgren-type monotonicity formula, blow-up analysis, and Fourier decomposition on eigenspaces of a spherical eigenvalue problem. A strong unique continuation principle follows as a corollar
Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity
Full text linkWe deal with a class on nonlinear Schr ̈odinger equations (NLS) with potentials V (x) \simeq
|x|^{-\alpha} , 0 < \alpha < 2, and K(x) \simeq |x|^{-\beta} , \beta > 0. Working in weighted Sobolev spaces, the existence
of ground states belonging to W^{1,2}(R^N) is proved under the assumption that \sigma < p <
(N + 2)/(N − 2) for some \sigma = \sigma_{N,\alpha,\beta} . Furthermore, it is shown that these are spikes concentrating
at a minimum point of A = V^\theta K^{−2/(p−1)}, where \theta = (p + 1)/(p − 1) − 1/2
Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications
Full text linkWe deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the singular part of the operator has two coalescing poles
On Aharonov–Bohm Operators with Two Colliding Poles
Full text linkAbstract
We consider Aharonov–Bohm operators with two poles and prove sharp asymptotics for simple eigenvalues as the poles collapse at an interior point out of nodal lines of the limit eigenfunction.</jats:p
Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations
Full text linkWe study local asymptotics of solutions to fractional elliptic equations at
boundary points, under some outer homogeneous Dirichlet boundary condition. Our
analysis is based on a blow-up procedure which involves some Almgren type
monotonicity formulae and provides a classification of all possible homogeneity
degrees of limiting entire profiles. As a consequence, we establish a strong
unique continuation principle from boundary points.Comment: 41 pages, 2 figure
Frequency-dependent time decay of Schrödinger flows
No full textWe show that the presence of negative eigenvalues in the spectrum of the angular component of an electromagnetic Schr\"odinger hamiltonian generically produces a lack of the classical time-decay for the associated Schr\"odinger flow . This is in contrast with the fact that dispersive estimates (Strichartz) still hold, in general, also in this case. We also observe an improvement of the decay for higher positive modes, showing that the time decay of the solution is due to the first nonzero term in the expansion of the initial datum as a series of eigenfunctions of a quantum harmonic oscillator with a singular potential. A completely analogous phenomenon is shown for the heat semigroup, as expected
Eigenvalues of the Laplacian with moving mixed boundary conditions: The case of disappearing Neumann region
Full text linkWe deal with eigenvalue problems for the Laplacian with varying mixed boundary conditions, consisting in homogeneous Neumann conditions on a vanishing portion of the boundary and Dirichlet conditions on the complement. By the study of an Almgren-type frequency function, we derive upper and lower bounds of the eigenvalue variation and sharp estimates in the case of a strictly star-shaped Neumann region
Time Decay of Scaling Invariant Electromagnetic Schrödinger Equations on the Plane
No full textWe prove the sharp time-decay estimate for the -Schr\"odinger equation with a general family of scaling critical electromagnetic potentials
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