On the Characters and the Plancherel Formula of Nilpotent Groups ...

More documents

Recommendations

Info

368 KATO the boundary between IR and @,I. It follows that Condition satisfied with the subscripts 1, 2 exchanged, with m = M = 1. 10.1 is By Theorem 10.5 (a), Condition 10.2 is also satisfied with the subscripts exchanged. Applying Theorem 10.3 with the subscripts exchanged, we see that W,( U, , U,; 9’) and W,( U, , U,; 9) exist, are complete and mutually adjoint partial isometries. Here 9’ is the projection of & onto $jl , which is now a subspace of J& , and p is the canonical injection of $r into sjB . Other remarks similar to the ones given in the preceding paragraphs apply also in this case. Finally we consider boundary conditions of the third kind au/&z + u(x) u = 0 on aJ2. We denote the corresponding selfadjoint operator in R’ = L2(sZ) by Ai . If we assume that o(x) > 0, then A; > 0 and it is associated with the nonnegative quadratic form 1) Bju ]I2 which is the sum of the Dirichlet form and the surface integral Jan u(x) 1 u I2 dS, where B; = Ai1i2. For convenience we define A: = Ai as above as the selfadjoint operator in a” = L2(Q,,) with the Neumann boundary condition, and set A, = Aj @ A:. Again Condition 9.1 is satisfied by Birman’s criterion, and Theorem 9.3 holds true. The form 11 B,u /I2 = 11 Bju /I2 + (I B& II2 associated with A, has domain 3(B3) = BL(Q) @ BL(Q,) = a(B,), since the surface integral is bounded with respect to the Dirichlet integral if u(x) is sufficiently smooth. Thus we have D(B,) 3 D(B,) and Condition 10.1 is satisfied with the subscripts 1, 2 replaced by 3, 1. But the corresponding constants are such that m < 1 = M if u(x) > 0 for some x E 8Q. Thus we do not know whether or not Condition 10.2 is satisfied. To avoid this difficulty, it is convenient to use the chain rule (Theorem 4.3). We consider three groups, the unperturbed group U, , the group U, considered above for the Neumann boundary condition, and the group U, for the third-kind boundary condition under consideration. Accordingly, we introduce the following identification operators. Jzr E B(& ,a,) and Ji2 E B(& , &) are the Jand J’ considered above for the wave operators for the pair U, , U, . Similarly we define Jal E B($j, , $a) and Jr, E B($& , $i) for the pair U, , U, . For the pair U, , U, , we define Ja2 E B(Jj, , a,) and -723 = hi1 E W, 9 52) as simple identifications between fj, and $a , which are identical as a vector space. In fact we have a(B,) = Ad and II B,u II < II B,u II < M’ II B,u II > u E a(B,) = 3)(B,). (11.7) It follows that not only Condition 10.1 but also Condition 10.2 is satisfied for the pair U, , U, , again by Theorem 10.5, (a). Hence
SCATTERING THEORY WITH TWO HILBERT SPACES 369 W,( U, , U,; JS2) and W+( U, , Ua; J2a) exist, are complete and are mutually adjoint partial isometries. Also we know that the same is true of w,( us , UC Jzl) and W,(u, , us; j&J. But 331 = 332321 9 both members being equal to the canonical injection of 5r into &. Thus it follows from Theorem 4.3 that W+( U, , Ul; y3r) exist, are complete and are partial isometries. Furthermore, since JrsJai = 1 (identity in $Q, J1a is a (U, , &)asymptotic left inverse to J3r . It follows from Theorem 5.3 that W+( U, , U3; Jra) exist, are complete, and are equal to W+( U, , Ul; 33,>*. REFERENCES 1. -TO, T. “Perturbation Theory for Linear Operators.” Springer, Berlin, 1966. 2. BIRMAN, M. SH., Existence conditions for wave operators. Izv. Akad. Nauk S.S.S.R., Ser. Mat. 21 (1963), 883-906; Am. Math. Sac. Transl., Ser. 2. 54 (1966), 91-117. 3. KURODA, S. T., An abstract stationary approach to perturbation of continuous spectra and scattering theory. J. Anal. Math. (to be published). 4. LAX, P. AND PHILLIPS, R. S., Scattering theory. Bull. Am. Math. Sot. 70 (1964), 130-142 (also, forthcoming book “Scattering Theory,” Academic Press, New York). 5. SHWK II, N. A., Eigenfunction expansions and scattering theory for the wave equation in an exterior region. Arch. Ratl. Meek. Anal. 21 (1966), 120-150. 6. SCHMIDT, G., Scattering theory for Maxwell’s equations in an exterior domain (Preprint, Stanford University). 7. THOE, D., Spectral theory for the wave equation with a potential term. Arch. Ratl. Meek. Anal. 22 (1966), 364-406. 8. WILCOX, C. H., Wave operators and asymptotic solutions of wave propagation problems of classical physics. Arch. RatZ. Mech. Anal. 22 (1966), 37-78. 9. PUTNAM, C. R., On normal operators in Hilbert space. Am. J. Math. 73 (1951), 357-362. 10. KATO, T., Wave operators and unitary equivalence. Pacific J. Math. 15 (1965), 171-180. 11. kEF3E, T., Eigenfunction expansions associated with the Schroedinger operators and their applications to scattering theory. Arch. Ratl. Mech. An&. 5 (1960), l-34. 12. KATO, T., Notes on some inequalities for linear operators. Math. Ann. 125 (1952), 208-212. 13. WILCOX, C. H., Uniform asymptotic estimates for wave packets in the quantum theory of scattering. J. Math. Phys. 6 (1965), 611-620. 14. BIRMAN, M. SH., Perturbations of the continuous spectrum of a singular elliptic operator under the change of the boundary and boundary conditions. Vest. Leningrad. Univ., Ser. Mat., Meh, Astron. 1 (1962), 22-55. 15. DENY, J. AND LIONS, J. L., L-es espaces du type de Beppo Levi. Ann. Inst. Fourier 5 (1953-54), 305-370. 16. SHIZUTA, Y., Eigenfunction expansion associated with the operator -A in the exterior domain. Proc. Japan Acad. 39 (1963), 656-660.
Page 1 and 2:
JOURNAL OF FUNCTIONAL ANALYSIS 1, 2
Page 3 and 4:
PLANCHEREL FORMULA OF NILPOTENT GRO
Page 5 and 6:
and PLANCHEREL FORMULA OF NILPOTENT
Page 7 and 8:
Page 9 and 10:
PLANCHERRL FORMULA OF NILPOTRNT GRO
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
PLANCHERRL FORMULA OF NILPOTENT GRO
Page 27 and 28:
JOURNAL OF FUNCTIONAL ANALYSIS 1, 2
Page 29 and 30:
VECTEURS SePARATEURs DES ALGhBRES D
Page 31 and 32:
VECTEURS SfiPARATEURS DES ALGkBRES
Page 33 and 34:
VECTEURS SltPARATEURS DES ALGhBRBS
Page 35 and 36:
VECTEURS SiPARATEURS DES ALGBBRES D
Page 37 and 38:
SIZES OF COMPACT SUBSETS OF HILBERT
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Thus where SIZES OF COMPACT SUBSETS
Page 59 and 60:
Page 61 and 62:
Page 63 and 64: SIZES OF COMPACT SUBSETS OF HILBERT
Page 77 and 78: JOURNAL OF FUNCTIONAL ANALYSIS 1, 3
Page 79 and 80: ALGEBRAS WITH THE SAME MULTIPLICATI
Page 89 and 90: SCATTERING THEORY WITH TWO HILBERT
Page 107 and 108: Condition 10.2. SCATTERING THEORY W
Page 113: SCATTERING THEORY WITH TWO HILBERT
Page 117 and 118: STRUCTURE SPACES AND EXTENSIONS 0~
Page 119 and 120: STRUCTURE SPACES AND EXTENSIONS OF
show all

On the Characters and the Plancherel Formula of Nilpotent Groups ...

Create successful ePaper yourself

Delete template?

Save as template?