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We consider the equation Δu = Vu in the half-space ${\open R}_ + ^d $, d ⩾ 2 where V has certain periodicity properties. In particular, we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schrödinger operators. The equation Δu = Vu is studied as part of a broader class of elliptic evolution equations.

We study the distribution of the discrete spectrumof the Schrödinger operator perturbed by a fast oscillating decaying potential depending on a small parameter $h$.

We study discrete spectrum in spectral gaps of an elliptic periodic second orderdifferential operator in L 2(ℝd )perturbed by a decaying potential. It is assumed that a perturbation is nonnegative andhas a power-like behavior at infinity. We find asymptotics in the large coupling constantlimit for the number of eigenvalues of the perturbed operator that have crossed a givenpoint inside the gap or the edge of the gap. The corresponding asymptotics is power-likeand depends on the observation point.

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Decay rates at infinity for solutions to periodic Schrödinger equations

Semi-classical Asymptotics for the Schrödinger Operator with Oscillating Decaying Potential

Discrete Spectrum of the Periodic Schrödinger Operator with aVariable Metric Perturbed by a Nonnegative Potential

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3 results

Decay rates at infinity for solutions to periodic Schrödinger equations

Semi-classical Asymptotics for the Schrödinger Operator with Oscillating Decaying Potential

Discrete Spectrum of the Periodic Schrödinger Operator with aVariable Metric Perturbed by a Nonnegative Potential