Acta Mathematica

The Schrödinger operator on the energy space: boundedness and compactness criteria

Vladimir G. Maz'ya and Igor E. Verbitsky

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Note

The second author was supported in part by NSF Grant DMS-0070623

Article information

Source
Acta Math. Volume 188, Number 2 (2002), 263-302.

Dates
Received: 23 October 2000
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485891491

Digital Object Identifier
doi:10.1007/BF02392684

Rights
2002 © Institut Mittag-Leffler

Citation

Maz'ya, Vladimir G.; Verbitsky, Igor E. The Schrödinger operator on the energy space: boundedness and compactness criteria. Acta Math. 188 (2002), no. 2, 263--302. doi:10.1007/BF02392684. https://projecteuclid.org/euclid.acta/1485891491


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References

  • [AdH] Adams, D. R. & Hedberg, L. I., Function Spaces and Potential Theory, Grundlehren Math. Wiss., 314. Springer-Verlag, Berlin, 1996.
  • [AiS] Aizenman, M. & Simon, B., Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math., 35 (1982), 209–273.
  • [An] Ancona, A., On strong barriers and an inequality of Hardy for domains in Rn. J. London Math. Soc. (2), 34 (1986), 274–290.
  • [BeS] Berezin, F. A. & Shubin, M. A., The Schrödinger Equation. Math. Appl. (Soviet Ser.), 66. Kluwer, Dordrecht, 1991.
  • [Bi] Birman, M. S., The spectrum of singular boundary problems. Mat. Sb. (N.S.), 55 (1961), 125–174. (Russian); English translation in Amer. Math. Soc. Transl. Ser. 2. 53 (1966), 23–80.
  • [BiS1] Birman, M. S. & Solomyak, M. Z., Spectral Theory of Self-Adjoint Operators in Hilbert Space. Math. Appl. (Soviet Ser.) Reidel, Dordrecht, 1987.
  • [BiS2]— Schrödinger operator. Estimates for number of bound states as function-theoretical problem, in Spectral Theory of Operators (Novgorod, 1989), pp. 1–54. Amer. Math. Soc. Transl. Ser. 2, 150. Amer. Math. Soc., Providence, RI, 1992.
  • Coifman, R. R. & Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals. Studia Math., 51 (1974), 241–250.
  • Combescure, M. & Ginibre, J., Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials. Ann. Inst. H. Poincaré Sect. A (N.S.), 24 (1976), 17–30.
  • Chang, S.-Y. A., Wilson, J. M. & Wolff, T. H., Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helv., 60 (1985), 217–246.
  • Chung, K. L. & Zhao, Z. X., From Brownian Motion to Schrödinger's Equation. Grundlehren Math. Wiss., 312. Springer-Verlag, Berlin, 1995.
  • Davies, E. B., Lp spectral theory of higher-order elliptic differential operators. Bull. London Math. Soc., 29 (1997), 513–546.
  • [D2]—, A review of Hardy inequalities, in The Maz'ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 55–67. Oper. Theory Adv. Appl., 110. Birkhäuser, Basel, 1999.
  • Edmunds, D. E. & Evans, W. D., Spectral Theory and Differential Operators. Oxford Math. Monographs. Clarendon Press, Oxford Univ. Press, New York, 1987.
  • Faris, W. G., Self-Adjoint Operators. Lecture Notes in Math., 433. Springer-Verlag, Berlin-New York, 1975.
  • Fefferman, C., The uncertainty principle. Bull. Amer. Math. Soc. (N.S.), 9 (1983), 129–206.
  • Hille, E., Non-oscillation theorems. Trans. Amer. Math. Soc., 64 (1948), 234–252.
  • Hansson, K., Maz'Ya, V. G. & Verbitsky, I. E., Criteria of solvability for multi-dimensional Riccati equations. Ark. Mat., 37 (1999), 87–120.
  • Kalton, N. J. & Verbitsky, I. E., Nonlinear equations and weighted norm inequalities. Trans. Amer. Math. Soc., 351 (1999), 3441–3497.
  • Kerman, R. & Sawyer, E., The trace inequality and eigenvalue estimates for Schrödinger operators. Ann. Inst. Fourier (Grenoble), 36 (1986), 207–228.
  • Kondratiev, V., Maz'ya, V. G. & Shubin, M., Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators. To appear.
  • [KoS] Kondratiev, V. & Shubin, M., Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry, in The Maz'ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 185–226. Oper. Theory Adv. Appl., 110. Birkhäuser, Basel, 1999.
  • [Le] Lewis, J., Uniformly fat sets. Trans. Amer. Math. Soc., 308 (1988), 177–196.
  • [Ma1] Maz'ya, V. G., On the theory of the n-dimensional Schrödinger operator. Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1145–1172 (Russian).
  • [Ma2]—, The (p, l)-capacity, embedding theorems, and the spectrum of a self-adjoint elliptic operator. Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 356–385 (Russian).
  • [Ma3]—, Sobolev Spaces. Springer Ser. Soviet Math, Springer-Verlag, Berlin, 1985.
  • [MaS] Maz'ya, V. G. & Shaposhnikova, T. O., Theory of Multipliers in Spaces of Differentiable Functions. Monographs Stud. Math., 23. Pitman, Boston, MA, 1985.
  • [MaV] Maz'ya, V. G. & Verbitsky, I. E., Capacitary estimates for fractional integrals, with applications to partial differential equations and Sobolev multipliers. Ark. Mat., 33 (1995), 81–115.
  • [MMP] Marcus, M., Mizel, V. J. & Pinchover, Y., On the best constant for Hardy's inequality in Rn. Trans. Amer. Math. Soc. 350 (1998), 3237–3255.
  • [Mo] Molchanov, A., On conditions for the discreteness of spectrum of self-adjoint secondorder differential equations. Trans. Moscow Math. Soc., 2 (1953), 169–200 (Russian).
  • [NaS] Naimark, K. & Solomyak, M., Regular and pathological eigenvalue behavior for the equation-λu″=Vu on the semiaxis. J. Funct. Anal., 151 (1997), 504–530.
  • [Ne] Nelson, E., Topics in Dynamics, I: Flows. Math. Notes. Princeton Univ. Press, Princeton, NJ, 1969.
  • [RS1] Reed, M. & Simon, B., Methods of Modern Mathematical Physics, I: Functional Analysis, 2nd edition. Academic Press, New York, 1980.
  • [RS2]—, Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness. Academic Press, New York-London, 1975.
  • [S1] Schechter, M., Operator Methods in Quantum Mechanics. North-Holland, New York-Amsterdam, 1981.
  • [S2]—, Spectra of Partial Differential Operators, 2nd edition. North-Holland Ser. Appl. Math. Mech., 14, North-Holland, Amsterdam, 1986.
  • [S3]—, Weighted norm inequalities for potential operators. Trans. Amer. Math. Soc., 308 (1988), 57–68.
  • [Si] Simon, B., Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447–526.
  • [St1] Stein, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser., 30. Princeton Univ. Press, Princeton, NJ, 1970.
  • [St2]—, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Math. Ser., 43. Princeton Univ. Press, Princeton, NJ, 1993.
  • [Stu] Sturm, K. T., Schrödinger operators with highly singular, oscillating potentials. Manuscripta Math., 76 (1992), 367–395.
  • [StW] Stein, E. M. & Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces. Princeton Math. Ser., 32. Princeton Univ. Press, Princeton, NJ, 1971.
  • [Ve] Verbitsky, I. E., Nonlinear potentials and trace inequalities, in The Maz'ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 323–343: Oper. Theory Adv. Appl., 110. Birkhäuser, Basel, 1999.