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Estimates for the spectral shift function of the polyharmonic operator. (English) Zbl 0954.47036

The author establishes some estimates for the spectral shift functions \(\xi(\lambda; H_0\pm V,H_0)\) for the pair of operators \(H_0= (-\Delta)^\ell\), \(\ell> 0\), and \(H_0\pm V\) in \(L^2(\mathbb{R}^d)\), where \(V\) is a multiplication operator. Let \[ F_\gamma(x)= 1+(\log_+|x|)^\gamma,\quad \gamma>0, \] and \(\chi= d/(2\ell)\). It is shown that if \(V>0\) is \(H_0\)-form compact, then the following estimates hold for \(\lambda> 0\) and \(\gamma> 2\): \[ \xi(\lambda; H_0+ V,H_0)\leq C(d,\ell,\gamma) \lambda^{\chi-1}\|V\|^{1/2}_{L_1}\|VF_\gamma\|^{1/2}_{L_1}+ C(d, \ell)\lambda^{\chi- 1}(\log_+ \lambda)\|V\|_{L_1}, \]
\[ \begin{split} |\xi(\lambda; H_0- V, H_0)|\leq C(d,\ell, \gamma) \lambda^{\chi- 1}\|V\|^{1/2}_{L_1}\|VF_\gamma\|^{1/2}_{L_1}+ C(d, \ell)\lambda^{\chi- 1}(\log_+\lambda)\|V\|_{L_1}+\\ |\xi(- \lambda; H_0- 6V, H_0)|.\end{split} \] {}.

MSC:

47F05 General theory of partial differential operators
35J40 Boundary value problems for higher-order elliptic equations
35P05 General topics in linear spectral theory for PDEs
47A55 Perturbation theory of linear operators
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