zbMATH — the first resource for mathematics

Unique continuation for Schrödinger operators in dimension three or less. (English) Zbl 0535.35007
We show that the differential inequality \(| \Delta u| \leq v| u|\) has the unique continuation property relative to the Sobolev space \(H^{2,1}\!\!\!_{loc}(\Omega)\), \(\Omega \subset R^ n\), \(n\leq 3\), if v satisfies the condition \[ (K_ n\!^{loc})\quad \lim_{r\to 0}\sup_{x\in K}\int_{| x-y|<r}| x-y|^{2-n}v(y)dy=0 \] for all compact \(K\subset \Omega\), where if \(n=2\), we replace \(| x-y|^{2-n}\) by -lo\(g| x-y|\). This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, \(H=-\Delta +v\), in the case \(n\leq 3\). The proof uses Carleman’s approach together with the following pointwise inequality valid for all \(N=0,1,2,..\). and any \(u\in H_ c\!^{2,1}(R^ 3-\{0\});\) \[ \frac{| u(x)|}{| x|^ N}\leq C\int_{R^ 3}| x-y|^{- 1}\frac{| \Delta u(y)|}{| y|^ N}dy \] for a.e. x in \(R^ 3\).

35B60 Continuation and prolongation of solutions to PDEs
35J10 Schrödinger operator, Schrödinger equation
35R45 Partial differential inequalities and systems of partial differential inequalities
Full Text: DOI Numdam EuDML
[1] W.O. AMREIN, A.M. BERTHIER and V. GEORGESCU, Lp inequalities for the Laplacian and unique continuation, Ann. Inst. Fourier, Grenoble, 31-3 (1981), 153-168. · Zbl 0468.35017
[2] A.M. BERTHIER, Sur le spectre ponctuel de l’opérateur de Schrödinger, C.R.A.S., Paris, 290A (1980), 393-395. · Zbl 0454.35070
[3] T. CARLEMAN, Sur un problème d’unicité pour LES systèmes d’équations aux dérivés partielles à deux variables indépendantes, Ark. Mat., 26B (1939), 1-9. · Zbl 0022.34201
[4] V. GEORGESCU, On the unique continuation property for Schrödinger Hamiltonians, Helv. Phys. Acta, 52 (1979), 655-670.
[5] E. HEINZ, Uber die eindeutigkeit beim Cauchy’schen anfangswertproblem einer elliptischen differentialgleichung zweiter ordnung, Nachr. Akad.-Wiss. Göttingen, II (1955), 1-12. · Zbl 0067.07503
[6] L. HORMANDER, Linear partial differential operators, Springer, Berlin, 1963. · Zbl 0108.09301
[7] R. KERMAN and E. SAWYER, Weighted norm inequalities of trace-type for potential operators, preprint. · Zbl 0673.47030
[8] V.G. MAZ’YA, Imbedding theorems and their applications, Baku Sympos. (1966), “Nauka”, Moscow, (1970), 142-154 (Russian).
[9] V.G. MAZ’YA, On some integral inequalities for functions of several variables, Problems in Math. Analysis, No 3 (1972) Leningrad U. (Russian).
[10] M. REED and B. SIMON, Methods of modern mathematical physics, IV. Analysis of Operators, Academic Press, New York, 1978. · Zbl 0401.47001
[11] J. SAUT and B. SCHEURER, Un théorème de prolongement unique pour des opérators elliptiques dont LES coefficients ne sont pas localement bornés, C.R.A.S., Paris, 290A (1980), 595-598. · Zbl 0429.35020
[12] M. SCHECHTER and B. SIMON, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl., 77 (1980), 482-492. · Zbl 0458.35024
[13] B. SIMON, Schrödinger semigroups, Bull. A.M.S., 7 (1982), 447-526. · Zbl 0524.35002
[14] E.M. STEIN, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, N.J. 1970. · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.