# Estimates for the spectral shift function of the polyharmonic operator

@article{Pushnitski1999EstimatesFT,
title={Estimates for the spectral shift function of the polyharmonic operator},
author={Alexander Pushnitski},
journal={Journal of Mathematical Physics},
year={1999},
volume={40},
pages={5578-5592}
}
• A. Pushnitski
• Published 26 October 1999
• Mathematics
• Journal of Mathematical Physics
The Lifshits–Krein spectral shift function is considered for the pair of operators H0=(−Δ)l, l>0 and H=H0+V in L2(Rd), d⩾1; here V is a multiplication operator. The estimates for this spectral shift function ξ(λ;H,H0) are obtained in terms of the spectral parameter λ>0 and the integral norms of V. These estimates are in a good agreement with the ones predicted by the classical phase space volume considerations.
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## References

SHOWING 1-10 OF 34 REFERENCES

### INTEGRAL ESTIMATES FOR THE SPECTRAL SHIFT FUNCTION

The spectral shift function ξ(λ) is considered for the pair of operators H0, H0 + V , where H0 is the Schrödinger operator with variable Riemannian metric and with electromagnetic field, and V is the

### Spectral Shift Function of the Schrödinger Operator in the Large Coupling Constant Limit

• Mathematics
• 2000
The spectral shift function of a Schrödinger operator with a perturbation of definite sign is considered. The asymptotics of the spectral shift function for large coupling constant is studied, and

### Efficient bounds for the spectral shift function

Let H 0 , H be a pair of selfadjoint operators in a separable Hilbert space whose difference V = H − H 0 belongs to the trace class and let Θ(λ) = Θ(λ; H 0 , H) be the spectral shift function for the

### The Ξ operator and its relation to Krein's spectral shift function

• Mathematics
• 1999
We explore connections between Krein's spectral shift function ζ(λ,H0, H) associated with the pair of self-adjoint operators (H0, H),H=H0+V, in a Hilbert spaceH and the recently introduced concept of

### Piecewise-polynomial approximation of functions fromHℓ((0, 1)d), 2ℓ=d, and applications to the spectral theory of the Schrödinger operator

For the selfadjoint Schrödinger operator −Δ−αV on ℝ2 the number of negative eigenvalues is estimated. The estimates obtained are based upon a new result on the weightedL2-approximation of functions

### Bounds on the eigenvalues of the Laplace and Schroedinger operators

If 12 is an open set in R", and if N(£l, X) is the number of eigenvalues of A (with Dirichlet boundary conditions on d£2) which are < X (k > 0), one has the asymptotic formula of Weyl [1] , [2] : l i

### SCATTERING THEORY APPROACH TO RANDOM SCHRÖDINGER OPERATORS IN ONE DIMENSION

• Mathematics
• 1999
Methods from scattering theory are introduced to analyze random Schrodinger operators in one dimension by applying a volume cutoff to the potential. The key ingredient is the Lifshitz–Krein spectral

### ESTIMATES OF SINGULAR NUMBERS OF INTEGRAL OPERATORS

• Mathematics
• 1977
ContentsIntroduction § 1. Operator spaces and function spaces § 2. Estimates of singular numbers based on the method of piecewise-polynomial approximation § 3. Interpolation methods § 4. General

### Estimates and asymptotics for discrete spectra of integral and differential equations

Estimates for the number of negative eigenvalues of the Schrodinger operator and its generalizations by M. Sh. Birman and M. Z. Solomyak Discrete spectrum in the gaps of a continuous one for

### THE STATIONARY METHOD IN THE ABSTRACT THEORY OF SCATTERING

• Mathematics
• 1967
The wave operators and scattering matrix for pairs of self-adjoint operators are constructed in an explicit and invariant form. It is assumed that the perturbation is nuclear or "relatively nuclear".