On Dirichlet-Schrödinger operators with strong potentials

Grillo, Gabriele

Studia Mathematica, Tome 113 (1995), p. 239-254 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider Schrödinger operators H = -Δ/2 + V (V≥0 and locally bounded) with Dirichlet boundary conditions, on any open and connected subdomain $D ⊂ ℝ^n$ which either is bounded or satisfies the condition $d(x,D^{c}) → 0$ as |x| → ∞. We prove exponential decay at the boundary of all the eigenfunctions of H whenever V diverges sufficiently fast at the boundary ∂D, in the sense that $d(x,D^C)^{2}V(x) → ∞$ as $d(x,D^C) → 0$. We also prove bounds from above and below for Tr(exp[-tH]), and in particular we give criterions for the finiteness of such trace. Applications to pointwise bounds for the integral kernel of exp[-tH] and to the computation of expected values of the Feynman-Kac functional with respect to Doob h-conditioned measures are given as well.

Détail

Publié le : 1995-01-01

Zbl 0848.35030

EUDML-ID : urn:eudml:doc:216231

Bibliographie

[00000] [A] S. Agmon, Lectures on Exponential Decay of Eigenfunctions, Princeton University Press, 1982. | Zbl 0503.35001

[00001] [B1] R. Bañuelos, On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains, Probab. Theory Related Fields 76 (1987), 311-323. | Zbl 0611.60071

[00002] [B2] R. Bañuelos, Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators, J. Funct. Anal. 100 (1991), 181-206. | Zbl 0766.47025

[00003] [BD1] R. Bañuelos and B. Davis, Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains, ibid. 89 (1989), 188-200. | Zbl 0676.60073

[00004] [BD2] R. Bañuelos and B. Davis, A geometrical characterization of intrinsic ultracontractivity for planar domains with boundaries given by the graph of functions, Indiana Univ. Math. J. 49 (1992), 885-913. | Zbl 0836.60082

[00005] [vdB] M. van der Berg, On the asymptotics of the heat equation and bounds on traces associated with the Dirichlet Laplacian, J. Funct. Anal. 71 (1987), 279-293. | Zbl 0632.35016

[00006] [CKS] E. A. Carlen, S. Kusuoka and D. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré 23 (1987), 245-287. | Zbl 0634.60066

[00007] [C] R. Carmona, Pointwise bounds for Schrödinger eigenstates, Comm. Math. Phys. 62 (1978), 97-106. | Zbl 0403.47016

[00008] [CS] R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, ibid. 80 (1981), 59-98.

[00009] [CG1] F. Cipriani and G. Grillo, Contractivity properties of Schrödinger semigroups on bounded domains, J. London Math. Soc., to appear. | Zbl 0856.35031

[00010] [CG2] F. Cipriani and G. Grillo, Pointwise lower bounds for positive solutions of elliptic equations and applications to intrinsic ultracontractivity of Schrödinger semigroups, SFB 237-Preprint 202, Ruhr-Universität Bochum, November 1993. | Zbl 0888.35019

[00011] [CG3] F. Cipriani and G. Grillo, Pointwise properties of Dirichlet-Schrödinger operators, preprint RR/12/94, Università di Udine, July 1994.

[00012] [CMC] M. Cranston and T. McConnell, The lifetime of conditioned Brownian motion, Z. Wahrsch. Verw. Gebiete 65 (1983), 1-11. | Zbl 0506.60071

[00013] [D1] E. B. Davies, Hypercontractive and related bounds for double well Schrödinger operators, Quart. J. Math. Oxford Ser. (2) 34 (1983), 407-421. | Zbl 0546.34021

[00014] [D2] E. B. Davies, Trace properties of the Dirichlet Laplacian, Math. Z. 188 (1985), 245-251. | Zbl 0573.35072

[00015] [D3] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.

[00016] [D4] E. B. Davies, Eigenvalue stability bounds in weighted Sobolev spaces, Math. Z. 214 (1993), 357-371. | Zbl 0799.58082

[00017] [DS] E. B. Davies and B. Simon, Ultracontractivity and heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), 335-395. | Zbl 0568.47034

[00018] [Da] B. Davis, Intrinsic ultracontractivity and the Dirichlet Laplacian, ibid. 100 (1991), 162-180. | Zbl 0766.47026

[00019] [Do] J. L. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, New York, 1984. | Zbl 0549.31001

[00020] [GHM] F. W. Gehring, K. Hag and O. Martio, Quasihyperbolic geodesics in John domains, Math. Scand. 65 (1989), 75-92. | Zbl 0702.30007

[00021] [GO] F. W. Gehring and B. G. Osgood, Uniform domains and the quasi-hyperbolic metric, J. Anal. Math. 36 (1979), 50-74. | Zbl 0449.30012

[00022] [GP] F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, ibid. 30 (1976), 172-199. | Zbl 0349.30019

[00023] [RS1] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-adjointness, Academic Press, New York, 1975.

[00024] [RS2] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators, Academic Press, New York, 1978.

[00025] [S] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447-526. | Zbl 0524.35002

[00026] [SS] W. Smith and D. A. Stegenga, Hölder domains and Poincaré domains, Trans. Amer. Math. Soc. 319 (1990), 67-100. | Zbl 0707.46028

[00027] [Ste] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Academic Press, San Diego, 1978.

[00028] [Sta] S. G. Staples, $L^p$-averaging domains and the Poincaré inequality, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 103-127. | Zbl 0706.26010

[00029] [V] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 1319, Springer, Berlin, 1988.