The number of eigenstates: Counting function and heat kernel

Journal of High Energy Physics

Volumn 2009, Issue 2, 2009, Pages

  Dai, Wu Sheng ^a,b   Xie, Mi ^a,b  

^a TIANJIN UNIVERSITY   (China)

^b NANKAI UNIVERSITY   (China)

Author keywords

Boundary quantum field theory; Differential and algebraic geometry

Indexed keywords

EID: 67649540732     PISSN: 11266708     EISSN: 10298479     Source Type: Journal    
DOI: 10.1088/1126-6708/2009/02/033     Document Type: Article

References (40)

1. M. Kac, Can one hear the shape of a drum? 1966 Am. Math. Monthly 73 1
2. M. Berger, A panoramic view of Riemannian geometry (Springer-Verlag, Berlin) Germany (2003)
3. D.V. Vassilevich, Heat kernel expansion: user's manual 2003 Phys. Rep. 388 279 [hep-th/0306138]
4. E. Elizalde and A. Romeo, Expressions for the zeta Function Regularized Casimir Energy 1989 J. Math. Phys. 30 1133 [Erratum 1990 J. Math. Phys. 31 771]
5. E. Elizalde, Multiple zeta functions with arbitrary exponents 1989 J. Phys. A: Math. Gen. 22 931
6. M. Bordag, E. Elizalde and K. Kirsten, Heat kernel coefficients of the Laplace operator on the D-dimensional ball 1996 J. Math. Phys. 37 895 [hep-th/9503023]
7. M. Bordag, E. Elizalde, K. Kirsten and S. Leseduarte, Casimir energies for massive fields in the bag 1997 Phys. Rev. D 56 4896 [hep-th/9608071]
8. E. Elizalde, M. Bordag and K. Kirsten, Casimir energy for a massive fermionic quantum field with a spherical boundary 1998 J. Phys. A: Math. Gen. 31 1743
9. V.V. Nesterenko, I.G. Pirozhenko and J. Dittrich, Nonsmoothness of the boundary and the relevant heat kernel coefficients 2003 Class. Quantum Grav. 20 431 [hep-th/0207038]
10. P.H. Bérard, Spectral geometry: direct and inverse problems, with an appendix by G. Besson (Springer-Verlag, Berlin) Germany (1986)
11. J.W. Milnor, Eigenvalues of the Laplace operator on certain manifolds 1964 Proc. Nat. Acad. Sci. U.S.A. 51 542
12. G. Gordon, D. Webb and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds 1992 Invent. Math. 110 1
13. V. Ivrii, Microlocal analysis and precise spectral asymptotics (Springer-Verlag, Berlin) Germany (1998)
14. R. Aurich, S. Lustig, F. Steiner and H. Then, Can one hear the shape of the Universe? 2005 Phys. Rev. Lett. 94 021301 [astro-ph/0412407]
15. F. Bastianelli, O. Corradini and P.A.G. Pisani, Worldline approach to quantum field theories on flat manifolds with boundaries 2007 J. High Energy Phys. JHEP02(2007)059 [hep-th/0612236]
16. I.G. Avramidi, Heat kernel approach in quantum field theory 2002 Nucl. Phys. (Proc. Suppl.) 104 3 [math-ph/0107018]
17. M. Bordag and N. Khusnutdinov, On the vacuum energy of a spherical plasma shell 2008 Phys. Rev. D 77 085026 [0801.2062]
18. V.N. Marachevsky, Casimir interaction of two plates inside a cylinder 2007 Phys. Rev. D 75 085019 [hep-th/0703158]
19. E. Elizalde, M. Minamitsuji and W. Naylor, Casimir effect in rugby-ball type flux compactifications 2007 Phys. Rev. D 75 064032 [hep-th/0702098]
20. V.N. Marachevsky and D.V. Vassilevich, Chiral anomaly for local boundary conditions 2004 Nucl. Phys. B 677 535 [hep-th/0309019]
21. I.L. Shapiro, Effective action of vacuum: the semiclassical approach 2008 Class. Quantum Grav. 25 103001 [0801.0216]
22. H.E. Camblong and C.R. Ordonez, Conformal enhancement of holographic scaling in black hole thermodynamics: a near-horizon heat-kernel framework 2007 J. High Energy Phys. JHEP12(2007)099 [0709.2942]
23. N.R. Khusnutdinov and I.V. Bakhmatov, Self-force for charged particle in the wormhole space-time 2007 Phys. Rev. D 76 124015 [0707.3396]
24. L.L. Salcedo, Derivative expansion of the heat kernel in curved space 2007 Phys. Rev. D 76 044009 [0706.1875]
25. C.P. Burgess and D. Hoover, UV sensitivity in supersymmetric large extra dimensions: the Ricci-flat case 2007 Nucl. Phys. B 772 175 [hep-th/0504004]
26. D. Hoover and C.P. Burgess, Ultraviolet sensitivity in higher dimensions 2006 J. High Energy Phys. JHEP01(2006)058 [hep-th/0507293]
27. Y.-b. Kim, O.-K. Kwon and J.-S. Sohn, Vacuum structure and global strings with conical singularities 2002 Phys. Lett. B 549 221 [hep-th/0208140]
28. A.O. Barvinsky and D.V. Nesterov, Quantum effective action in spacetimes with branes and boundaries 2006 Phys. Rev. D 73 066012 [hep-th/0512291]
29. M. Minamitsuji, W. Naylor and M. Sasaki, Volume stabilization in a warped flux compactification model 2006 J. High Energy Phys. JHEP12(2006)079 [hep-th/0606238]
30. R. Fresneda, D.M. Gitman and D.V. Vassilevich, Nilpotent noncommutativity and renormalization 2008 Phys. Rev. D 78 025004 [0804.1566]
31. O.D. Azorkina, A.T. Banin, I.L. Buchbinder and N.G. Pletnev, Construction of the effective action in nonanticommutative supersymmetric field theories 2006 Phys. Lett. B 633 389 [hep-th/0509193]
32. V. Gayral, B. Iochum and D.V. Vassilevich, Heat kernel and number theory on NC-torus 2007 Comm. Math. Phys. 273 415 [hep-th/0607078]
33. I.G. Pirozhenko and V.V. Nesterenko, Integral equations for heat kernel in compound media 2005 J. Math. Phys. 46 042305 [hep-th/0409289]
34. 3/2 heat kernel coefficient for oblique boundary conditions, 1999 Class. Quantum Grav. 16 1917 [hep-th/9806168]
35. th edition, (Dover, New York) U.S.A. (1972)
36. M. Stone, The physics of quantum fields Springer-Verlag, New York U.S.A. (2000)
37. H. Weyl, Gesammelte Abhandlungen. Volume I Springer, Berlin Germany (1968)
38. A. Pleijel, A study of certain Green's functions with applications in the theory of vibrating membranes 1954 Ark. Mat. 2 553
39. S.S. Chern, W.H Chen and K.S. Lam, Lectures on differential geometry (World Scientific, Singapore) Singapore (2000)
40. R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II Wiley Interscience, New York U.S.A. (1962)