Compactness of isospectral potentials. (English) Zbl 1062.58033
Let \(P_V:=\Delta+V\) be the Schrödinger operator on a compact Riemannian manifold \((M,g)\) of dimension \(m\); here \(\Delta=\delta d\) is the scalar Laplace operator and \(V\) is a smooth potential. Let \(Iso(V_0)\) be the set of smooth potentials on \(V\) so that \(P_V\) and \(P_{V_0}\) have the spectrum.
The author improves an estimate of J. Brüning [Commun. Partial Differ. Equations 9, 687–698 (1984; Zbl 0547.58039)] and shows that Iso\((V_0)\) is compact given a bound for the Sobolev norm \(| V| _{s,2}\) of \(s\) derivatives in \(L^2\). This recovers the compactness of Iso\((V_0)\) when \(m\leq3\). The author also establishes the compactness of non-negative isospectral potentials in dimensions \(m\leq 9\) and for the flat torus establishes the compactness of isospectral potentials with non-negative Fourier coefficients.
The proofs are based on the heat equation asymptotics and the Sobolev embedding theorems.
The author improves an estimate of J. Brüning [Commun. Partial Differ. Equations 9, 687–698 (1984; Zbl 0547.58039)] and shows that Iso\((V_0)\) is compact given a bound for the Sobolev norm \(| V| _{s,2}\) of \(s\) derivatives in \(L^2\). This recovers the compactness of Iso\((V_0)\) when \(m\leq3\). The author also establishes the compactness of non-negative isospectral potentials in dimensions \(m\leq 9\) and for the flat torus establishes the compactness of isospectral potentials with non-negative Fourier coefficients.
The proofs are based on the heat equation asymptotics and the Sobolev embedding theorems.
Reviewer: Peter B. Gilkey (Eugene)
MSC:
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 58J37 | Perturbations of PDEs on manifolds; asymptotics |
Citations:
Zbl 0547.58039References:
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| [3] | Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). · Zbl 0223.53034 |
| [4] | Jochen Brüning, On the compactness of isospectral potentials, Comm. Partial Differential Equations 9 (1984), no. 7, 687 – 698. · Zbl 0547.58039 · doi:10.1080/03605308408820344 |
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