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Register a new userAbsence of Embedded Eigenvalues for Non-Local Schrodinger Operators
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We consider non-local Schrodinger operators with kinetic terms given by several different types of functions of the Laplacian and potentials decaying to zero at infinity, and derive conditions ruling embedded eigenvalues out. These results contrast and complement recent work on showing the existence of such eigenvalues occurring for the same types of operators under different conditions. Our goal in this paper is to advance techniques based on virial theorems, Mourre estimates, and an extended version of the Birman-Schwinger principle, previously developed for classical Schrodinger operators but thus far not used for non-local operators. We also present a number of specific cases by choosing particular classes of kinetic and potential terms of immediate interest.
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In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued function. The proof uses a natural decomposition of Pushnitskis $mu$-invariant into absolutely continuous and singular parts. As a corollary, the Birman-Krein formula follows.
We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrodinger operators in $mathbb{R}^d,, dgeq 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.
Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n geq 3$, the absence of resonances at $pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n geq 5$, and recall the well-known case of absence of zero-energy resonances for Schrodinger operators in dimension $n geq 5$.
In the presence of the homogeneous electric field ${bf E}$ and the homogeneous perpendicular magnetic field ${bf B}$, the classical trajectory of a quantum particle on ${mathbb R}^2$ moves with drift velocity $alpha$ which is perpendicular to the electric and magnetic fields. For such Hamiltonians the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper one proves this conjecture for the perturbations $V(x, y)$ which have sufficiently small support in direction of drift velocity.
We revisit an archive submission by P. B. Denton, S. J. Parke, T. Tao, and X. Zhang, arXiv:1908.03795, on $n times n$ self-adjoint matrices from the point of view of self-adjoint Dirichlet Schrodinger operators on a compact interval.
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