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The Schrödinger operator on the energy space: Boundedness and compactness criteria. (English) Zbl 1013.35021

This paper deals with the property of the Schrödinger operator on the energy space. The authors present an complete solution to the problem of the relative form-boundedness of the potential energy operator V with respect to the Laplacian -Δ, which is fundamental to quantum mechanics. Moreover, the authors give both boundedness and compactness criteria for Sobolev spaces on domains Ω d under mild restrictions on Ω. They obtain also criteria for the classical inequality

d u ( x ) 2 V(x)dxC * d u ( x ) 2 dx,uC 0 ( d ),

to be hold, where the “indefinite” weight V may change sign, or even be a complex-valued distribution on d , d3.


MSC:
35J10Schrödinger operator
35B35Stability of solutions of PDE
47F05Partial differential operators
47H50Potential operators (MSC2000)
46N50Applications of functional analysis in quantum physics
References:
[1][AdH]Adams, D. R. &Hedberg, L. I.,Function Spaces and Potential Theory, Grundlehren Math. Wiss., 314. Springer-Verlag, Berlin, 1996.
[2][AiS]Aizenman, M. &Simon, B., Brownian motion and Harnack inequality for Schrödinger operators.Comm. Pure Appl. Math., 35 (1982), 209–273. · Zbl 0475.60063 · doi:10.1002/cpa.3160350206
[3][An]Ancona, A., On strong barriers and an inequality of Hardy for domains inR n.J. London Math. Soc. (2), 34 (1986), 274–290. · Zbl 0629.31002 · doi:10.1112/jlms/s2-34.2.274
[4][BeS]Berezin, F. A. &Shubin, M. A.,The Schrödinger Equation. Math. Appl. (Soviet Ser.), 66. Kluwer, Dordrecht, 1991.
[5][Bi]Birman, M. S., The spectrum of singular boundary problems.Mat. Sb. (N.S.), 55 (1961), 125–174. (Russian); English translation inAmer. Math. Soc. Transl. Ser. 2. 53 (1966), 23–80.
[6][BiS1]Birman, M. S. &Solomyak, M. Z.,Spectral Theory of Self-Adjoint Operators in Hilbert Space. Math. Appl. (Soviet Ser.) Reidel, Dordrecht, 1987.
[7][BiS2]– Schrödinger operator. Estimates for number of bound states as function-theoretical problem, inSpectral Theory of Operators (Novgorod, 1989), pp. 1–54. Amer. Math. Soc. Transl. Ser. 2, 150. Amer. Math. Soc., Providence, RI, 1992.
[8]Coifman, R. R. &Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals.Studia Math., 51 (1974), 241–250.
[9]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.
[10]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. · Zbl 0575.42025 · doi:10.1007/BF02567411
[11]Chung, K. L. &Zhao, Z. X.,From Brownian Motion to Schrödinger’s Equation. Grundlehren Math. Wiss., 312. Springer-Verlag, Berlin, 1995.
[12]Davies, E. B.,L p spectral theory of higher-order elliptic differential operators.Bull. London Math. Soc., 29 (1997), 513–546. · Zbl 0955.35019 · doi:10.1112/S002460939700324X
[13][D2]–, A review of Hardy inequalities, inThe Maz’ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 55–67. Oper. Theory Adv. Appl., 110. Birkhäuser, Basel, 1999.
[14]Edmunds, D. E. &Evans, W. D.,Spectral Theory and Differential Operators. Oxford Math. Monographs. Clarendon Press, Oxford Univ. Press, New York, 1987.
[15]Faris, W. G.,Self-Adjoint Operators. Lecture Notes in Math., 433. Springer-Verlag, Berlin-New York, 1975.
[16]Fefferman, C., The uncertainty principle.Bull. Amer. Math. Soc. (N.S.), 9 (1983), 129–206. · Zbl 0526.35080 · doi:10.1090/S0273-0979-1983-15154-6
[17]Hille, E., Non-oscillation theorems.Trans. Amer. Math. Soc., 64 (1948), 234–252. · doi:10.1090/S0002-9947-1948-0027925-7
[18]Hansson, K., Maz’Ya, V. G. &Verbitsky, I. E., Criteria of solvability for multi-dimensional Riccati equations.Ark. Mat., 37 (1999), 87–120. · Zbl 1087.35513 · doi:10.1007/BF02384829
[19]Kalton, N. J. &Verbitsky, I. E., Nonlinear equations and weighted norm inequalities.Trans. Amer. Math. Soc., 351 (1999), 3441–3497. · Zbl 0948.35044 · doi:10.1090/S0002-9947-99-02215-1
[20]Kerman, R. &Sawyer, E., The trace inequality and eigenvalue estimates for Schrödinger operators.Ann. Inst. Fourier (Grenoble), 36 (1986), 207–228.
[21]Kondratiev, V., Maz’ya, V. G. & Shubin, M., Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators. To appear.
[22][KoS]Kondratiev, V. &Shubin, M., Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry, inThe Maz’ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 185–226. Oper. Theory Adv. Appl., 110. Birkhäuser, Basel, 1999.
[23][Le]Lewis, J., Uniformly fat sets.Trans. Amer. Math. Soc., 308 (1988), 177–196. · doi:10.1090/S0002-9947-1988-0946438-4
[24][Ma1]Maz’ya, V. G., On the theory of then-dimensional Schrödinger operator.Izv. Akad. Nauk SSSR Ser. Mat., 28 (1964), 1145–1172 (Russian).
[25][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).
[26][Ma3]–,Sobolev Spaces. Springer Ser. Soviet Math, Springer-Verlag, Berlin, 1985.
[27][MaS]Maz’ya, V. G. &Shaposhnikova, T. O.,Theory of Multipliers in Spaces of Differentiable Functions. Monographs Stud. Math., 23. Pitman, Boston, MA, 1985.
[28][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. · Zbl 0834.31006 · doi:10.1007/BF02559606
[29][MMP]Marcus, M., Mizel, V. J. &Pinchover, Y., On the best constant for Hardy’s inequality inR n.Trans. Amer. Math. Soc. 350 (1998), 3237–3255. · Zbl 0917.26016 · doi:10.1090/S0002-9947-98-02122-9
[30][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).
[31][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. · Zbl 0895.34063 · doi:10.1006/jfan.1997.3149
[32][Ne]Nelson, E.,Topics in Dynamics, I:Flows. Math. Notes. Princeton Univ. Press, Princeton, NJ, 1969.
[33][RS1]Reed, M. &Simon, B.,Methods of Modern Mathematical Physics, I:Functional Analysis, 2nd edition. Academic Press, New York, 1980.
[34][RS2]–,Methods of Modern Mathematical Physics, II:Fourier Analysis, Self-Adjointness. Academic Press, New York-London, 1975.
[35][S1]Schechter, M.,Operator Methods in Quantum Mechanics. North-Holland, New York-Amsterdam, 1981.
[36][S2]–,Spectra of Partial Differential Operators, 2nd edition. North-Holland Ser. Appl. Math. Mech., 14, North-Holland, Amsterdam, 1986.
[37][S3]–, Weighted norm inequalities for potential operators.Trans. Amer. Math. Soc., 308 (1988), 57–68. · doi:10.1090/S0002-9947-1988-0946429-3
[38][Si]Simon, B., Schrödinger semigroups.Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447–526. · Zbl 0524.35002 · doi:10.1090/S0273-0979-1982-15041-8
[39][St1]Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser., 30. Princeton Univ. Press, Princeton, NJ, 1970.
[40][St2]–,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Math. Ser., 43. Princeton Univ. Press, Princeton, NJ, 1993.
[41][Stu]Sturm, K. T., Schrödinger operators with highly singular, oscillating potentials.Manuscripta Math., 76 (1992), 367–395. · Zbl 0767.35054 · doi:10.1007/BF02567767
[42][StW]Stein, E. M. &Weiss, G.,Introduction to Fourier Analysis on Euclidean Spaces. Princeton Math. Ser., 32. Princeton Univ. Press, Princeton, NJ, 1971.
[43][Ve]Verbitsky, I. E., Nonlinear potentials and trace inequalities, inThe Maz’ya Anniversary Collection, Vol. 2 (Rostock, 1998), pp. 323–343: Oper. Theory Adv. Appl., 110. Birkhäuser, Basel, 1999.