zbMATH — the first resource for mathematics

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 \(-\Delta\), which is fundamental to quantum mechanics. Moreover, the authors give both boundedness and compactness criteria for Sobolev spaces on domains \(\Omega\subset \mathbb{R}^d\) under mild restrictions on \(\partial \Omega\). They obtain also criteria for the classical inequality \[ \left |\int_{\mathbb{R}^d} \bigl|u(x)\bigr |^2 V(x)dx\right |\leq C_* \int_{ \mathbb{R}^d} \bigl|\nabla u(x) \bigr|^2dx,\;u\in C_0^\infty (\mathbb{R}^d), \] to be hold, where the “indefinite” weight \(V\) may change sign, or even be a complex-valued distribution on \(\mathbb{R}^d\), \(d\geq 3\).

35J10 Schrödinger operator, Schrödinger equation
35B35 Stability in context of PDEs
47F05 General theory of partial differential operators
47H50 Potential operators (MSC2000)
46N50 Applications of functional analysis in quantum physics
Full Text: DOI
[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
[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
[4] [BeS]Berezin, F. A. &Shubin, M. A.,The Schrödinger Equation. Math. Appl. (Soviet Ser.), 66. Kluwer, Dordrecht, 1991. · Zbl 0749.35001
[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. · Zbl 0744.47017
[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. · Zbl 0291.44007
[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. · Zbl 0336.47007
[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
[11] Chung, K. L. &Zhao, Z. X.,From Brownian Motion to Schrödinger’s Equation. Grundlehren Math. Wiss., 312. Springer-Verlag, Berlin, 1995. · Zbl 0819.60068
[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
[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. · Zbl 0628.47017
[15] Faris, W. G.,Self-Adjoint Operators. Lecture Notes in Math., 433. Springer-Verlag, Berlin-New York, 1975. · Zbl 0317.47016
[16] Fefferman, C., The uncertainty principle.Bull. Amer. Math. Soc. (N.S.), 9 (1983), 129–206. · Zbl 0526.35080
[17] Hille, E., Non-oscillation theorems.Trans. Amer. Math. Soc., 64 (1948), 234–252. · Zbl 0031.35402
[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
[19] Kalton, N. J. &Verbitsky, I. E., Nonlinear equations and weighted norm inequalities.Trans. Amer. Math. Soc., 351 (1999), 3441–3497. · Zbl 0948.35044
[20] Kerman, R. &Sawyer, E., The trace inequality and eigenvalue estimates for Schrödinger operators.Ann. Inst. Fourier (Grenoble), 36 (1986), 207–228. · Zbl 0591.47037
[21] Kondratiev, V., Maz’ya, V. G. & Shubin, M., Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators. To appear. · Zbl 1140.35300
[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. · Zbl 0668.31002
[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. · Zbl 0727.46017
[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
[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
[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-{\(\lambda\)}u”=Vu on the semiaxis.J. Funct. Anal., 151 (1997), 504–530. · Zbl 0895.34063
[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. · Zbl 0308.47002
[35] [S1]Schechter, M.,Operator Methods in Quantum Mechanics. North-Holland, New York-Amsterdam, 1981. · Zbl 0456.47012
[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. · Zbl 0654.42022
[38] [Si]Simon, B., Schrödinger semigroups.Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447–526. · Zbl 0524.35002
[39] [St1]Stein, E. M.,Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser., 30. Princeton Univ. Press, Princeton, NJ, 1970. · Zbl 0207.13501
[40] [St2]–,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Math. Ser., 43. Princeton Univ. Press, Princeton, NJ, 1993. · Zbl 0821.42001
[41] [Stu]Sturm, K. T., Schrödinger operators with highly singular, oscillating potentials.Manuscripta Math., 76 (1992), 367–395. · Zbl 0767.35054
[42] [StW]Stein, E. M. &Weiss, G.,Introduction to Fourier Analysis on Euclidean Spaces. Princeton Math. Ser., 32. Princeton Univ. Press, Princeton, NJ, 1971. · Zbl 0232.42007
[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.
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.