H Witten- ¾. 1956, Payne-póyla Weinberger [15] Ó ĐË È : (1) λ k+1 λ r 4. λ r. (2) n k. λ k , Yang [19] ÅĐ «Yang ¾. (λ k+1 λ r )λ r 1+ 4 ) 1

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1 44Ñ Vol.44, No Ù ADVANCES IN MATHEMATICSCHINA Mar., 015 H Witten- ¾ É ÁÅ ³ Ý 1,, Õ doi: /sxjz b 0 1. Æ Þ ÆÔÅ Ø, Æ,, 5300;. Þ Ê, Æ,, : Ë Ñ H- ÔÖ Witten- ÐÒÐÛÜÅ G+ G, Gϕ Þ Đß. Ð Ù Ú ÏÎÑ ÜÅÞ Yang Í Ø, ÏÎÑ Yang Í Ø. Ë Ê Ñ [16], Ì [14] Ó. : H- Ô; Þ ; Witten- ÐÒÐÛ MR010 Ç : 35P15; 53C0 / Æ : O186»: A À : 1956, Payne-póyla Weinberger [15] Ó ĐË È : { u = λu, x, u = 0, x, 1 Đ ¾, Ã n Æ, Ä PPW ¾ λ k+1 λ k 4 nk λ r. Ü Payne-póyla Weinberger, ± ÈÒ½ ºÈ. 1980, Hile Protter [6] ĐË HP ¾ λ r nk λ k+1 λ r , Yang [19] ÅĐ «Yang ¾ λ k+1 λ r 4 n λ k+1 λ r λ r ³ Yang ¾ λ k n k λ r. ¾³ : »¾ ³ : Á Ï : É Ô Á No »½ Ô Á No. 14KJD ystsy@163.com

2 ß 44Ñ Ôº Þ, Õ ÅÓ [1 4, 6 9]. Ù, È Đ Û M,g,e φ dv Witten- È [11 13]. { + ϕ, u = λu, x, 3 u = 0, x. Ð ϕ. 013, Xia Xu [18] Ó ĐÛ º Æ, Đ Yang ¾. 003, Niu [14] Đ Hesienberg H n È : { H nu = λu, x, 4 u = 0, x, ĐË ¾ λ k+1 λ k nk λ r. 5 Đ È 4, Soufi, Harrel II lias [16] ÐÑ µ ĐË λ k+1 λ r n λ k+1 λ r λ r. 6 01, Sun [17] Ó Đ H n ÆÈ, [16] «Ã Ê Ç Ó¼ «dµ := e ϕw dg Heisenberg G Witten- Æ, «ϕ dw ¼, 1 4 Gϕ 1 Gϕ. dw G ² w ÐÜ. Ò [11] Õ, Witten- L = G + G ϕ, G L dµ, É ÏÓ Ó Ö º Dirichlet È : { Lu = λu, w, 7 u = 0. Ò [5, 10] Õ, º ÓÜ : 0 < λ 1 λ λ r ր. 1 ± Ä Ü n+m Æ ÖÌ R n R n R m Ë Ú z,t z,t = x,y,t x,y,t = z i +z i,t j +t j + 1 z,uj z, 8 i = 1,,,n; j = 1,,,m, «z = x,y R n, t R m,, Â, «U 1,U,,U n Ë ÀÎ: 1 U j m m, j = 1,,,n; U i U j +U j U i = 0, i,j {1,,,n}, i j. Ð Ú n+m Æ ÖÌ Å Heisenberg, Í Å H. ß g À Å X j = x j + 1 m k=1 n l=1 z l U k l,j, Y j = + 1 t k y j m k=1 n l=1 z l U k l,j+n t k, T k = t k, 9

3 º, ±: H Witten- Ø Ø Ç 3 «U k l,j n n = U k. Ó¼ «dµ := e ϕw dg H G Witten Å L = G + G ϕ, G, 10 Ð G = n X j +Y j, G = X 1,X,,X n,y 1,Y,,Y n. G ² w = z,t Õ ÐÜ Å dw = z t 1 4. H Í «Heisenberg. n+1 Æ Heisenberg H n Ë Ú x,y,tx,y,t = x+y,x+y,t+t + 1 x,y x,y, «x,y,x,y R n, t,t R,, R n «Â. H n ß À Å «X j = x j + yj {X j,y j,t}, j = 1,,,n. t, Y j = y j xj t, T = t ¹. 1.1 Ë¹ G Ó Ö, Ý f,g C k, k, f = g = 0, G fg = f G g+g G f; 11 G fg = f G g + G f, G g +g G f. 1 X i fx j gdµ = fx i X j gdµ; fdiv G G gdµ = G f, G g dµ; f G gdµ = g G fdµ. ¹ 1. Ë¹ Å G Ó Ö, f,g C k, k f = g = 0, ÝÓ G f, G g dµ = glfdµ = flgdµ. 13 Â¼ Ò Ý 1.1 Ã ÝÕ glfdµ = g G f + G ϕ, G f dµ = ge ϕ fdµ+ g G ϕ, G f dµ = G fe ϕ G g ge ϕ G ϕdg+ g G ϕ, G f dµ = G f, G g e ϕ dg = G f, G g dµ. 14

4 4 ß 44Ñ È µ ²Ã½ ¹.1 ¹ λ i ÅÈ 7 i, i = 1,,,n, u i Å λ i ÎÐ, ÐÔ Â Î, ÅÓ Ö, Ý λ k+1 λ i 1 n λ k+1 λ i u i G ϕ +4 G u i +4u i G ϕ, G u i dµ. 15 Â¼ Ì, u i Ë : φ i = x α u i G u i + ϕ, u i = λu i, x, u = 0, u iu j dµ = δ i,j. a i,j u j, i = 1,,,k, α = 1,,,n. 16 «a i,j = x αu i u j dµ. Á Í Æ Õ Í φ i = 0, φ i u j dµ = 0, i,j = 1,,,k. 17 ² φ i x α u i dµ = Æ Õ Lφ i = X α u j +x α Lu i +u i X α ϕ 19 ¾ µ «Rayleigh-Ritz ¾ λ k+1 φ idµ Õ ÝÓ Û¹ ¹ φ idµ. 18 λ j a i,j u j, 19 φ i Lφ i dµ, 0 λ k+1 λ i φ idµ φ i X α u i +u i X α ϕdµ. 1 φ i X α u i +u i X α ϕdµ = Q i, λ k+1 λ i φ i dµ Q i. 3 b i,j = X α u j +u j X α ϕu i dµ, 4

5 º, ±: H Witten- Ø Ø Ç 5 Ý b i,j = b j,i, λ i λ j a i,j = b i,j. Ó 18, Õ Q i = φ i X α u i +u i X α ϕ 1 φ i dµ b i,j u j dµ X α u i +u i X α ϕ b i,j u j dµ 1. 5 ² ² λ k+1 λ i Q i λ k+1 λ i φ i dµ λ k+1 λ i Q i λ k+1 λ i X α u i +u i X α ϕ X α u i +u i X α ϕ b i,j u j dµ, b i,j u j dµ. 6, Æ Õ X α u i +u i X α ϕ b i,j u j dµ = X α u i +u i X α ϕ dµ ÐÑ 3 Á ¹ Â¹Õ Q i = Æ Õ x α u i a i,j u j X α u i +u i X α ϕdµ = λ k+1 λ i λ i λ j a i,j = i, Ï 7 9 µ 6, Õ λ k+1 λ i Ï ¾ Ô α 1 Ê n, Õ n λ k+1 λ i Ú, Õ φ i = y β u i u idµ+ λ i λ j a i,j. 7 λ i λ j a i,j. 8 λ k+1 λ i λ i λ j a i,j. 9 i, λ k+1 λ i X α u i +u i X α ϕ dµ. 30 λ k+1 λ i n X α u i +u i X α ϕ dµ. 31 α=1 a i,j u j, i = 1,,,k, β = 1,,,n,

6 6 ß 44Ñ «a i,j = y βu i u j dµ. Ú, Õ n λ k+1 λ i Ó 31 3, Õ n λ k+1 λ i λ k+1 λ i n Y β u i +u i X β ϕ dµ. 3 β=1 λ k+1 λ i u i Gϕ +4 G u i +4u i G ϕ, G u i dµ. 33 Å, Ý.1. ¹. Ü Ý.1 ÀÎË, Ë¹ G ϕ c, Ý Â¼ Å λ k+1 λ i 1 n λ k+1 λ i c +4λ i +4cλ 1 i. 34 u i G ϕ, G u i dµ u i ϕ G u i dµ 1 c u i dµ = c G u i dµ 1, G u i dµ 1 35 G u i dµ = G u i, G u i dµ = Ý Ï µ 15, ÄÕÄÆ Ý. Å. º.1 Ü Ý.1 ÀÎË, Ë¹ ϕ=, Ý u i Lu i dµ = λ i, 36 u i G ϕ, G u i dµ cλ 1 i. 37 λ k+1 λ i n λ k+1 λ i λ i. 38 É Õ ÍÃ.1 [16] «6, º [14] 5. Á Ý. «¾, Õ ³ Yang ¾. º. Ü Ý.1 ÀÎË, { λ k+1 1 ka k n + λ i + ka n + 1 } λ i ka n λ i, 39

7 º, ±: H Witten- Ø Ø Ç 7 «A = c +4λ i +4cλ 1 i. Á ºÃ, É Õ ÐÑ k Ù Æ k +1. [1] Ashbaugh, M.S., Universal eigenvalue bounds of Payne-Pólya- Weinberger, Hile-Protter and H. C. Yang, Proc. Indian Acad. Sci. Math. Sci., 00, 11: [] Chen, D.G. and Cheng, Q.M., Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan, 008, 60: [3] Cheng, Q.M. and Yang, H.C., Estimates on Eigenvalues of Laplacian, Math. Ann., 005, 331: [4] Cheng, Q.M. and Yang, H.C., Inequalities for eigenvalues of Laplacian on domains and compact hypersurfaces in complex projective spaces, J. Math. Soc. Japan, 006, 58: [5] Fridrichs, K.O., Spectral theory of operators in Hilbert space, New York: Springer-Verlag, [6] Hile, G.N. and Protter, M.H., Inequalities for eigenvalues of the Laplacian, Indiana Univ. Math. J, 1980, 9: [7] Harrell II, E.M., Some geometric bounds on eigenvalue gaps, Comm. in Part. Diff. Eqns, 1993, 18: [8] Harrell II, E.M., Commutators, eigenvalue gaps, and mean curvature in the theory of Schröinger operators, Commun. Part. Diff. Eqns, 007, [9] Harrell II, E.M., and Michel, P.L., Commutator bouds for eigenvalues, with applications to spectral geometry,comm. Part. Diff. Eqns, 1994, 19: [10] Inglis, J., Spectral inequalities for operators on H-type groups, J. Spectr. Theory, 01, : [11] Ma, L. and Liu, B.Y., Convex eigenfunction of a drifting Laplacian operator and the fundamental gap. Pacific J. Math,009, 40: [1] Ma, L. and Liu, B.Y., Convexity of the first eigenfunction of the drifting Laplacian operator and its applications. New Yo rk J. Math., 008, 14: [13] Ma, L. and Du, S.H., Extension of Reilly formula with applications to eigenvalue estimates for drifting Laplacians, C. R. Math. Acad. Sci. Paris, 010, 3481/: [14] Niu, P.C. and Zhang H.Q., Payne-Pólya-Weinberger type inequalities for eigenvalues of nonelliptic operators, Pacific J Math, 003, 08: [15] Payne, L.E. Polya, G. and Weinberger, H.F., On the ratio of consecutive eigenvalues, J. Math. and Phis, 1956, 35: [16] Soufi, E.A., Harrell II, E.M. and Ilias, S., Universal inequalities for the eigenvalues of Laplace and Schrodinger operators on submanifolds, Trans. Amer. Math. Soc, 009, 361: [17] Sun, H.J., Universal inequalities and bounds for weighted eigenvalues of the Schrodinger operator on the Heisenberg group, Turk J Math, 011, 35: [18] Xia, C.Y. and Xu, H.W., Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds, Ann Glob Anal Geom, 013. [19] Yang, H.C., An estimate of the difference between consecutive eigenvalues, IC/91/60 of ICTP, Trieste, Estimates for Eigenvalues of the Witten-Laplacian on Heisenberg-type Group TAN Shenyang 1, HUANG Tiren 1. Tai zhou institute of science and technology., NUST., Taizhou, Jiangsu, 5300, P.R. China;. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, , P. R. China Abstract: In this paper, we investigate the Dirichlet eigenvalue problem of the witten Laplacian G + G, G ϕ on Heisenberg-type group. we obtain a Yang-type inequality for eigenvalues of this problem which contain the result in [16]. we also get second yang type inequality. Keywords: Heisenberg-type group; Eigenvalue estimates; witten-laplacian

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