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Register a new userSpatial asymptotics of Greens function and applications
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Authors
Sergey A. Denisov
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We study the spatial asymptotics of Greens function for the 1d Schrodinger operator with operator-valued decaying potential. The bounds on the entropy of the spectral measures are obtained. They are used to establish the presence of a.c. spectrum
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We consider metric perturbations of the Landau Hamiltonian. We investigate the asymptotic behaviour of the discrete spectrum of the perturbed operator near the Landau levels, for perturbations with power-like decay, exponential decay or compact support.
We consider the Dirichlet Laplacian in a three-dimensional waveguide that is a small deformation of a periodically twisted tube. The deformation is given by a bending and an additional twisting of the tube, both parametrized by a coupling constant $delta$. We expand the resolvent of the perturbed operator near the bottom of its essential spectrum and we show the existence of exactly one resonance, in the asymptotic regime of $delta$ small. We are able to perform the asymptotic expansion of the resonance in $delta$, which in particular permits us to give a quantitative geometric criterion for the existence of a discrete eigenvalue below the essential spectrum. In the particular case of perturbations of straight tubes, we are able to show the existence of resonances not only near the bottom of the essential spectrum but near each threshold in the spectrum. We also obtain the asymptotic behavior of the resonances in this situation, which is generically different from the first case.
We consider the twisted waveguide $Omega_theta$, i.e. the domain obtained by the rotation of the bounded cross section $omega subset {mathbb R}^{2}$ of the straight tube $Omega : = omega times {mathbb R}$ at angle $theta$ which depends on the variable along the axis of $Omega$. We study the spectral properties of the Dirichlet Laplacian in $Omega_theta$, unitarily equivalent under the diffeomorphism $Omega_theta to Omega$ to the operator $H_{theta}$, self-adjoint in ${rm L}^2(Omega)$. We assume that $theta = beta - epsilon$ where $beta$ is a $2pi$-periodic function, and $epsilon$ decays at infinity. Then in the spectrum $sigma(H_beta)$ of the unperturbed operator $H_beta$ there is a semi-bounded gap $(-infty, {mathcal E}_0^+)$, and, possibly, a number of bounded open gaps $({mathcal E}_j^-, {mathcal E}_j^+)$. Since $epsilon$ decays at infinity, the essential spectra of $H_beta$ and $H_{beta - epsilon}$ coincide. We investigate the asymptotic behaviour of the discrete spectrum of $H_{beta - epsilon}$ near an arbitrary fixed spectral edge ${mathcal E}_j^pm$. We establish necessary and quite close sufficient conditions which guarantee the finiteness of $sigma_{rm disc}(H_{beta-epsilon})$ in a neighbourhood of ${mathcal E}_j^pm$. In the case where the necessary conditions are violated, we obtain the main asymptotic term of the corresponding eigenvalue counting function. The effective Hamiltonian which governs the the asymptotics of $sigma_{rm disc}(H_{beta-epsilon})$ near ${mathcal E}_j^pm$ could be represented as a finite orthogonal sum of operators of the form $-mufrac{d^2}{dx^2} - eta epsilon$, self-adjoint in ${rm L}^2({mathbb R})$; here, $mu > 0$ is a constant related to the so-called effective mass, while $eta$ is $2pi$-periodic function depending on $beta$ and $omega$.
In three-dimensional case, we consider two classical operators: Schrodinger operator and an operator in the divergence form. For slowly-decaying oscillating potentials, we establish spatial asymptotics of the Greens function. The main term in this asymptotics involves vector-valued analytic function whose behavior is studied away from the spectrum. The absolute continuity of the spectrum is established as a corollary. For the operator in the divergence form, we consider the wave equation and establish existence of wave operators.
This paper is a continuation of my previous work on absolutely continuous and singular spectral shift functions, where it was in particular proved that the singular part of the spectral shift function is an a.e. integer-valued function. It was also shown that the singular spectral shift function is a locally constant function of the coupling constant $r,$ with possible jumps only at resonance points. Main result of this paper asserts that the jump of the singular spectral shift function at a resonance point is equal to the so-called resonance index, --- a new (to the best of my knowledge) notion introduced in this paper. Resonance index can be described as follows. For a fixed $lambda$ the resonance points $r_0$ of a path $H_r$ of self-adjoint operators are real poles of a certain meromorphic function associated with the triple $(lambda+i0; H_0,V).$ When $lambda+i0$ is shifted to $lambda+iy$ with small $y>0,$ that pole get off the real axis in the coupling constant complex plane and, in general, splits into some $N_+$ poles in the upper half-plane and some $N_-$ poles in the lower half-plane (counting multiplicities). Resonance index of the triple $(lambda; H_{r_0},V)$ is the difference $N_+-N_-.$ Based on the theorem just described, a non-trivial example of singular spectral shift function is given.
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