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1、 - relationM. Krishna Institute of Mathematical Sciences Taramani, Chennai 600 113, India E-mail: krishnaimsc.ernet.in7 January 1999AbstractIn this note we prove a relation between the Riemann Zeta func- tion, and the function (Krein spectral shift) associated with the Harmonic Oscillator in one dim
2、ension. This gives a new integral repre- sentation of the zeta function and also a reformulation of the Riemann hypothesis as a question in L1(R).Part of talk presented at the Conference on Harmonic Analysis, 13-15 March 1997,Ramanujan Institute, University of Madras, Chennai11IntroductionInverse sp
3、ectral theory in one dimension involves recovering a Schr odinger operator from the knowledge of spectrum and a spectral function as doneby Gelfand-Levitan in the fifties. In the recent years there is a great deal of progress achieved in parametrizing iso-spectral classes of potentials (see the revi
4、ews of Simon 13, Gesztesy 4, and the papers of Levitan 7, Kotani- Krishna 9, Craig 1 and Sodin-Yuditskii 14 for Schr odinger operators). One of the consequences of a general formulation obtained via using the Krein spectral shift function by Gesztesy-Simon 5 is given in this paper. The Riemann zeta
5、function is a well studied object, for example, Titch- march 15 gives a detailed exposition of this function. There are several expressions for , and in this note we present an integral representation for , that comes from the Krein spectral shift formula of Krein 10, 11. Recently Gesztesy-Simon 5 g
6、eneralized the trace formulae for Schr odinger operators using the Krein spectral shift function, which they named the function, as it is central to inverse spectral theories in one dimension and had several im- portant applications in spectral theories of operators in one dimension. This work used
7、the proof of the Krein formula, given in Simon 13, theorem I.10 and its generalizations. A proof of the formula for a slightly larger class is shown in Mohapatra-Sinha 8. We refer to these papers for the history and other work on the Krein spectral shift function. Finally we note that the reformulat
8、ion we obtain for the Riemann hy- pothesis is a closure problem in the space L1(R). Though, via the powerfulWieners theorem, the verification only requires exhibiting a single function g(for each (1/2,1) to lie in an explicit subspace X. A. Beurling provided an equivalent condition earlier in his pa
9、per 2, (also see Donoghue 3) which reformulates the Riemann hypothesis as a complete- ness problem in L2(0,1). Lee Jungseob 12 gave another such reformulation.2 function of the Harmonic oscillatorIn this section we recall the function of the Harmonic oscillator (from Gesztesy-Simon 5). We consider t
10、he Harmonic oscillator, H =1 2(d2 dx2+ x2+ 1), acting on L2(R) with the set C0(R) its domain of essential self-adjointness and nor- malized so that its spectrum is the positive integers Z+. We consider the2operator H,xdefined on L2(,x) + L2(x,) as H together with the Dirichlet condition f(x) = 0 at
11、x, using the notation of 5, 6, 13. We denote by Hthe operator H,0in the following. Then the spectrum of His even integers 2Z+, with uniform multiplicity 2. The Krein spectral shift function () for the pair of operators (H, H) is given by() =Xm=12m1,2m)()where Xdenotes the indicator function of X. In
12、 terms of the function, Gesztesy-Simon 5, Simon 13 theorem I.10 (case = ), and Mohapatra- Sinha theorem 4.2 , proved the trace formula,Tr(f(H) f(H) = Z (f()0()d(1)with different smoothness and decay conditions on f. Fix s = + it and consider the smooth function f() = s, on 1,), and zero in (,0), the
13、n by functional calculus, it follows that,f(H) =Z f()dEH(),and f(H) =Z f()dEH()are both trace class for 2, sinceXm=1mandXm=1(2m)converge. Here is the primary relation between the and functions.Theorem 2.1. Let () denote the Krein spectral shift function for the pair of operators (H, H), defined abov
14、e. Then the Riemann zeta function (s) is related to through the relation,(1 2(1s)(s) = sZ1s1()d(2)valid for any s = + it, with 0.3Proof: . We consider s = + it with 2, then we take f(x) = x(s). Then bydefinition (s) = Trace(f(H). Now we rewrite this as,(s) = Trace(f(H) f(H) + Trace(f(H)by the linear
15、ity of the trace. Now we notice that since the spectral multi- plicity of His 2 and the spectrum is the even integers we have,Trace(f(H) = 2Xn=1(2n)(s)= 2(1s)Xn=1(n)(s)= 2(1s)(s).Using the above relations, and the trace formula in terms of the Krein spectral shift, we immediately see that for 2, the theorem is valid, since then the function f() = sis C2and satisfies (1 + |2)fj() L2(R+),j = 1,2, while its extension to 0 follows from the analyticity of the left and right hand sides of equation 2. A simple change of variables ln = x in the expression for given in the above theorem gives the fol
