Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators
Guglielmo Fucci, Fritz Gesztesy, Klaus Kirsten, Jonathan Stanfill
aa r X i v : . [ m a t h . SP ] J a n SPECTRAL ζ -FUNCTIONS AND ζ -REGULARIZEDFUNCTIONAL DETERMINANTS FOR REGULARSTURM–LIOUVILLE OPERATORS GUGLIELMO FUCCI, FRITZ GESZTESY, KLAUS KIRSTEN, AND JONATHAN STANFILL
Abstract.
The principal aim in this paper is to employ a recently developedunified approach to the computation of traces of resolvents and ζ -functionsto efficiently compute values of spectral ζ -functions at positive integers as-sociated to regular (three-coefficient) self-adjoint Sturm–Liouville differentialexpressions τ . Depending on the underlying boundary conditions, we expressthe ζ -function values in terms of a fundamental system of solutions of τy = zy and their expansions about the spectral point z = 0. Furthermore, we give thefull analytic continuation of the ζ -function through a Liouville transformationand provide an explicit expression for the ζ -regularized functional determinantin terms of a particular set of this fundamental system of solutions.An array of examples illustrating the applicability of these methods is pro-vided, including regular Schr¨odinger operators with zero, piecewise constant,and a linear potential on a compact interval. Contents
1. Introduction 22. Background on Self-Adjoint Regular Sturm–Liouville Operators 33. Expansion in z for Fundamental Solutions, Asymptotic Expansion, andthe Zeta Regularized Functional Determinant 103.1. Expansion in z for Fundamental Solutions 103.2. Asymptotic Expansion of the Characteristic Function 113.3. Analytic Continuation of the Spectral Zeta Function and the ZetaRegularized Functional Determinant 204. Computing Spectral Zeta Function Values and Traces for RegularSturm–Liouville Operators 224.1. Computing Spectral Zeta Function Values and Traces for SeparatedBoundary Conditions 234.2. Computing Spectral Zeta Function Values and Traces for CoupledBoundary Conditions 265. Examples 295.1. The Example q =0 295.2. Examples of Nonnegative (Piecewise) Constant Potentials 345.3. Example of a Negative Constant Potential 38 Date : February 1, 2021.2020
Mathematics Subject Classification.
Primary: 47A10, 47B10, 47G10. Secondary: 34B27,34L40.
Key words and phrases. ζ -function, Sturm–Liouville operators, Traces, (modified) Fredholmdeterminants, zeta regularized functional determinants. Introduction
The principal motivation for this paper is to illustrate how a recently developedunified approach to the computation of Fredholm determinants, traces of resolvents,and ζ -functions in [33] can be used to efficiently compute certain values of spec-tral ζ -functions associated to regular Sturm–Liouville operators as well as give thefull analytic continuation of the ζ -function through a Liouville transformation andfinally provide an explicit expression for the ζ -regularized functional determinant.In Section 2 we begin by outlining the background for regular self-adjoint Sturm–Liouville operators on bounded intervals, that is, operators in L (( a, b ); rdx ) withseparated and coupled boundary conditions and the associated spectral ζ -functions.Under appropriate hypotheses on the Sturm–Liouville operator associated withthree-coefficient differential expressions of the type τ = r − [ − ( d/dx ) p ( d/dx ) + q ],certain values of the spectral ζ -function can be found via complex contour integra-tion techniques to be equal to residues of explicit functions involving a canonicalsystem of fundamental solutions φ ( z, · , a ) and θ ( z, · , a ) of τ y = zy for separated orcoupled boundary conditions. Moreover, the zeros with respect to the parameter z of φ , θ , and some of their (boundary condition dependent) linear combinationsare precisely the eigenvalues corresponding to the underlying operator, includingmultiplicity.In Section 3 we provide a series expansion for φ ( z, · , a ) and θ ( z, · , a ) about z = 0using the Volterra integral equation associated with the general three-coefficientregular self-adjoint Sturm–Liouville operator. This method leads to an expansionin powers of z of the fundamental solutions and their z -derivative involving theirvalues at z = 0 and the appropriate Volterra Green’s function. We also investigatethe | z | → ∞ asymptotic expansion of the characteristic function appearing in thecomplex integral representation of the spectral ζ -function given in Section 2. Thisasymptotic expansion is then exploited in order to construct the analytic contin-uation of the spectral ζ -function and to obtain an explicit expression for the zetaregularized functional determinant.Section 4 contains the main theorems that allow for the calculation of the valuesof spectral ζ -functions of general regular Sturm–Liouville operators on boundedintervals as ratios of series expansions of (boundary condition dependent) solutionsof τ y = zy about z = 0. In particular, we consider separated boundary conditionswhen zero is not an eigenvalue, or, when it is (necessarily) a simple eigenvalue, andcoupled boundary conditions when either zero is not an eigenvalue, or, an eigenvalueof multiplicity (necessarily) at most two. (For more details in this context see [33]as well as [34, Ch. 3], [76, Sect. 8.4], [77, Sect. 13.2], and [78, Ch. 4].)We continue by providing some examples in Section 5 illustrating the main the-orems and corollaries of Section 4 and the zeta regularized functional determinantgiven in Section 3. In particular, we present the case of Schr¨odinger operators withzero potential imposing Dirichlet, Neumann, periodic, antiperiodic, and Krein–vonNeumann boundary conditions. We then consider positive (piecewise) constant and PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 3 negative constant potentials for Dirichlet boundary conditions, and finally the caseof a linear potential.Here we summarize some of the basic notation used in this manuscript. If A is alinear operator mapping (a subspace of) a Hilbert space into another, then dom( A )and ker( A ) denote the domain and the kernel (i.e., null space) of A . The spectrum,point spectrum, and resolvent set of a closed linear operator in a separable complexHilbert space, H , will be denoted by σ ( · ) , σ p ( · ) , and ρ ( · ) respectively. If S is self-adjoint in H , the multiplicity of an eigenvalue z ∈ σ p ( S ) is denoted m ( z ; S ) (thegeometric and algebraic multiplicities of S coincide in this case). The proper settingfor our investigations is the Hilbert space L (( a, b ); rdx ), which we will occasionallyabbreviate as L r (( a, b )). The spectral ζ -function of a self-adjoint linear operator S is denoted by ζ ( s ; S ). In addition, tr H ( T ) denotes the trace of a trace class operator T ∈ B ( H ) and det H ( I H − T ) the Fredholm determinant of I H − T .For consistency of notation, throughout this manuscript we will follow the con-ventional notion that derivatives annotated with superscripts are understood aswith respect to x and derivatives with respect to ξ will be abbreviated by . = d/dξ .We also employ the notation N = N ∪ { } .2. Background on Self-Adjoint Regular Sturm–Liouville Operators
In the first part of this section we briefly recall basic facts on regular Sturm–Liouville operators and their self-adjoint boundary conditions. This material isstandard and well-known, hence we just refer to some of the standard monographson this subject, such as, [9, Sect. 6.3], [34, Ch. 3], [41, Sect. II.5], [61, Ch. V], [76,Sect. 8.4], [77, Sect. 13.2], [78, Ch. 4]. In the second part we discuss Fredholmdeterminants, traces of resolvents, and spectral ζ -functions associated with theseregular Sturm–Liouville problems. For background as well as relevant material inthis context we refer to [3], [7], [12], [13], [17], [19], [20], [25], [26], [27], [28], [29],[31], [33], [38], [39], [42], [49], [50], [51], [53], [57], [58], [59], [62], [64], [71], [72,Sects. 5.4, 5.5, 6.3], [75].Throughout our discussion of regular Sturm–Liouville operators we make thefollowing assumptions: Hypothesis 2.1.
Let ( a, b ) ⊂ R be a finite interval and suppose that p, q, r are ( Lebesgue ) measurable functions on ( a, b ) such that the following items ( i ) – ( iii ) hold: ( i ) r > a.e. on ( a, b ) , r ∈ L (( a, b ); dx ) . ( ii ) p > a.e. on ( a, b ) , /p ∈ L (( a, b ); dx ) . ( iii ) q is real-valued a.e. on ( a, b ) , q ∈ L (( a, b ); dx ) . Given Hypothesis 2.1, we now study Sturm–Liouville operators associated withthe general, three-coefficient differential expression τ of the type, τ = 1 r ( x ) (cid:20) − ddx p ( x ) ddx + q ( x ) (cid:21) for a.e. x ∈ ( a, b ) ⊆ R . (2.1)We start with the notion of minimal and maximal L (( a, b ); rdx )-realizationsassociated with the regular differential expression τ on the finite interval ( a, b ) ⊂ R .Here, and elsewhere throughout this manuscript, the inner product in L (( a, b ); rdx ) G. FUCCI, F. GESZTESY, K. KIRSTEN, AND J. STANFILL is defined by( f, g ) L (( a,b ); rdx ) = ˆ ba r ( x ) dx f ( x ) g ( x ) , f, g ∈ L (( a, b ); rdx ) . (2.2)Assuming Hypothesis 2.1, the differential expression τ of the form (2.1) on thefinite interval ( a, b ) ⊂ R is called regular on [ a, b ]. The corresponding maximaloperator T max in L (( a, b ); rdx ) associated with τ is defined by T max f = τ f,f ∈ dom( T max ) = (cid:8) g ∈ L (( a, b ); rdx ) (cid:12)(cid:12) g, g [1] ∈ AC ([ a, b ]); (2.3) τ g ∈ L (( a, b ); rdx ) (cid:9) , and the corresponding minimal operator T min in L (( a, b ); rdx ) associated with τ is given by T min f = τ f,f ∈ dom( T min ) = (cid:8) g ∈ L (( a, b ); rdx ) (cid:12)(cid:12) g, g [1] ∈ AC ([ a, b ]); (2.4) g ( a ) = g [1] ( a ) = g ( b ) = g [1] ( b ) = 0; τ g ∈ L (( a, b ); rdx ) (cid:9) . Here (with ′ := d/dx ) y [1] ( x ) = p ( x ) y ′ ( x ) , (2.5)denotes the first quasi-derivative of a function y on ( a, b ), assuming that y, py ′ ∈ AC loc (( a, b )).Assuming Hypothesis 2.1 so that τ is regular on [ a, b ], the following is well-known(see, e.g., [9, Sect. 6.3], [34, Sect. 3.2], [41, Sect. II.5], [61, Ch. V], [76, Sect. 8.4], [77,Sect. 13.2], [78, Ch. 4]): T min is a densely defined, closed operator in L (( a, b ); rdx ),moreover, T max is densely defined and closed in L (( a, b ); rdx ), and T ∗ min = T max , T min = T ∗ max . (2.6)Moreover, T min ⊂ T max = T ∗ min , and hence T min is symmetric, while T max is not.The next theorem describes all self-adjoint extensions of T min (cf., e.g., [77,Sect. 13.2], [78, Ch. 4]). Theorem 2.2.
Assume Hypothesis so that τ is regular on [ a, b ] . Then thefollowing items ( i ) – ( iii ) hold: ( i ) All self-adjoint extensions T α,β of T min with separated boundary conditions areof the form T α,β f = τ f, α, β ∈ [0 , π ) ,f ∈ dom( T α,β ) = (cid:8) g ∈ dom( T max ) (cid:12)(cid:12) g ( a ) cos( α ) + g [1] ( a ) sin( α ) = 0; (2.7) g ( b ) cos( β ) − g [1] ( b ) sin( β ) = 0 (cid:9) . Special cases: α = 0 ( i.e., g ( a ) = 0) is called the Dirichlet boundary condition at a ; α = π , ( i.e., g [1] ( a ) = 0) is called the Neumann boundary condition at a ( analogousfacts hold at the endpoint b ) . ( ii ) All self-adjoint extensions T ϕ,R of T min with coupled boundary conditions are PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 5 of the type T ϕ,R f = τ f,f ∈ dom( T ϕ,R ) = (cid:26) g ∈ dom( T max ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) g ( b ) g [1] ( b ) (cid:19) = e iϕ R (cid:18) g ( a ) g [1] ( a ) (cid:19) (cid:27) , (2.8) where ϕ ∈ [0 , π ) , and R is a real × matrix with det( R ) = 1 ( i.e., R ∈ SL (2 , R )) .Special cases: ϕ = 0 , R = I ( i.e., g ( b ) = g ( a ) , g [1] ( b ) = g [1] ( a )) are called periodicboundary conditions; similarly, ϕ = π , R = I ( i.e., g ( b ) = − g ( a ) , g [1] ( b ) = − g [1] ( a )) are called antiperiodic boundary conditions. ( iii ) Every self-adjoint extension of T min is either of type ( i ) ( i.e., separated ) or oftype ( ii ) ( i.e., coupled ) . Next we state some of the most pertinent concepts and results summarized from[33] (in particular, Section 3) and will then illustrate how this permits one toeffectively calculate certain values for the spectral ζ -functions of the regular Sturm–Liouville operators considered.For this purpose we introduce the fundamental system of solutions θ ( z, x, a ), φ ( z, x, a ) of τ y = zy defined by θ ( z, a, a ) = φ [1] ( z, a, a ) = 1 , θ [1] ( z, a, a ) = φ ( z, a, a ) = 0 , (2.9)such that W ( θ ( z, · , a ) , φ ( z, · , a )) = 1 , (2.10)noting that for fixed x, each is entire with respect to z . Here the Wronskian of f and g , for f, g ∈ AC loc (( a, b )), is defined by W ( f, g )( x ) = f ( x ) g [1] ( x ) − f [1] ( x ) g ( x ) . (2.11)Furthermore, we introduce the boundary values for g, g [1] ∈ AC ([ a, b ]), see [60,Ch. I], [78, Sect. 3.2], U α,β, ( g ) = g ( a ) cos( α ) + g [1] ( a ) sin( α ) ,U α,β, ( g ) = g ( b ) cos( β ) − g [1] ( b ) sin( β ) , (2.12)in the case ( i ) of separated boundary conditions in Theorem 2.2, and V ϕ,R, ( g ) = g ( b ) − e iϕ R g ( a ) − e iϕ R g [1] ( a ) ,V ϕ,R, ( g ) = g [1] ( b ) − e iϕ R g ( a ) − e iϕ R g [1] ( a ) , (2.13)in the case ( ii ) of coupled boundary conditions in Theorem 2.2. Moreover, we definethe characteristic functions F α,β ( z ) = det (cid:18) U α,β, ( θ ( z, · , a )) U α,β, ( φ ( z, · , a )) U α,β, ( θ ( z, · , a )) U α,β, ( φ ( z, · , a )) (cid:19) , (2.14)and F ϕ,R ( z ) = det (cid:18) V ϕ,R, ( θ ( z, · , a )) V ϕ,R, ( φ ( z, · , a )) V ϕ,R, ( θ ( z, · , a )) V ϕ,R, ( φ ( z, · , a )) (cid:19) . (2.15) Notational Convention.
To describe all possible self-adjoint boundary conditionsassociated with self-adjoint extensions of T min effectively, we will frequently employthe notation T A,B , F A,B , λ A,B,j , j ∈ J , etc., where A, B represents α, β in thecase of separated boundary conditions and ϕ, R in the context of coupled boundaryconditions.
G. FUCCI, F. GESZTESY, K. KIRSTEN, AND J. STANFILL
By construction, eigenvalues of T A,B are determined via F A,B ( z ) = 0, withmultiplicity of eigenvalues of T A,B corresponding to multiplicity of zeros of F A,B ,and F A,B ( z ) is entire with respect to z . In particular, for T α,β , that is, for separatedboundary conditions, one has F α,β ( z ) = cos( α )[ − sin( β ) φ [1] ( z, b, a ) + cos( β ) φ ( z, b, a )] − sin( α )[ − sin( β ) θ [1] ( z, b, a ) + cos( β ) θ ( z, b, a )] , α, β ∈ [0 , π ) , (2.16)and for T ϕ,R , that is, for coupled boundary conditions, one has for ϕ ∈ [0 , π ) and R ∈ SL (2 , R ), F ϕ,R ( z ) = e iϕ (cid:0) R θ [1] ( z, b, a ) − R θ ( z, b, a ) + R φ ( z, b, a ) − R φ [1] ( z, b, a ) (cid:1) + e iϕ + 1 . (2.17)Next we will demonstrate that F A,B ( · ) is an entire function of order 1 / ζ -function for large values of the spectral parameter z .For this purpose we recall the following facts (see, e.g., [10, Ch. 2], [52, Ch. I]):Supposing that F ( · ) is entire, one introduces M F ( R ) = sup | z | = R | F ( z ) | , R ∈ [0 , ∞ ) . (2.18)Then the order ρ F of F is defined by ρ F = lim sup R →∞ ln(ln( M F ( R ))) / ln( R ) ∈ [0 , ∞ ) ∪ {∞} . (2.19)In addition, if ρ F >
0, the type τ F of F is defined as τ F = lim sup R →∞ ln( M F ( R )) /R ρ F ∈ [0 , ∞ ) ∪ {∞} , (2.20)and, in obvious notation, F is called of order ρ F > τ F if τ F ∈ [0 , ∞ ).Thus, F is of finite order ρ F ∈ [0 , ∞ ) if and only if for every ε >
0, but for no ε < M F ( R ) = R →∞ O (cid:0) exp (cid:0) R ρ F + ε (cid:1)(cid:1) , (2.21)and F is of positive and finite order ρ F ∈ (0 , ∞ ) and finite type τ F ∈ [0 , ∞ ) if andonly if for every ε >
0, but for no ε < M F ( R ) = R →∞ O (cid:0) exp (cid:0) ( τ F + ε ) R ρ F (cid:1)(cid:1) . (2.22)By definition, if F j are entire of order ρ j , j = 1 ,
2, then the order of F F doesnot exceed the larger of ρ and ρ .For F entire we also introduce the zero counting function N F ( R ) = (cid:0) Z F ∩ D (0; R ) (cid:1) , R ∈ (0 , ∞ ) , (2.23)where Z F represents the set of zeros of F countingmultiplicity (i.e., N F ( R ) counts the number of zeros of F in the closed disk ofradius R >
PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 7 Remark . Assuming Hypothesis 2.1, then all solutions ψ ( z, · ) of the regularSturm–Liouville problem ( τ y )( z, x ) = zy ( z, x ), z ∈ C , x ∈ [ a, b ], satisfying z -independent initial conditions ψ ( z, x ) = c , ψ [1] ( z, x ) = c , (2.24)for some x ∈ [ a, b ] and some ( c , c ) ∈ C , together with ψ [1] ( z, · ), for any fixed x ∈ [ a, b ], are entire functions of z of order at most 1 /
2. Indeed, as shown in[5, Sect. 8.2] (see also [56], [78, Theorem 2.5.3]), upon employing a Pr¨ufer-typetransformation, one obtains | z || ψ ( z, x ) | + (cid:12)(cid:12) ψ [1] ( z, x ) (cid:12)(cid:12) C ( x ) exp (cid:18) | z | / ˆ max( x ,x )min( x ,x ) dt (cid:2) | p ( t ) | − + | r ( t ) | (cid:3) + | z | − / ˆ max( x ,x )min( x ,x ) dt | q ( t ) | (cid:19) , z ∈ C , x , x ∈ [ a, b ] . (2.25)In particular, (2.16) and (2.17) yield that F A,B is an entire function of order atmost 1 / A, B , that is, ρ F A,B / . (2.26)Given Hypothesis 2.1, one infers that T A,B > Λ A,B I L r (( a,b )) for some Λ A,B ∈ R ,with purely discrete spectrum, and hence Z F A,B ( R ) ⊂ [Λ A,B , R ] the elements of Z F A,B ( R ) being precisely the eigenvalues of T A,B in the interval [max( − R, Λ A,B ) , R ].Employing the theory of Volterra operators in Hilbert spaces (and under some ad-ditional lower boundedness hypotheses on q ) in [36, Chs. VI, VII], alternatively,using oscillation theoretic methods in [6], it is shown that the eigenvalue countingfunction N F A,B associated with T A,B satisfies N F A,B ( λ ) = λ →∞ π − ˆ ba dx [ r ( x ) /p ( x )] / λ / [1 + o (1)] . (2.27)Ignoring finitely many nonpositive eigenvalues of T A,B , equivalently, splitting offthe factors in the infinite product representation associated with nonpositive zerosof F A,B , that is, replacing F A,B by e F A,B ( z ) = C A,B Y j ∈ N ,λ A,B,j > [1 − ( z/λ A,B,j )] (2.28)with N e F A,B ( λ ) = λ →∞ π − ˆ ba dx [ r ( x ) /p ( x )] / λ / [1 + o (1)] , (2.29)implies (cf. [10, Theorem 4.1.1], [73], [74]),ln (cid:0) e F A,B ( λ ) (cid:1) = λ →∞ ˆ ba dx [ r ( x ) /p ( x )] / λ / [1 + o (1)] . (2.30)Thus, ρ F A,B = ρ e F A,B > / , (2.31)and hence by (2.26), ρ F A,B = 1 / . (2.32) Upon closer inspection, the additional condition stated on [36, p. 305, 306] just ensures lowersemiboundedness of T A,B , which is independently known to hold in our present scalar context.
G. FUCCI, F. GESZTESY, K. KIRSTEN, AND J. STANFILL
Moreover, by (2.25), F A,B is of order 1/2 and finite type. Finally, we also mentionthat (2.27) implies that λ A,B,j = j →∞ (cid:20) ˆ ba dx [ r ( x ) /p ( x )] / (cid:21) − π j [1 + o (1)] (2.33)(cf. also the discussion in [65, Sects. 1.11, 9.1], [78, Sect. 4.3]). ⋄ The following theorem (see [33, Thm. 3.4]) directly relates the function F A,B toFredholm determinants and traces (see [35, Ch. IV], [63, Sect. XIII.17], [68], [69,Ch. 3], [70, Ch. 3] for background).
Theorem 2.4.
Assume Hypothesis and denote by T α,β and T ϕ,R the self-adjointextensions of T min as described in cases ( i ) and ( ii ) of Theorem , respectively. ( i ) Suppose z ∈ ρ ( T α,β ) , then det L r (( a,b )) (cid:0) I L r (( a,b )) − ( z − z )( T α,β − z I L r (( a,b )) ) − (cid:1) = F α,β ( z ) /F α,β ( z ) , z ∈ C . (2.34) In particular, tr L r (( a,b )) (cid:0) ( T α,β − zI L r (( a,b )) ) − (cid:1) = − ( d/dz )ln( F α,β ( z )) , z ∈ ρ ( T α,β ) . (2.35)( ii ) Suppose z ∈ ρ ( T ϕ,R ) , then det L r (( a,b )) (cid:0) I L r (( a,b )) − ( z − z )( T ϕ,R − z I L r (( a,b )) ) − (cid:1) = F ϕ,R ( z ) /F ϕ,R ( z ) , z ∈ C . (2.36) In particular, tr L r (( a,b )) (cid:0) ( T ϕ,R − zI L r (( a,b )) ) − (cid:1) = − ( d/dz )ln( F ϕ,R ( z )) , z ∈ ρ ( T ϕ,R ) . (2.37)Given these preparations, we let T A,B denote the self-adjoint extension of T min with either separated ( T α,β ) or coupled ( T ϕ,R ) boundary conditions as described incases ( i ) and ( ii ) of Theorem 2.2. One recalls (see, e.g., [33]), the spectral ζ -functionof the operator, T A,B , is defined as ζ ( s ; T A,B ) := X j ∈ J λj =0 λ − sA,B,j , (2.38)with J ⊂ Z an appropriate index set counting eigenvalues according to their mul-tiplicity and Re( s ) > s ) > ζ ( s ; T A,B ) = 12 πi ‰ γ dz z − s (cid:18) ddz ln( F A,B ( z )) − z − m (0; T A,B ) (cid:19) = 12 πi ‰ γ dz z − s (cid:18) ddz ln( F A,B ( z )) − z − m (cid:19) , (2.39)where m (0; T A,B ) = m is the multiplicity of zero as an eigenvalue of T A,B and γ isa simple contour enclosing σ ( T A,B ) \{ } in a counterclockwise manner so as to dipunder (and hence avoid) the point 0 (cf. Figure 1). Here, following [47] (see also[48]), we take R ψ = { z = te iψ : t ∈ [0 , ∞ ) } , ψ ∈ ( π/ , π ) , (2.40) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 9 to be the branch cut of z − s , and, once again, eigenvalues will be determined via F A,B ( z ) = 0, with the multiplicity of eigenvalues of T A,B corresponding to themultiplicity of zeros of F A,B . ✲✻ ✛✡ ✠☞✛✚ ✛❆❆❆❆❆❆❆❆❆ The cut R ψ for z − s s s s s s s s s s ss s s z -plane γ Figure 1.
Contour γ in the complex z -plane. ✲✻ ✝ ✆ ❆❆❆❆❆❆ ❑ ❆❆❆❆❆❆❯ ❆❆❆❆❆❆❆❆ The cut R ψ for z − s r r r r r r r r r rr r r z -plane γ Figure 2.
Deforming γ . ✲✻ ❧ ✠ r r r r r r r r r rr r r z -plane C ε Figure 3.
Contour C ε .To continue the computation of (2.39) and deform the contour γ as to “hug”the branch cut R ψ (cf. Figure 2) requires knowledge of the asymptotic behavior of F A,B ( z ) as | z | → ∞ , which in turn demands Re( s ) > / z convergence(cf. Remark 2.3). Furthermore, if one is interested in the calculation of the value ofthe spectral zeta function at positive integers, the following method provides a verysimple way of obtaining those values. In fact, by letting s = n , n ∈ N , in (2.39),one no longer needs a branch cut for the fractional powers of z − s given in Figures 1and 2. This reduces the integral along the curve γ to a clockwise oriented integralalong the circle C ε , centered at zero with radius ε > s = n also ensures that m (the multiplicity of zero as an eigenvalue of T A,B ) does notcontribute to the integral in (2.39). Hence, ζ ( n ; T A,B ) = − πi ‰ C ε dz z − n ddz ln( F A,B ( z ))= − Res (cid:20) z − n ddz ln( F A,B ( z )); z = 0 (cid:21) , n ∈ N . (2.41) Thus, determining an expansion of F A,B ( z ) about z = 0 enables one to effectivelycompute ζ ( n ; T A,B ). In addition, by (2.16), (2.17), F A,B ( z ) is a linear combinationof θ , θ [1] , φ , and φ [1] for each boundary condition considered, so it suffices to findthe expansion of each of these functions individually.3. Expansion in z for Fundamental Solutions, Asymptotic Expansion,and the Zeta Regularized Functional Determinant Expansion in z for Fundamental Solutions. Assuming Hypothesis 2.1 throughout this section, we discuss next the expansionin z about z = 0 for the solutions φ ( z, · , a ) and θ ( z, · , a ) of τ y = zy , φ ( z, x, a ) = φ (0 , x, a ) + z ˆ xa r ( x ′ ) dx ′ g (0 , x, x ′ ) φ ( z, x ′ , a ) , (3.1) θ ( z, x, a ) = θ (0 , x, a ) + z ˆ xa r ( x ′ ) dx ′ g (0 , x, x ′ ) θ ( z, x ′ , a ) , (3.2) z ∈ C , x ∈ [ a, b ] , employing the following expression for the Volterra Green’s function g (0 , x, x ′ ) = θ (0 , x, a ) φ (0 , x ′ , a ) − θ (0 , x ′ , a ) φ (0 , x, a ) , x, x ′ ∈ [ a, b ] . (3.3)That (3.1) and (3.2) indeed represent solutions of τ y = zy is clear from applying τ to either side, moreover, the initial conditions (2.9) are readily verified.Iterating these integral equations establishes the power series expansions φ ( z, x, a ) = ∞ X m =0 z m φ m ( x ) , z ∈ C , x ∈ [ a, b ] , (3.4)where φ ( x ) = φ (0 , x, a ) ,φ ( x ) = ˆ xa r ( x ) dx g (0 , x, x ) φ (0 , x , a ) ,φ k ( x ) = ˆ xa r ( x ) dx g (0 , x, x ) ˆ x a r ( x ) dx g (0 , x , x ) . . .. . . ˆ x k − a r ( x k ) dx k g (0 , x k − , x k ) φ (0 , x k , a ) , k ∈ N , (3.5)and θ ( z, x, a ) = ∞ X m =0 z m θ m ( x ) , z ∈ C , x ∈ [ a, b ] , (3.6)where θ ( x ) = θ (0 , x, a ) ,θ ( x ) = ˆ xa r ( x ) dx g (0 , x, x ) θ (0 , x , a ) ,θ k ( x ) = ˆ xa r ( x ) dx g (0 , x, x ) ˆ x a r ( x ) dx g (0 , x , x ) . . .. . . ˆ x k − a r ( x k ) dx k g (0 , x k − , x k ) θ (0 , x k , a ) , k ∈ N . (3.7) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 11 Analogously one obtains φ [1] ( z, x, a ) = ∞ X m =0 z m φ [1] m ( x ) , z ∈ C , x ∈ [ a, b ] , (3.8)where φ [1]0 ( x ) = φ [1] (0 , x, a ) ,φ [1]1 ( x ) = ˆ xa r ( x ) dx g [1] (0 , x, x ) φ (0 , x , a ) ,φ [1] k ( x ) = ˆ xa r ( x ) dx g [1] (0 , x, x ) ˆ x a r ( x ) dx g (0 , x , x ) . . .. . . ˆ x k − a r ( x k ) dx k g (0 , x k − , x k ) φ (0 , x k , a ) , k ∈ N , (3.9)using the abbreviation g [1] (0 , x, x ) = θ [1] (0 , x, a ) φ (0 , x , a ) − θ (0 , x , a ) φ [1] (0 , x, a ) . (3.10)Similarly, one finds from (3.6) θ [1] ( z, x, a ) = ∞ X m =0 z m θ [1] m ( x ) , z ∈ C , x ∈ [ a, b ] , (3.11)where θ [1]0 ( x ) = θ [1] (0 , x, a ) ,θ [1]1 ( x ) = ˆ xa r ( x ) dx g [1] (0 , x, x ) θ (0 , x , a ) ,θ [1] k ( x ) = ˆ xa r ( x ) dx g [1] (0 , x, x ) ˆ x a r ( x ) dx g (0 , x , x ) . . .. . . ˆ x k − a r ( x k ) dx k g (0 , x k − , x k ) θ (0 , x k , a ) , k ∈ N . (3.12)3.2. Asymptotic Expansion of the Characteristic Function.
Next we investigate the | z | → ∞ asymptotic expansion of the function F A,B ( z )in order to provide an analytic continuation of the spectral ζ -function, ζ ( s ; T A,B ),and compute the zeta regularized functional determinant. We first strengthen Hy-pothesis 2.1 by introducing the following assumptions on p, q, r following [33, Sect.3]. These additional assumptions are necessary in order to perform a Liouville-typetransformation.
Hypothesis 3.1.
Let ( a, b ) ⊂ R be a finite interval and suppose that p, q, r are ( Lebesgue ) measurable functions on ( a, b ) such that the following items ( i ) – ( iv ) hold: ( i ) r > a.e. on ( a, b ) , r ∈ L (( a, b ); dx ) , /r ∈ L ∞ (( a, b ); dx ) . ( ii ) p > a.e. on ( a, b ) , /p ∈ L (( a, b ); dx ) . ( iii ) q is real-valued a.e. on ( a, b ) , q ∈ L (( a, b ); dx ) . ( iv ) pr and ( pr ) ′ /r are absolutely continuous on [ a, b ] , and for some ε > , pr > ε on [ a, b ] . The variable transformations (cf. [54, p. 2]), ξ ( x ) = 1 c ˆ xa dt [ r ( t ) /p ( t )] / , ξ ( x ) ∈ [0 ,
1] for x ∈ [ a, b ] , (3.13) ξ ′ ( x ) = c − [ r ( x ) /p ( x )] / > a, b ), (3.14) u ( z, ξ ) = [ p ( x ( ξ )) r ( x ( ξ ))] / y ( z, x ( ξ )) , (3.15)with c > c = ˆ ba dt [ r ( t ) /p ( t )] / , (3.16)transform the Sturm–Liouville problem ( τ y ( z, · ))( x ) = zy ( z, x ), x ∈ ( a, b ), into − .. u ( z, ξ ) + V ( ξ ) u ( z, ξ ) = c zu ( z, ξ ) , ξ ∈ (0 , , (3.17)and abbreviating ν ( ξ ) = [ p ( x ( ξ )) r ( x ( ξ ))] / , (3.18)one verifies that V ( ξ ) = .. ν ( ξ ) ν ( ξ ) + c q ( x ) r ( x )= − c
16 1 p ( x ) r ( x ) (cid:20) ( p ( x ) r ( x )) ′ r ( x ) (cid:21) + c r ( x ) (cid:20) ( p ( x ) r ( x )) ′ r ( x ) (cid:21) ′ + c q ( x ) r ( x ) , (3.19)and V ∈ L ((0 , dξ ) , (3.20)as guaranteed by Hypothesis 3.1.In order to construct the asymptotic expansion of F A,B ( z ) we begin by assum-ing Hypothesis 3.1, but note that throughout the construction of the expansionstronger assumptions will be necessary, all of which will be addressed once the finalasymptotic expansion is given.When applying the Liouville transformation the boundary conditions undergo asimilar transformation. In fact, setting Q ( ξ ) = [( pr ) ′ /r ]( x ( ξ )) (3.21)one can write (cid:18) u ( z, ξ ) . u ( z, ξ ) (cid:19) = M ( ξ ) (cid:18) y ( z, x ( ξ )) y [1] ( z, x ( ξ )) (cid:19) , (3.22)where M ( ξ ) = (cid:18) ν ( ξ ) 0( c/ ν ( ξ ) − Q ( ξ ) cν ( ξ ) − (cid:19) , ξ ∈ [0 , , det C ( M ( · )) = c. (3.23)Employing relation (3.22), the separated boundary conditions for the function g ( · ) in Theorem 2.2 ( i ) transform into separated boundary conditions for the trans-formed function v ( · ) as follows, (cid:18) cos( α ) sin( α )0 0 (cid:19) M (0) − (cid:18) v (0) . v (0) (cid:19) + (cid:18) β ) − sin( β ) (cid:19) M (1) − (cid:18) v (1) . v (1) (cid:19) , (3.24)where α, β ∈ [0 , π ), and the inverse matrix M − ( · ) has the form M ( ξ ) − = (cid:18) ν ( ξ ) − − (1 / ν ( ξ ) − Q ( ξ ) c − ν ( ξ ) (cid:19) , ξ ∈ [0 , , (3.25) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 13 or, more explicitly, c − ν (0) sin( α ) . v (0) + ν (0) − (cid:2) cos( α ) − − sin( α ) Q (0) (cid:3) v (0) = 0 , − c − ν (1) sin( β ) . v (1) + ν (1) − (cid:2) cos( β ) + 4 − sin( β ) Q (1) (cid:3) v (1) = 0 . (3.26)With the help of relation (3.22) the coupled boundary conditions for g ( · ) inTheorem 2.2 ( ii ) transform into coupled boundary conditions for v ( · ) via (cid:18) v (1) . v (1) (cid:19) = e iϕ e R (cid:18) v (0) . v (0) (cid:19) , ϕ ∈ [0 , π ) , (3.27)where e R = M (1) − RM (0) ∈ SL (2 , R ) (3.28)is of the form e R = ν (0) − ν (1) (cid:2) R − − Q (0) R (cid:3) , e R = c − ν (0) ν (1) R , e R = cν (0) − ν (1) − (cid:2) R − − Q (0) R + 4 − Q (1) R − (16) − Q (0) Q (1) R (cid:3) , e R = ν (0) ν (1) − (cid:2) R + 4 − Q (1) R (cid:3) . (3.29)The fundamental system of solutions φ ( z, · , a ) and θ ( z, · , a ) of τ y = zy satisfying(2.9) is transformed into the set of solutions Φ( z, · ,
0) and Θ( z, · ,
0) of (3.17)satisfying the conditionsΦ( z, ,
0) = 0 , . Φ( z, ,
0) = cν (0) − , (3.30)Θ( z, ,
0) = ν (0) , . Θ( z, ,
0) = 4 − cν (0) − Q (0) , (3.31)where, once again, the derivatives of Φ( z, ξ,
0) and Θ( z, ξ,
0) are understood withrespect to the variable ξ (cf. (3.17)) and one notes that for fixed ξ , each is entirewith respect to z . By writing a generic solution of (3.17) as a linear combinationof Φ( z, ξ,
0) and Θ( z, ξ,
0) and by imposing the separated boundary conditions in(3.26) one obtains the following characteristic function F α,β ( z ) = sin( α ) (cid:8) c − ν (1) sin( β ) . Θ( z, , − ν (1) − (cid:2) cos( β ) + 4 − sin( β ) Q (1) (cid:3) Θ( z, , (cid:9) + cos( α ) (cid:8) − c − ν (1) sin( β ) . Φ( z, , ν (1) − (cid:2) cos( β ) + 4 − sin( β ) Q (1) (cid:3) Φ( z, , (cid:9) , z ∈ C . (3.32)The zeros of F α,β ( z ) represent, including multiplicity, the eigenvalues λ A,B,j , j ∈ J , of the original Sturm–Liouville problem τ y = zy endowed with the separatedboundary conditions in (2.7). By repeating this argument for coupled boundaryconditions (2.8) one obtains the characteristic function F ϕ, e R ( z ) = e iϕ (cid:8) ϕ ) − (cid:2) c − ν (0) e R + 4 − ν (0) − Q (0) e R (cid:3) . Φ( z, , (cid:2) c − ν (0) e R + 4 − ν (0) − Q (0) e R (cid:3) Φ( z, , e R ν (0) − . Θ( z, , − e R ν (0) − Θ( z, , (cid:9) , z ∈ C . (3.33) Remark . Explicit computations confirm that in the case of separated as well ascoupled boundary conditions one finds F α,β ( z ) = F α,β ( z ) , z ∈ C , (3.34) F ϕ,R ( z ) = F ϕ, e R ( z ) , z ∈ C . (3.35) ⋄ As an example we now consider the case of the Krein–von Neumann extension(see, e.g., [30] and the literature cited therein for details):
Example 3.3.
The Krein–von Neumann boundary conditions in terms of the vari-able x ∈ [ a, b ] are characterized by imposing the coupled boundary conditions ϕ = 0 , R = R K ( cf., e.g., [33, eq. (3.35)]) with R K = (cid:18) θ (0 , b, a ) φ (0 , b, a ) θ [1] (0 , b, a ) φ [1] (0 , b, a ) (cid:19) . (3.36) In terms of the variable ξ ∈ [0 , , these boundary conditions are transformed into ϕ = 0 and e R = e R K with e R K = ν (0) − (cid:2) Θ(0 , , − − Q (0)Φ(0 , , (cid:3) c − ν (0)Φ(0 , , ν (0) − (cid:2) . Θ(0 , , − − Q (0) . Φ(0 , , (cid:3) c − ν (0) . Φ(0 , , ! . (3.37) By using these parameters in (3.33) one obtains the transformed characteristic func-tion F , e R K ( z ) = 2 − c − (cid:2) . Φ(0 , , z, ,
0) + Θ(0 , , . Φ( z, , − Φ(0 , , . Θ( z, , − . Θ(0 , , z, , (cid:3) , z ∈ C , (3.38) to be compared with ( see [33, eq. (3.36), (3.37)]) F ,R K ( z ) = 2 − (cid:2) φ [1] (0 , b, a ) θ ( z, b, a ) + θ (0 , b, a ) φ [1] ( z, b, a ) − φ (0 , b, a ) θ [1] ( z, b, a ) − θ [1] (0 , b, a ) φ ( z, b, a ) (cid:3) , z ∈ C . (3.39)In order to obtain a large- z asymptotic expansion of the functions (3.32) and(3.33), we need the asymptotic expansion of the transformed fundamental set ofsolutions Φ( z, ξ,
0) and Θ( z, ξ, ξ ∈ [0 , V ( · ) is continued in a sufficiently smooth and compactly supportedmanner to a function on R (by a slight abuse of notation still abbreviated by V ), V ∈ C N ( R ) ∩ C ∞ (( −∞ , − ∪ (2 , ∞ )) , (3.40)for N ∈ N to be determined later on. In addition, we consider the associatedWeyl–Titchmarsh (resp., Jost) solutions u ± ( z, · ) such that for all x ∈ R , u + ( z, · ) ∈ L ([ x , ∞ ); dξ ) , u − ( z, · ) ∈ L (( −∞ , x ]; dξ ) , Im (cid:0) z / (cid:1) > . (3.41)Writing u ± ( z, ξ ) = exp (cid:26) ˆ ξ dt S ± ( z, t ) (cid:27) , S ± ( z, ξ ) = . u ± ( z, ξ ) u ± ( z, ξ ) , ξ ∈ R , Im (cid:0) z / (cid:1) > V on R , more generally, a suitable short-range,i.e., integrability assumption on V , permits the continuous extension of S ± ( z, · ) toIm (cid:0) z / (cid:1) > S ± ( z, · ) satisfy the Riccati differential equation . S ( z, ξ ) + S ± ( z, ξ ) − V ( ξ ) + c z = 0 , ξ ∈ R , Im (cid:0) z / (cid:1) > . (3.43) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 15 In addition, S ± ( z, ξ ) represent the half-line Weyl–Titchmarsh functions on [ ξ, + ∞ ),respectively, ( −∞ , ξ ], in particular, for each ξ ∈ R , ± S ± ( · , ξ ) are Nevanlinna–Herglotz functions on C + (i.e., analytic on C + with strictly positive imaginary parton C + ).Inserting the formal asymptotic expansion S ± ( z, · ) = | z |→∞ Im( z / ) > ± icz / + ∞ X j =1 ( ∓ j S j ( · ) z − j/ (3.44)into the Riccati equation (3.43) yields the recursion relation S ( ξ ) = [ i/ (2 c )] V ( ξ ) , S ( ξ ) = [1 / c ] . V ( ξ ) ,S j +1 ( ξ ) = − [ i/ (2 c )] (cid:20) . S j ( ξ ) + j − X k =1 S k ( ξ ) S j − k ( ξ ) (cid:21) , j ∈ N , ξ ∈ R . (3.45)The first few terms S j ( · ) explicitly read S ( ξ ) = (cid:2) i (cid:14)(cid:0) c (cid:1)(cid:3)(cid:2) V ( ξ ) − .. V ( ξ ) (cid:3) ,S ( ξ ) = − (cid:2) / c (cid:3)(cid:2) V (3) ( ξ ) − V ( ξ ) . V ( ξ ) (cid:3) ,S ( ξ ) = (cid:2) i (cid:14)(cid:0) c (cid:1)(cid:3)(cid:2) V ( ξ ) − . V ( ξ ) − V ( ξ ) .. V ( ξ ) + V (4) ( ξ ) (cid:3) , etc. (3.46)See [32, Sects. 5, 6] for a variety of closely related asymptotic expansions.Assuming (3.40), the formal asymptotic expansion (3.43) turns into an actualasymptotic expansion of the the type (see [11]), S ± ( z, ξ ) = | z |→∞ Im( z / ) > ± icz / + N X j =1 ( ∓ j S j ( ξ ) z − j/ + o (cid:0) | z | − N/ (cid:1) , (3.47)with the o (cid:0) | z | − N/ (cid:1) -term uniform with respect to ξ ∈ [0 , Remark . There is an enormous literature available in connection with asymp-totic high-energy expansions of Weyl–Titchmarsh m -functions (see, e.g., the de-tailed list in [14]) and the associated spectral function, however, much less can befound in connection with (local) uniformity of the error term o (cid:0) | z | − N/ (cid:1) with re-spect to x in expansions of the type (3.47). Notable exceptions are, for instance,[11], [16], [40], [43], [66], [67]. In particular, [11] (see [55, Sects. 1.4, 3.1]) and [16]use the theory of transformation operators, while [40] and [43] employ a detailedanalysis of the Riccati equation (3.43), and [66], [67] iterate an underlying Volterraintegral equation. In addition we note that the compact support hypothesis on V can be relaxed to the condition ˆ R (1 + | x | ) dx (cid:12)(cid:12) V ( ℓ ) ( x ) (cid:12)(cid:12) < ∞ , ℓ N. (3.48) ⋄ The correct asymptotic behavior as | z | → ∞ of any solution u ( z, · ) to (3.17) isgiven as a linear combination of u ± ( z, · ), u ( z, ξ ) = A ( z ) u + ( z, ξ ) + B ( z ) u − ( z, ξ ) , Im( z ) > , ξ ∈ [0 , , (3.49) and one notices that the solutions u ± ( z, · ) satisfy the initial conditions u ± ( z,
0) = 1 , . u ± ( z,
0) = S ± ( z, , Im( z ) > . (3.50)Since W ( u + ( z, · ) , u − ( z, · ))( ξ ) = 0, ξ ∈ [0 , S + ( z, − S − ( z, = 0 , Im( z ) > . (3.51)By imposing the initial conditions (3.30) and (3.31) on the function (3.49), oneobtains an expression for Φ( z, · ,
0) and Θ( z, · ,
0) suitable for an asymptotic expan-sion. For instance, in the case of Φ( z, ξ,
0) one obtainsΦ( z, ξ,
0) = cν (0) − S − ( z, − S + ( z,
0) exp (cid:18) ˆ ξ dη S − ( z, η ) (cid:19) × (cid:20) − exp (cid:18) ˆ ξ dη [ S + ( z, η ) − S − ( z, η )] (cid:19)(cid:21) . (3.52)Furthermore, for large values of z , with Im( z ) >
0, (3.47) impliesexp (cid:18) ˆ ξ dη [ S + ( z, η ) − S − ( z, η )] (cid:19) = | z |→∞ Im( z / ) > exp (cid:0) icz / ξ (cid:1) exp (cid:18) − N X n =1 z − n +(1 / ˆ ξ dη S n − ( η ) (cid:19) (3.53) × [1 + o (cid:0) z − N +1 / (cid:1) ] . Since the integrals on the right-hand side of (3.53) are finite, one findsexp (cid:18) − N X n =1 z − n +1 / ˆ ξ dη S n − ( η ) (cid:19) = | z |→∞ Im( z / ) > O (1) , (3.54)uniformly in ξ ∈ [0 , (cid:18) ˆ ξ dη [ S + ( z, η ) − S − ( z, η )] (cid:19) = | z |→∞ Im( z / ) > O (cid:0) e icz / (cid:1) , (3.55)uniformly for ξ ∈ [0 , z, ξ,
0) = | z |→∞ Im( z / ) > cν (0) − S − ( z, − S + ( z,
0) exp (cid:18) ˆ ξ dη S − ( z, η ) (cid:19)(cid:2) O (cid:0) e icz / (cid:1)(cid:3) . (3.56)Similar arguments permit one to derive the following expressions:Θ( z, ξ,
0) = | z |→∞ Im( z / ) > ( c/ ν (0) − Q (0) − ν (0) S + ( z, S − ( z, − S + ( z,
0) exp (cid:18) ˆ ξ dη S − ( z, η ) (cid:19) × (cid:2) O (cid:0) e icz / (cid:1)(cid:3) , (3.57) . Φ( z, ξ,
0) = | z |→∞ Im( z / ) > cν (0) − S − ( z, S − ( z, − S + ( z,
0) exp (cid:18) ˆ ξ dη S − ( z, η ) (cid:19)(cid:2) O (cid:0) e icz / (cid:1)(cid:3) , (3.58) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 17 . Θ( z, ξ,
0) = | z |→∞ Im( z / ) > (cid:2) ( c/ ν (0) − Q (0) − ν (0) S + ( z, (cid:3) S − ( z, S − ( z, − S + ( z, × exp (cid:18) ˆ ξ dη S − ( z, η ) (cid:19)(cid:2) O (cid:0) e icz / (cid:1)(cid:3) , (3.59)uniformly with respect to ξ ∈ [0 , F A,B ( z ) = | z |→∞ Im( z / ) > S − ( z, − S − ( z,
0) exp (cid:18) ˆ dη S − ( z, η ) (cid:19) (3.60) × [ j A,B + k A,B S + ( z,
0) + ℓ A,B S − ( z,
1) + m A,B S + ( z, S − ( z, × (cid:2) O (cid:0) e icz / (cid:1)(cid:3) . The first line on the right-hand side of (3.60) is entirely independent of boundaryconditions, in particular, it does not distinguish between separated and coupledboundary conditions. In contrast, the terms j A,B , k
A,B , ℓ
A,B , and m A,B in thesecond line on the right-hand side of (3.60) encode the specific information aboutthe boundary conditions imposed. In the case of separated boundary conditions,where
A, B represents α, β as in (2.7) one obtains j α,β = − cν (0) ν (1) [cos( β ) + (1 /
4) sin( β ) Q (1)] [cos( α ) − (1 /
4) sin( α ) Q (0)] ,k α,β = − ν (0) ν (1) sin( α ) [cos( β ) + (1 /
4) sin( β ) Q (1)] ,ℓ α,β = ν (1) ν (0) sin( β ) [cos( α ) − (1 /
4) sin( α ) Q (0)] ,m α,β = (1 /c ) ν (0) ν (1) sin( α ) sin( β ) . (3.61)In the case of coupled boundary conditions, where A, B represents ϕ, e R as in (3.27),(3.29), one infers j ϕ, e R = − e iϕ e R , k ϕ, e R = − e iϕ e R , ℓ ϕ, e R = e iϕ e R , m ϕ, e R = e iϕ e R . (3.62)For the purpose of the analytic continuation of the spectral ζ -function one needsthe large- z asymptotic expansion of ln( F A,B ( z )) rather then the one for F A,B ( z ).For this reason we will focus next on the derivation of the large- z asymptotic ex-pansion of the expressionln( F A,B ( z )) = | z |→∞ Im( z / ) > − ln (cid:0) S + ( z, − S − ( z, (cid:1) + ˆ dη S − ( z, η )+ ln (cid:0) j A,B + k A,B S + ( z,
0) + ℓ A,B S − ( z,
1) + m A,B S + ( z, S − ( z, (cid:1) (3.63)+ O (cid:0) e icz / (cid:1) . We can now use the expansion (3.43) in (3.60) to obtain a large- z asymptoticexpansion of (3.63). We start with the part of (3.63) that is independent of the boundary conditions. For the integral in (3.63) one finds ˆ dη S − ( z, η ) = | z |→∞ Im( z / ) > − iz / c + N X m =1 z − m/ ˆ dη S m ( η ) + o (cid:0) z − N/ (cid:1) . (3.64)For the first term in (3.63) one concludes that S + ( z, − S − ( z,
0) = | z |→∞ Im( z / ) > icz / (cid:18) i/c ) N X j =1 S j − (0) z − j (cid:19) + o (cid:0) z − N +1 / (cid:1) . (3.65)Relation (3.65) permits one to writeln (cid:0) S + ( z, − S − ( z, (cid:1) = | z |→∞ Im( z / ) > ln(2 ic ) + 2 − ln( z ) + N X m =1 D m − z − m + o (cid:0) z − N (cid:1) , (3.66)where the terms D m − are determined through the formal asymptotic expansionln (cid:18) i/c ) ∞ X m =1 S m − (0) z − m (cid:19) = ∞ X j =1 D j z − j . (3.67)We refer to (4.7)–(4.9) for a recursive formula for D j in terms of ( i/c ) S m − (0).The first few D j explicitly read D = − V (0) (cid:14)(cid:2) c (cid:3) , D = (cid:2) .. V (0) − V (0) (cid:3)(cid:14)(cid:2) c (cid:3) ,D = − (cid:2) V (4) (0) − V (0) .. V (0) − . V (0) + 16 V (0) (cid:3)(cid:14)(cid:2) c (cid:3) ,D = (cid:0) c (cid:1) − (cid:2) V (6) (0) + 48 V (0) .. V (0) − .. V (0) − V (0) V (4) (0)+ 60 V (0) . V (0) − V (3) (0) . V (0) − V (0) (cid:3) , etc. (3.68)Computing the asymptotic expansion of the last logarithmic term in (3.63),namely the term which depends on the boundary conditions, is somewhat moreinvolved. By using the asymptotic expansion (3.43) it is not difficult to find j A,B + k A,B S − ( z,
0) + ℓ A,B S + ( z, | z |→∞ Im( z / ) > − icz / ( ℓ A,B − k A,B ) + N X m =0 ∆ m z − m/ + o (cid:0) z − N/ (cid:1) , (3.69)where ∆ = j A,B , ∆ m = ℓ A,B S m (1) + ( − m k A,B S m (0) , m ∈ N , (3.70)and m A,B S − ( z, S + ( z,
1) = | z |→∞ Im( z / ) > m A,B c z (cid:18) N X m =2 Λ m z − m/ (cid:19) + o (cid:0) z − ( N − / (cid:1) , (3.71) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 19 where Λ m = m X ℓ =0 Ω − ℓ (0)Ω + m − ℓ (1) , m ∈ N , m > , (3.72)withΩ − (0) = Ω +0 (1) = 1 , Ω + j ( x ) = ( − j Ω − j ( x ) = ( i/c ) S j − ( x ) , j ∈ N . (3.73)The first few Λ m have the explicit form,Λ = − − c − [ V (1) + V (0)] , Λ = − i − c − (cid:2) . V (1) + . V (0) (cid:3) , Λ = 8 − c − (cid:2) .. V (1) + .. V (0) − V (0) − V (1) + 2 V (1) V (0) (cid:3) , Λ = i (16) − c − h V (3) (0) − V (3) (1) − V (0) (cid:0) . V (0) + . V (1) (cid:1) + 2 V (1) (cid:2) . V (0)+ 2 . V (1) (cid:3)i , etc. (3.74)This finally implies j A,B + k A,B S − ( z,
0) + ℓ A,B S + ( z,
1) + m A,B S − ( z, S + ( z, | z |→∞ Im( z / ) > N X m = − Γ m z − m/ + o (cid:0) z − N/ (cid:1) , (3.75)where Γ − = m A,B c , Γ − = − ic ( ℓ A,B − k A,B ) , Γ m = ∆ m + m A,B c Λ m +2 , m ∈ N . (3.76)Let Γ k with k ∈ Z and k > −
2, be the first non-vanishing term of the seriesin (3.75). Since Γ k = 0 one can writeln (cid:0) j A,B + k A,B S − ( z,
0) + ℓ A,B S + ( z,
1) + m A,B S − ( z, S + ( z, (cid:1) (3.77)= | z |→∞ Im( z / ) > ln(Γ k ) − ( k / z ) + ln (cid:18) N X m =1 [Γ m + k / Γ k ] z − m/ + o (cid:0) z − N/ (cid:1)(cid:19) , which, in turn, yieldsln (cid:0) j A,B + k A,B S − ( z,
0) + ℓ A,B S + ( z,
1) + m A,B S − ( z, S + ( z, (cid:1) = | z |→∞ Im( z / ) > ln(Γ k ) − ( k / z ) + N X j =1 Π j z − j/ + o (cid:0) z − N/ (cid:1) , (3.78)where the terms Π j are obtained via the formal asymptotic expansionln (cid:18) ∞ X m =1 [Γ m + k / Γ k ] z − m/ (cid:19) = ∞ X j =1 Π j z − j/ . (3.79)Once again we refer to (4.7)–(4.9) for a recursive determination of Π j in terms ofΓ m + k / Γ k . The first few Π m are explicitly of the form,Π = Γ k / Γ k , Π = 2 − Γ − k (cid:2) k Γ k +2 − Γ k +1 (cid:3) , Π = 3 − Γ − k (cid:2) Γ k +1 − k Γ k +2 Γ k +1 + 3Γ k Γ k +3 (cid:3) , (3.80) Π = − − Γ − k (cid:2) Γ k +1 − k Γ k +2 Γ k +1 + 4Γ k Γ k +3 Γ k +1 + 2Γ k (cid:0) Γ k +2 − k Γ k +4 (cid:1) (cid:3) , etc. (3.81)More explicit expressions for Π m in terms of the potential V and its derivativescan be obtained with a simple computer program once the index k has been de-termined.Finally, we can provide the large- z asymptotic expansion of the logarithm of thecharacteristic function in the formln( F A,B ( z )) = | z |→∞ Im( z / ) > − icz / − − ( k + 1)ln( z ) + ln(Γ k / (2 ic ))+ N X m =1 Ψ m z − m/ + o (cid:0) z − N/ (cid:1) , (3.82)where Ψ n = ˆ dη S n ( η ) − D n − + Π n , n ∈ N , Ψ n +1 = ˆ dη S n +1 ( η ) + Π n +1 , n ∈ N . (3.83)3.3. Analytic Continuation of the Spectral Zeta Function and the ZetaRegularized Functional Determinant.
In order to perform the analytic continuation of the spectral ζ -function, weneed to investigate the specific behavior for z ↓ | z | → ∞ . The characteris-tic function F A,B ( z ) is constructed as a linear combination of the basis functions φ ( z, · , a ) and θ ( z, · , a ) (or equivalently the transformed basis functions Φ( z, · , z, · , φ ( z, · , a ) and θ ( z, · , a ), and consequently Φ( z, · ,
0) and Θ( z, · , z asymptoticexpansion in the form of a power series in the variable z in Section 3.1. This im-plies that, in general, the characteristic function F A,B ( z ) has a small- z asymptoticexpansion of the form F A,B ( z ) = F m z m + ∞ X m = m +1 F m z m , (3.84)where m ∈ { , , } represents the multiplicity of the zero eigenvalue and F m = 0.The asymptotic expansion (3.84) suggests that the appropriate characteristic func-tion to use in the integral representation of the spectral ζ -function is z − m F A,B ( z )rather than simply F A,B ( z ) (obviously the two coincide when no zero eigenvalue ispresent). In this case it is easy to verify that ddz ln (cid:0) F A,B ( z ) z − m (cid:1) = | z |↓ O (1) . (3.85) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 21 From the large- z asymptotic expansion (3.82) of the characteristic function,namely,ln( F A,B ( z )) = | z |→∞ Im( z / ) > − icz / − [( k + 1) / z ) + ln(Γ k / (2 ic ))+ N X m =1 Ψ m z − m/ + o (cid:0) | z | − N/ (cid:1) , (3.86)one readily infers that ddz ln( F A,B ( z ) z − m ) = | z |→∞ Im( z / ) > O (cid:0) | z | − / (cid:1) . (3.87)The asymptotic behaviors in (3.85) and (3.87) justify deforming the contour γ inthe integral representation (2.39) to one that surrounds the branch cut R ψ as shownin Figure 2. This contour deformation leads to the following integral representation(with ψ introduced in (2.40)) ζ ( s ; T A,B ) = e is ( π − ψ ) π − sin( πs ) ˆ ∞ dt t − s ddt ln (cid:0) F A,B ( te iψ ) t − m e − im ψ (cid:1) , (3.88)which is valid in the region 1 / < Re( s ) <
1. To obtain the analytic continuation of(3.88) to the left of the abscissa of convergence Re( s ) = 1 / N terms of the large- z asymptotic expansion of ln (cid:0) F A,B ( te iψ ) t − m e − im ψ (cid:1) .This process leads to the following expression of the spectral ζ -function ζ ( s ; T A,B ) = Z ( s, A, B ) + N X j = − h j ( s, A, B ) , (3.89)which is valid in the region − ( N + 1) / < Re( s ) <
1. The explicit form of thefunctions in the analytically continued expression of ζ ( s ; T A,B ) in (3.89) is Z ( s, A, B ) = e is ( π − ψ ) π − sin( πs ) ˆ ∞ dt t − s ddt (cid:26) ln (cid:0) F A,B ( te iψ ) t − m e − im ψ (cid:1) − H ( t − (cid:20) − ict / e iψ/ − [(( k + 1) /
2) + m ]ln( t ) (3.90) − (cid:2) (( k + 1) /
2) + m (cid:3) iψ + ln(Γ k / (2 ic )) + N X n =1 Ψ n e − inψ/ t − n/ (cid:21)(cid:27) , where H ( s ) = ( , s > , , s < , represents the Heaviside function, and h − ( s, A, B ) = − ie is ( π − ψ ) π − sin( πs ) c e iψ/ / (2 s − ,h ( s, A, B ) = − ( k + 1 + 2 m ) e is ( π − ψ ) (2 πs ) − sin( πs ) ,h n ( s, A, B ) = − e is ( π − ψ ) π − sin( πs )[ n/ (2 s + n )] e − inψ/ Ψ n , n ∈ N . (3.91)Thanks to the expression (3.89) we are now able to compute the zeta regularizedfunctional determinant in terms of ζ ′ (0; T A,B ) as in [33, Thm. 2.9]. For the purposeof computing ζ ′ (0; T A,B ), it is sufficient to set N = 0 in (3.89) to obtain ζ ′ (0; T A,B ) = Z ′ (0 , A, B ) + h ′− (0 , A, B ) + h ′ (0 , A, B ) . (3.92) By computing the derivative with respect to s of (3.90) and the first two expres-sions in (3.91) at s = 0 one obtains the remarkably simple formula ζ ′ (0; T A,B ) = iπn − ln(2 c |F m / Γ k | ) , (3.93)where n is the number of strictly negative eigenvalues of T A,B .4.
Computing Spectral Zeta Function Values and Traces forRegular Sturm–Liouville Operators
We have now completed the necessary preparations to give the main theorem forcomputing values of the spectral ζ -function for self-adjoint regular Sturm–Liouvilleoperators when imposing either separated or coupled boundary conditions. Whenzero is not an eigenvalue we also find an expression for computing the trace of theinverse Sturm–Liouville operator. Theorem 4.1.
Assume Hypothesis , denote by T A,B the self-adjoint extension of T min with either separated or coupled boundary conditions as described in Theorem , and let m = 0 , , , denote the multiplicity of zero as an eigenvalue of T A,B ( with m = 0 denoting zero is not an eigenvalue ) . Suppose that F A,B ( z ) given in (2.39) has the series expansion F A,B ( z ) = ∞ X j =0 a j z j , | z | sufficiently small . (4.1) Then, ζ ( n ; T A,B ) = − Res (cid:20) z − n ddz ln( F A,B ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.2) where b = a m /a m ,b j = [ a j + m /a m ] − j − X ℓ =1 [ ℓ/j ][ a j − ℓ + m /a m ] b ℓ , j ∈ N , j > . (4.3) In particular, if zero is not an eigenvalue of T A,B , then tr L r (( a,b )) (cid:0) T − A,B (cid:1) = ζ (1; T A,B ) = − a /a . (4.4) Proof.
The residue in equation (4.2) coincides with the z − coefficient of the Laurentexpansion, in the neighborhood of z = 0, of the integrand in (2.41). By using theexpansion (4.1) one obtains, for | z | > n ∈ N , that z − n ddz ln( F A,B ( z )) = z − n ddz ln (cid:18) ∞ X j =0 a j z j (cid:19) . (4.5)Since z = 0 can be an eigenvalue of multiplicity at most 2, the expansion can berewritten as follows z − n ddz ln( F A,B ( z )) = z − n ddz ln (cid:18) ∞ X j = m a j z j (cid:19) = z − n ddz (cid:18) ln (cid:0) a m z m (cid:1) + ln (cid:18) ∞ X j =1 [ a j + m /a m ] z j (cid:19)(cid:19) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 23 = m z − n − + z − n ddz ln (cid:18) ∞ X j =1 [ a j + m /a m ] z j (cid:19) . (4.6)Since n ∈ N , the term m z − n − never contributes to the residue and the onlycontribution comes from the z n coefficient of the small- | z | asymptotic expansion ofthe logarithm on the right-hand side. This expansion can be obtained by makinguse of the fact that if F has the analytic expansion F ( z ) = ∞ X m =1 c m z m , | z | sufficiently small , (4.7)then ln(1 + F ( z )) = ∞ X m =1 d m z m , | z | sufficiently small , (4.8)where d = c , d j = c j − j − X ℓ =1 [ ℓ/j ] c j − ℓ d ℓ , j ∈ N , j > . (4.9)By using (4.8) one obtainsln (cid:18) ∞ X j =1 [ a j + m /a m ] z j (cid:19) = ∞ X j =1 b j z j , (4.10)with the coefficients b j given by equation (4.3). From the last expansion one finallyobtains z − n ddz ln( F A,B ( z )) = z − n ddz ln (cid:18) ∞ X j =1 a j z j (cid:19) = ∞ X j =1 jb j z j − n − . (4.11)This is the Laurent expansion, and from it one can easily deduce thatRes (cid:20) z − n ddz ln( F A,B ( z )); z = 0 (cid:21) = n b n , n ∈ N , (4.12)proving (4.2).Assertion (4.4) about the trace of the inverse operator when z = 0 is not aneigenvalue is obtained by noting − ddz ln( F A,B ( z )) (cid:12)(cid:12)(cid:12)(cid:12) z =0 = − d = − a /a (4.13)from the analytic expansions (4.7) and (4.8), and upon applying Theorem 2.4. (cid:3) This theorem allows one to utilize the series expansions found in the previoussection in order to express the ζ -function values for each of the boundary conditionsconsidered.4.1. Computing Spectral Zeta Function Values and Traces for SeparatedBoundary Conditions.
We begin by applying Theorem 4.1 to find an expression for values of ζ ( n ; T α,β )when imposing separated boundary conditions. Theorem 4.2.
Assume Hypothesis , consider T α,β as described in Theorem i ) , and let m = 0 , , denote the multiplicity of zero as an eigenvalue of T α,β .Then, ζ ( n ; T α,β ) = − Res (cid:20) z − n ddz ln( F α,β ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.14) where b = cos( α ) (cid:2) cos( β ) φ m ( b ) − sin( β ) φ [1]1+ m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ m ( b ) − sin( β ) θ [1]1+ m ( b ) (cid:3) cos( α ) (cid:2) cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ m ( b ) − sin( β ) θ [1] m ( b ) (cid:3) , b j = cos( α ) (cid:2) cos( β ) φ j + m ( b ) − sin( β ) φ [1] j + m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ j + m ( b ) − sin( β ) θ [1] j + m ( b ) (cid:3) cos( α ) (cid:2) cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ m ( b ) − sin( β ) θ [1] m ( b ) (cid:3) − j − X ℓ =1 (cid:18) ℓj (cid:19) cos( α ) (cid:2) cos( β ) φ j − ℓ + m ( b ) − sin( β ) φ [1] j − ℓ + m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ j − ℓ + m ( b ) − sin( β ) θ [1] j − ℓ + m ( b ) (cid:3) cos( α ) (cid:2) cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ m ( b ) − sin( β ) θ [1] m ( b ) (cid:3) b ℓ , j ∈ N , j > . (4.15) In particular, if zero is not an eigenvalue of T α,β , then tr L r (( a,b )) (cid:0) T − α,β (cid:1) = ζ (1; T α,β ) (4.16) = − cos( α ) (cid:2) cos( β ) φ ( b ) − sin( β ) φ [1]1 ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ ( b ) − sin( β ) θ [1]1 ( b ) (cid:3) cos( α ) (cid:2) cos( β ) φ ( b ) − sin( β ) φ [1]0 ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ ( b ) − sin( β ) θ [1]0 ( b ) (cid:3) . Proof.
One substitutes (3.4), (3.6), (3.8), and (3.11) into equation (2.16) for α, β ∈ [0 , π ) to find F α,β ( z ) = ∞ X m =0 (cid:8) cos( α ) (cid:2) cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) (cid:3) − sin( α ) (cid:2) cos( β ) θ m ( b ) − sin( β ) θ [1] m ( b ) (cid:3)(cid:9) z m . (4.17)From (4.17) one proves the assertion by applying Theorem 4.1 with a k = cos( α ) (cid:2) cos( β ) φ k ( b ) − sin( β ) φ [1] k ( b ) (cid:3) − sin( α ) (cid:2) − sin( β ) θ [1] k ( b ) + cos( β ) θ k ( b ) (cid:3) , k ∈ N . (4.18) (cid:3) We now give a few corollaries that will be of use in the context of specific bound-ary conditions. One notes that for Dirichlet boundary conditions one has α = β = 0and for Neumann boundary conditions one has α = β = π/ Corollary 4.3 (Dirichlet boundary conditions) . Assume Hypothesis , consider T , as described in case Theorem i ) , and let m = 0 , , denote the multiplicityof zero as an eigenvalue of T , . Then, ζ ( n ; T , ) = − Res (cid:20) z − n ddz ln( F , ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.19) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 25 where b = φ m ( b ) /φ m ( b ) ,b j = [ φ j + m ( b ) /φ m ( b )] − j − X ℓ =1 [ ℓ/j ][ φ j − ℓ + m ( b ) /φ m ( b )] b ℓ , j ∈ N , > . (4.20) In particular, if zero is not an eigenvalue of T , , then tr L r (( a,b )) (cid:0) T − , (cid:1) = ζ (1; T , ) = − φ ( b ) /φ ( b ) . (4.21) Proof.
Take α = β = 0 in Theorem 4.2. (cid:3) In particular, one finds explicitly for n = 2 , ,
4, when zero is not an eigenvalueof T , : ζ (2; T , ) = − b = − (cid:20) φ ( b ) φ ( b ) − [ φ ( b )] φ ( b )] (cid:21) ,ζ (3; T , ) = − b = − " φ ( b ) φ ( b ) − φ ( b ) φ ( b )[ φ ( b )] + [ φ ( b )] φ ( b )] , (4.22) ζ (4; T , ) = − b = − " φ ( b ) φ ( b ) − φ ( b ) φ ( b )[ φ ( b )] − [ φ ( b )] φ ( b )] + [ φ ( b )] φ ( b )[ φ ( b )] − [ φ ( b )] φ ( b )] . One also finds explicitly for n = 2 , ,
4, when zero is a simple eigenvalue of T , : ζ (2; T , ) = − b = − (cid:20) φ ( b ) φ ( b ) − [ φ ( b )] φ ( b )] (cid:21) ,ζ (3; T , ) = − b = − " φ ( b ) φ ( b ) − φ ( b ) φ ( b )[ φ ( b )] + [ φ ( b )] φ ( b )] , (4.23) ζ (4; T , ) = − b = − " φ ( b ) φ ( b ) − φ ( b ) φ ( b )[ φ ( b )] − [ φ ( b )] φ ( b )] + [ φ ( b )] φ ( b )[ φ ( b )] − [ φ ( b )] φ ( b )] . Corollary 4.4 (Dirichlet boundary condition at a ) . Assume Hypothesis , con-sider T ,β as described in Theorem i ) , and let m = 0 , , denote the multiplicityof zero as an eigenvalue of T ,β . Then, ζ ( n ; T ,β ) = − Res (cid:20) z − n ddz ln( F ,β ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.24) where b = cos( β ) φ m ( b ) − sin( β ) φ [1]1+ m ( b )cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) ,b j = cos( β ) φ j + m ( b ) − sin( β ) φ [1] j + m ( b )cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) − j − X ℓ =1 [ ℓ/j ] cos( β ) φ j − ℓ + m ( b ) − sin( β ) φ [1] j − ℓ + m ( b )cos( β ) φ m ( b ) − sin( β ) φ [1] m ( b ) b ℓ , j ∈ N , j > . (4.25) In particular, if zero is not an eigenvalue of T ,β , then tr L r (( a,b )) (cid:0) T − ,β (cid:1) = ζ (1; T ,β ) = − cos( β ) φ ( b ) − sin( β ) φ [1]1 ( b )cos( β ) φ ( b ) − sin( β ) φ [1]0 ( b ) . (4.26) Proof.
Take α = 0 in Theorem 4.2. (cid:3) Corollary 4.5 (Dirichlet boundary condition at b ) . Assume Hypothesis , con-sider T α, as described in Theorem i ) , and let m = 0 , , denote the multiplicityof zero as an eigenvalue of T α, . Then, ζ ( n ; T α, ) = − Res (cid:20) z − n ddz ln( F α, ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.27) where b = cos( α ) φ m ( b ) − sin( α ) θ m ( b )cos( α ) φ m ( b ) − sin( α ) θ m ( b ) ,b j = cos( α ) φ j + m ( b ) − sin( α ) θ j + m ( b )cos( α ) φ m ( b ) − sin( α ) θ m ( b ) − j − X ℓ =1 [ ℓ/j ] cos( α ) φ j − ℓ + m ( b ) − sin( α ) θ j − ℓ + m ( b )cos( α ) φ m ( b ) − sin( α ) θ m ( b ) b ℓ , j ∈ N , j > . (4.28) In particular, if zero is not an eigenvalue of T α, , then tr L r (( a,b )) (cid:0) T − α, (cid:1) = ζ (1; T α, ) = − cos( α ) φ ( b ) − sin( α ) θ ( b )cos( α ) φ ( b ) − sin( α ) θ ( b ) . (4.29) Proof.
Take β = 0 in Theorem 4.2. (cid:3) Corollary 4.6 (Neumann boundary conditions) . Assume Hypothesis , consider T π/ ,π/ as described in Theorem i ) , and let m = 0 , , denote the multiplicityof zero as an eigenvalue of T π/ ,π/ . Then, ζ ( n ; T π/ ,π/ ) = − Res (cid:20) z − n ddz ln( F π/ ,π/ ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.30) where b = θ [1]1+ m ( b ) (cid:14) θ [1] m ( b ) , (4.31) b j = θ [1] j + m (cid:14) ( b ) θ [1] m ( b ) − j − X ℓ =1 [ ℓ/j ] (cid:2) θ [1] j − ℓ + m ( b ) (cid:14) θ [1] m ( b ) (cid:3) b ℓ , j ∈ N , j > . In particular, if zero is not an eigenvalue of T π/ ,π/ , then tr L r (( a,b )) (cid:0) T − π/ ,π/ (cid:1) = ζ (1; T π/ ,π/ ) = − θ [1]1 ( b ) (cid:14) θ [1]0 ( b ) . (4.32) Proof.
Take α = β = π/ (cid:3) These are only a few of the most considered separated boundary conditions thathave been singled out. One can also consider Neumann boundary conditions atonly one endpoint, or any other combination of separated boundary conditions, byreferring back to Theorem 4.2 with the appropriate values chosen for α, β ∈ [0 , π ).4.2. Computing Spectral Zeta Function Values and Traces for CoupledBoundary Conditions.
We now apply Theorem 4.1 to find values of ζ ( n ; T ϕ,R ) when imposing coupledboundary conditions. Notice that according to [30], zero is an eigenvalue of multi-plicity 2 only for the Krein–von Neumann extension. PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 27 Theorem 4.7.
Assume Hypothesis , consider T ϕ,R as described in Theorem ii ) , and let m = 0 , , denote the multiplicity of zero as an eigenvalue of T ϕ,R .Then, ζ ( n ; T ϕ,R ) = − Res (cid:20) z − n ddz ln( F ϕ,R ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.33) where for m = 0 , b = e iϕ (cid:0) R θ [1]1 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]1 ( b ) (cid:1) e iϕ (cid:0) R θ [1]0 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]0 ( b ) (cid:1) + e iϕ + 1 ,b j = e iϕ (cid:0) R θ [1] j ( b ) − R θ j ( b ) + R φ j ( b ) − R φ [1] j ( b ) (cid:1) e iϕ (cid:0) R θ [1]0 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]0 ( b ) (cid:1) + e iϕ + 1 (4.34) − j − X ℓ =1 ℓj e iϕ (cid:0) R θ [1] j − ℓ ( b ) − R θ j − ℓ ( b ) + R φ j − ℓ ( b ) − R φ [1] j − ℓ ( b ) (cid:1) e iϕ (cid:0) R θ [1]0 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]0 ( b ) (cid:1) + e iϕ + 1 b ℓ ,j ∈ N , j > , and for m = 1 , b = e iϕ (cid:0) R θ [1]2 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]2 ( b ) (cid:1) e iϕ (cid:0) R θ [1]1 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]1 ( b ) (cid:1) ,b j = e iϕ (cid:0) R θ [1] j +1 ( b ) − R θ j +1 ( b ) + R φ j +1 ( b ) − R φ [1] j +1 ( b ) (cid:1) e iϕ (cid:0) R θ [1]1 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]1 ( b ) (cid:1) (4.35) − j − X ℓ =1 ℓj e iϕ (cid:0) R θ [1] j − ℓ +1 ( b ) − R θ j − ℓ +1 ( b ) + R φ j − ℓ +1 ( b ) − R φ [1] j − ℓ +1 ( b ) (cid:1) e iϕ (cid:0) R θ [1]1 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]1 ( b ) (cid:1) b ℓ ,j ∈ N , j > . In particular, if zero is not an eigenvalue of T ϕ,R , then tr L r (( a,b )) (cid:0) T − ϕ,R (cid:1) = ζ (1; T ϕ,R )= − e iϕ (cid:0) R θ [1]1 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]1 ( b ) (cid:1) e iϕ (cid:0) R θ [1]0 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]0 ( b ) (cid:1) + e iϕ + 1 . (4.36) Proof.
Substituting (3.4), (3.6), (3.8), and (3.11) into equation (2.17) yields F ϕ,R (0) = e iϕ (cid:0) R θ [1]0 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]0 ( b ) (cid:1) + e iϕ + 1 . (4.37)Thus, the coefficient of the z m term for m > e iϕ (cid:0) R θ [1] m ( b ) − R θ m ( b ) + R φ m ( b ) − R φ [1] m ( b ) (cid:1) . (4.38)Hence, assertions (4.34) and (4.35) follow from Theorem 4.1 with a = e iϕ (cid:0) R θ [1]0 ( b ) − R θ ( b ) + R φ ( b ) − R φ [1]0 ( b ) (cid:1) + e iϕ + 1 ,a k = e iϕ (cid:0) R θ [1] k ( b ) − R θ k ( b ) + R φ k ( b ) − R φ [1] k ( b ) (cid:1) , k ∈ N . (4.39) (cid:3) Next, we provide corollaries regarding the most common coupled boundary con-ditions, periodic and antiperiodic as well as the Krein-von Neumann extension.
Corollary 4.8 (Periodic boundary conditions) . Assume Hypothesis , consider T ,I as described in Theorem ii ) , and let m = 0 , , denote the multiplicity ofzero as an eigenvalue of T ,I . Then, ζ ( n ; T ,I ) = − Res (cid:20) z − n ddz ln( F ,I ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.40) where for m = 0 , b = (cid:2) − θ ( b ) − φ [1]1 ( b ) (cid:3)(cid:14)(cid:2) − θ ( b ) − φ [1]0 ( b ) + 2 (cid:3) , (4.41) b j = − θ j ( b ) − φ [1] j ( b ) − θ ( b ) − φ [1]0 ( b ) + 2 − j − X ℓ =1 ℓj − θ j − ℓ ( b ) − φ [1] j − ℓ ( b ) − θ ( b ) − φ [1]0 ( b ) + 2 b ℓ , j ∈ N , j > , and for m = 1 , b = (cid:2) θ ( b ) + φ [1]2 ( b ) (cid:3)(cid:14)(cid:2) θ ( b ) + φ [1]1 ( b ) (cid:3) , (4.42) b j = θ j +1 ( b ) + φ [1] j +1 ( b ) θ ( b ) + φ [1]1 ( b ) − j − X ℓ =1 ℓj θ j − ℓ +1 ( b ) + φ [1] j − ℓ +1 ( b ) θ ( b ) + φ [1]1 ( b ) b ℓ , j ∈ N , j > . In particular, if zero is not an eigenvalue of T ,I , then tr L r (( a,b )) (cid:0) T − ,I (cid:1) = ζ (1; T ,I ) = (cid:2) θ ( b ) + φ [1]1 ( b ) (cid:3)(cid:14)(cid:2) − θ ( b ) − φ [1]0 ( b ) + 2 (cid:3) . (4.43) Proof.
Take ϕ = 0 and R = I in Theorem 4.7. (cid:3) Corollary 4.9 (Antiperiodic boundary conditions) . Assume Hypothesis , con-sider T π,I as described in Theorem ii ) , and let m = 0 , , denote the multi-plicity of zero as an eigenvalue of T π,I . Then, ζ ( n ; T π,I ) = − Res (cid:20) z − n ddz ln( F π,I ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.44) where for m = 0 , b = (cid:2) θ ( b ) + φ [1]1 ( b ) (cid:3)(cid:14)(cid:2) θ ( b ) + φ [1]0 ( b ) + 2 (cid:3) , (4.45) b j = θ j ( b ) + φ [1] j ( b ) θ ( b ) + φ [1]0 ( b ) + 2 − j − X ℓ =1 ℓj θ j − ℓ ( b ) + φ [1] j − ℓ ( b ) θ ( b ) + φ [1]0 ( b ) + 2 b ℓ , j ∈ N , j > , and for m = 1 , b = (cid:2) θ ( b ) + φ [1]2 ( b ) (cid:3)(cid:14)(cid:2) θ ( b ) + φ [1]1 ( b ) (cid:3) , (4.46) b j = θ j +1 ( b ) + φ [1] j +1 ( b ) θ ( b ) + φ [1]1 ( b ) − j − X ℓ =1 ℓj θ j − ℓ +1 ( b ) + φ [1] j − ℓ +1 ( b ) θ ( b ) + φ [1]1 ( b ) b ℓ , j ∈ N , j > . In particular, if zero is not an eigenvalue of T π,I , then tr L r (( a,b )) (cid:0) T − π,I (cid:1) = ζ (1; T π,I ) = − (cid:2) θ ( b ) + φ [1]1 ( b ) (cid:3)(cid:14)(cid:2) θ ( b ) + φ [1]0 ( b ) + 2 (cid:3) . (4.47) Proof.
Take ϕ = π and R = I in Theorem 4.7. (cid:3) Corollary 4.10 (Krein-von Neumann extension) . Assume Hypothesis , con-sider T ,R K the Krein-von Neumann extension of T min with ϕ = 0 , R K = (cid:18) θ (0 , b, a ) φ (0 , b, a ) θ [1] (0 , b, a ) φ [1] (0 , b, a ) (cid:19) , (4.48) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 29 and let m = 2 , denote the multiplicity of zero as an eigenvalue of T ,R K . Then, ζ ( n ; T ,R K ) = − Res (cid:20) z − n ddz ln( F ,R K ( z )); z = 0 (cid:21) = − n b n , n ∈ N , (4.49) where b = φ ( b ) θ [1]3 ( b ) − φ [1]0 ( b ) θ ( b ) + θ [1]0 ( b ) φ ( b ) − θ ( b ) φ [1]3 ( b ) φ ( b ) θ [1]2 ( b ) − φ [1]0 ( b ) θ ( b ) + θ [1]0 ( b ) φ ( b ) − θ ( b ) φ [1]2 ( b ) ,b j = φ ( b ) θ [1] j +2 ( b ) − φ [1]0 ( b ) θ j +2 ( b ) + θ [1]0 ( b ) φ j +2 ( b ) − θ ( b ) φ [1] j +2 ( b ) φ ( b ) θ [1]2 ( b ) − φ [1]0 ( b ) θ ( b ) + θ [1]0 ( b ) φ ( b ) − θ ( b ) φ [1]2 ( b ) − j − X ℓ =1 ℓj φ ( b ) θ [1] j − ℓ +2 ( b ) − φ [1]0 ( b ) θ j − ℓ +2 ( b ) + θ [1]0 ( b ) φ j − ℓ +2 ( b ) − θ ( b ) φ [1] j − ℓ +2 ( b ) φ ( b ) θ [1]2 ( b ) − φ [1]0 ( b ) θ ( b ) + θ [1]0 ( b ) φ ( b ) − θ ( b ) φ [1]2 ( b ) b ℓ , j ∈ N , j > . (4.50) Proof.
As shown in [15, Example 3.3], the resulting operator T ,R K represents theKrein–von Neumann extension of T min . Take ϕ = 0 and R = R K (as defined by(4.48)) in Theorem 4.7, denoting φ (0 , b, a ) = φ ( b ), φ [1]0 ( b ) = φ [1] (0 , b, a ), θ ( b ) = θ (0 , b, a ), and θ [1]0 ( b ) = θ [1] (0 , b, a ) as before, for simplicity. (cid:3) Examples
In this section, we provide an array of examples illustrating our approach forcomputing spectral ζ -function values of regular Schr¨odinger operators starting withthe simplest case of q = 0, then a positive (piecewise) constant potential, followedby a constant negative potential, and ending with the case of a linear potential.Throughout this section we suppose that p = r = 1 a.e. on ( a, b ) (5.1)which leaves the potential coefficient q ∈ L (( a, b ); dx ), q real-valued, and henceleaves us with the differential expression τ = − (cid:0) d /dx (cid:1) + q ( x ) , x ∈ ( a, b ) . (5.2)5.1. The Example q =0. We start by providing examples for calculating spectral ζ -function values for thesimple case q ( x ) = 0, x ∈ ( a, b ), imposing various boundary conditions. In this case τ y = − y ′′ = zy has the following linearly independent solutions, φ ( z, x, a ) = z − / sin (cid:0) z / ( x − a ) (cid:1) , θ ( z, x, a ) = cos (cid:0) z / ( x − a ) (cid:1) , z ∈ C . (5.3)Hence, φ ( z, b, a ) = ∞ X m =0 z m φ m ( b ) , z ∈ C , φ k ( b ) = ( − k (2 k + 1)! ( b − a ) k +1 , k ∈ N ,θ ( z, b, a ) = ∞ X m =0 z m θ m ( b ) , z ∈ C , θ k ( b ) = ( − k (2 k )! ( b − a ) k , k ∈ N , (5.4) φ ′ ( z, b, a ) = ∞ X m =0 z m φ ′ m ( b ) , z ∈ C , φ ′ k ( b ) = − ( − k ( k + 1)! ( b − a ) k +1 , k ∈ N ,θ ′ ( z, b, a ) = ∞ X m =0 z m θ ′ m ( b ) , z ∈ C , θ ′ k ( b ) = ( − k k ! ( b − a ) k , k ∈ N . (5.5)One can explicitly write the corresponding expressions for F α,β ( z ) and F ϕ,R ( z )for this example to find for α, β ∈ [0 , π ), F α,β ( z ) = cos( α ) (cid:2) − sin( β ) cos (cid:0) z / ( b − a ) (cid:1) + cos( β ) z − / sin (cid:0) z / ( b − a ) (cid:1)(cid:3) − sin( α ) (cid:2) sin( β ) z / sin (cid:0) z / ( b − a ) (cid:1) + cos( β ) cos (cid:0) z / ( b − a ) (cid:1)(cid:3) , (5.6)and for ϕ ∈ [0 , π ) , R ∈ SL (2 , R ), F ϕ,R ( z ) = e iϕ (cid:2) − R z / sin (cid:0) z / ( b − a ) (cid:1) − R cos (cid:0) z / ( b − a ) (cid:1) + R z − / sin (cid:0) z / ( b − a ) (cid:1) − R cos (cid:0) z / ( b − a ) (cid:1)(cid:3) + e iϕ + 1 . (5.7)We provide an explicit expression for ζ (1; T A,B ) since it only involves the first fewcoefficients of the small- z expansion. In the case of separated boundary conditionsone obtains a = cos( α )(( b − a ) cos( β ) − sin( β )) − sin( α ) cos( β ) ,a = cos( α ) (cid:18)
12 ( b − a ) sin( β ) −
16 ( b − a ) cos( β ) (cid:19) + sin( α ) (cid:18)
12 ( b − a ) cos( β ) − ( b − a ) sin( β ) (cid:19) ,a = sin( α ) (cid:18)
16 ( b − a ) sin( β ) −
124 ( b − a ) cos( β ) (cid:19) + cos( α ) (cid:18) b − a ) cos( β ) −
124 ( b − a ) sin( β ) (cid:19) . (5.8)If T α,β does not have a zero eigenvalue, then a = 0 and, hence, one finds from(4.4),tr L r ((0 ,b )) (cid:0) T − α,β (cid:1) = ζ (1; T α,β ) = (5.9) cos( α ) (cid:0) b − a ) sin( β ) − ( b − a ) cos( β ) (cid:1) + sin( α ) (cid:0) b − a ) cos( β ) − b − a ) sin( β ) (cid:1) α ) cos( β ) − α )(( b − a ) cos( β ) − sin( β )) . If, instead, T α,β has a zero eigenvalue then a = 0 and one finds ζ (1; T α,β ) = (5.10) − sin( α ) (cid:0) b − a ) sin( β ) − b − a ) cos( β ) (cid:1) − cos( α ) (cid:0) ( b − a ) cos( β ) − b − a ) sin( β ) (cid:1) cos( α ) (60( b − a ) sin( β ) − b − a ) cos( β )) + sin( α ) (60( b − a ) cos( β ) − b − a ) sin( β )) . In the case of coupled boundary conditions one finds a = e iϕ (( b − a ) R − R − R ) + e iϕ + 1 ,a = e iϕ (cid:18) −
16 ( b − a ) R + 12 ( b − a ) R + 12 ( b − a ) R + ( a − b ) R (cid:19) , (5.11) a = e iϕ (cid:18) b − a ) R −
124 ( b − a ) R −
124 ( b − a ) R + 16 ( b − a ) R (cid:19) . PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 31 Once again, if zero is not an eigenvalue of T ϕ,R , a = 0 and one findstr L r ((0 ,b )) (cid:0) T − ϕ,R (cid:1) = ζ (1; T ϕ,R )= e iϕ (cid:0) R ( b − a ) − b − a ) R − b − a ) R + 6( b − a ) R (cid:1) e iϕ (( b − a ) R − R − R ) + 6 e iϕ + 6 . (5.12)If, on the other hand, zero is an eigenvalue of T ϕ,R with multiplicity one, then a = 0 and ζ (1; T ϕ,R ) = ( b − a ) R − b − a ) R − b − a ) R + 20( b − a ) R b − a ) R − b − a ) R − b − a ) R + 120( b − a ) R . (5.13)If zero is an eigenvalue of T ϕ,R with multiplicity two, we refer to the Krein–vonNeumann extension, see Example 5.5.Finally we give the form of the zeta regularized functional determinant for thisexample. As z ↓
0, one obtains F α,β ( z ) = ( b − a ) cos( α ) cos( β ) − sin( α + β ) + O (cid:0) z / (cid:1) , (5.14)which implies that for particular values of α and β one finds a zero eigenvalue. Fornow we will assume that no zero eigenvalue is present and hence we consider thefollowing set of parameters A = { α, β ∈ (0 , π ) | ( b − a ) cos( α ) cos( β ) − sin( α + β ) = 0 } . (5.15)For α, β ∈ A we have, by construction, that m = 0 and the product sin( α ) sin( β ) =0. The latter condition implies that in (3.93) one must set k = −
2. By using (5.14),one obtains ζ ′ (0; T α,β ) = − ln (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) F α,β (0)sin( α ) sin( β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) = − ln (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) b − a ) cos( α ) cos( β ) − α + β )sin( α ) sin( β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) , (5.16)which coincides with [33, Eq. (3.72)].Furthermore, as z ↓
0, one obtains F ϕ,R ( z ) = e iϕ [( b − a ) R − R − R ] + e iϕ + 1 + O (cid:0) z / (cid:1) , (5.17)which implies that for particular choices of ϕ and R one finds a zero eigenvalue.For now we will assume that no zero eigenvalue is present and hence we considerthe following set of parameters B = { ϕ ∈ (0 , π ) , R ∈ SL (2 , R ) | e iϕ [( b − a ) R − R − R ] + e iϕ + 1 = 0 } . (5.18)For ϕ, R ∈ B we have, by construction, that m = 0. Making the additionalassumption R = 0 implies that in (3.93) one must set k = −
2. By using (5.17),one obtains ζ ′ (0; T ϕ, e R ) = − ln (cid:16)(cid:12)(cid:12)(cid:12) F ϕ, e R (0) /R (cid:12)(cid:12)(cid:12)(cid:17) = − ln (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) b − a ) R − R − R ] + 4 cos( ϕ ) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (5.19) If R = 0, then since R ∈ SL (2 , R ), by assumption R = − R which impliesthat in (3.93) one must set k = −
1. By once again using (5.17), one obtains ζ ′ (0; T ϕ, e R ) = − ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ϕ, e R (0) R + R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! = − ln (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) b − a ) R − R − R ] + 4 cos( ϕ ) R + R (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (5.20)The following examples, each with different boundary conditions, will illustratehow the main theorems and corollaries of the previous section can be used to effec-tively compute the spectral ζ -function values of the operator, T A,B , for n ∈ N . Example 5.1 (Dirichlet boundary conditions) . Consider the case α = β = 0 .Then the operator T , has eigenvalues and eigenfunctions given by λ k = k π (cid:14) ( b − a ) , y k ( x ) = λ − / k sin (cid:0) λ / k ( x − a ) (cid:1) , k ∈ N (5.21)( in particular, z = 0 is not an eigenvalue of T , ) , and F , ( z ) = z − / sin (cid:0) z / ( b − a ) (cid:1) , z ∈ C . (5.22) Applying Corollary with m = 0 one finds for n = 1 , , , , ζ (1; T , ) = ( b − a ) π − ∞ X k =1 k − = tr L r (( a,b )) (cid:0) T − , (cid:1) = ( b − a ) / ,ζ (2; T , ) = ( b − a ) / ,ζ (3; T , ) = ( b − a ) / ,ζ (4; T , ) = ( b − a ) / . (5.23) Next, we explicitly compute the zeta regularized functional determinant withDirichlet boundary conditions. Since no zero eigenvalue is present and Γ = − ( b − a ) , one easily obtains ζ ′ (0; T , ) = − ln[2 F , (0)] = − ln[2( b − a )] . (5.24)One can corroborate the values found in Example 5.1 by utilizing the follow-ing relation of ζ ( s ; T , ) with the Riemann ζ -function (see, e.g., [8], [18] for somebackground) ζ ( s ; T , ) = ( b − a ) s π − s ζ (2 s ) , Re( s ) > / . (5.25)By using [37, 0.2333], the last expression allows us to find for s = n ∈ N , ζ ( n ; T , ) = 2 n − ( b − a ) n | B n | / [(2 n )!] , (5.26)where B n is the 2 n th Bernoulli number (cf. [1, Ch. 23]). Example 5.2 (Neumann boundary conditions) . Consider the case α = β = π/ .Then the operator T π/ ,π/ has eigenvalues and eigenfunctions given by λ k = k π / ( b − a ) , y k ( x ) = cos (cid:0) λ / k ( x − a ) (cid:1) , k ∈ N (5.27)( in particular, z = 0 is a simple eigenvalue of T π/ ,π/ ) and F π/ ,π/ ( z ) = − z / sin (cid:0) z / ( b − a ) (cid:1) , z ∈ C . (5.28) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 33 Applying Corollary with m = 1 one finds for n = 1 , , , , ζ (1; T π/ ,π/ ) = ( b − a ) π − ∞ X k =1 k − = ( b − a ) / ,ζ (2; T π/ ,π/ ) = ( b − a ) / ,ζ (3; T π/ ,π/ ) = ( b − a ) / ,ζ (4; T π/ ,π/ ) = ( b − a ) / . (5.29)Noting that the series expression for ζ ( s ; T π/ ,π/ ) in (2.38) sums only over non-zero eigenvalues, and that the eigenvalues for Dirichlet and Neumann boundaryconditions only differ by zero being an eigenvalue for the latter, but not the former,the same expressions apply as in Example 5.1, which is reflected in equations (5.23)and (5.29) yielding the same values. Example 5.3 (Periodic boundary conditions) . Consider the case ϕ = 0 , R = I .Then the operator T ,I has eigenvalues given by λ k = (2 k ) π / ( b − a ) , k ∈ N . (5.30) In particular, z = 0 is a simple eigenvalue of T ,I and all other eigenvalues of T ,I are of multiplicity 2, and F ,I ( z ) = − (cid:0) z / ( b − a ) (cid:1) + 2 , z ∈ C . (5.31) Applying Corollary with m = 1 one finds for n = 1 , , , , ζ (1; T ,I ) = 2( b − a ) π − ∞ X k =1 (2 k ) − = ( b − a ) / ,ζ (2; T ,I ) = ( b − a ) / ,ζ (3; T ,I ) = ( b − a ) / ,ζ (4; T ,I ) = ( b − a ) / . (5.32)Here, once again, one can verify the values found in Example 5.3 by utilizing thefollowing relation of ζ ( s ; T ,I ) with the Riemann ζ -function, ζ ( s ; T ,I ) = 2 − s π − s ( b − a ) s ζ (2 s ) , Re( s ) > / . (5.33)By using [37, 0.2333], the last expression allows one to find for s = n ∈ N , ζ ( n ; T ,I ) = ( b − a ) n | B n | / [(2 n )!] . (5.34) Example 5.4 (Antiperiodic boundary conditions) . Consider the case ϕ = π, R = I . Then the operator T π,I has eigenvalues given by λ k = (2 k − π / ( b − a ) , k ∈ N . (5.35) In particular, z = 0 is not an eigenvalue of T π,I and all eigenvalues of T π,I are ofmultiplicity 2, and F π,I ( z ) = 2 cos (cid:0) z / ( b − a ) (cid:1) + 2 , z ∈ C . (5.36) Applying Corollary with m = 0 one finds for n = 1 , , , , ζ (1; T π,I ) = 2( b − a ) π − ∞ X k =1 (2 k − − = tr L r (( a,b )) (cid:0) T − π,I (cid:1) = ( b − a ) / ,ζ (2; T π,I ) = ( b − a ) / ,ζ (3; T π,I ) = ( b − a ) / ,ζ (4; T π,I ) = [17 / b − a ) . (5.37)One can verify the values found in Example 5.4 by utilizing the following relation, ζ ( s ; T π,I ) = 2( b − a ) s π − s X k ∈ N (2 k − − s = (cid:0) − − s (cid:1) b − a ) s π − s ζ (2 s ) , Re( s ) > / , (5.38)which in turn by using either [37, 0.2335] on the first equality or [37, 0.2333] on thesecond allows one to find for s = n ∈ N , ζ ( n ; T π,I ) = (2 n − b − a ) n | B n | / [(2 n )!] . (5.39) Example 5.5 (Krein–von Neumann boundary conditions) . Consider the case ϕ =0 , R = R K , with R K = (cid:18) θ (0 , b, a ) φ (0 , b, a ) θ [1] (0 , b, a ) φ [1] (0 , b, a ) (cid:19) = (cid:18) b − a (cid:19) . (5.40) As shown in [15, Example 3.3] , the resulting operator T ,R K represents the Krein–von Neumann extension of T min . For more on the Krein–von Neumann extension,including an extensive discussion of eigenvalues and eigenfunctions, see [2] or [4] .From (2.17) with ϕ = 0 , R = R K defined as in (5.40) , F ,R K ( z ) = ( a − b ) z / sin (cid:0) z / ( b − a ) (cid:1) − (cid:0) z / ( b − a ) (cid:1) + 2 , z ∈ C . (5.41) Using the series expansions in (5.41) , one finds F ,R K ( z ) = z ↓ (cid:2) ( b − a ) / (cid:3) z + O (cid:0) z (cid:1) , (5.42) so that z = 0 is a zero of multiplicity two of F ,R K ( z ) and hence an eigenvalue ofmultiplicity two of T ,R K ( coinciding with what was found in [4] and noted in [33,Example 3.7]) . Applying Corollary with m = 2 gives ζ (1; T ,R K ) = ( b − a ) / ,ζ (2; T ,R K ) = [11 / b − a ) ,ζ (3; T ,R K ) = ( b − a ) / ,ζ (4; T ,R K ) = [457 / b − a ) . (5.43)5.2. Examples of Nonnegative (Piecewise) Constant Potentials.
Next we provide examples for calculating spectral ζ -function values consideringa positive (piecewise) constant potential q , imposing Dirichlet boundary conditions. Example 5.6.
Let V ∈ (0 , ∞ ) , consider q ( x ) = V , x ∈ ( a, b ) , and denote by T , the associated Schr¨odinger operator with Dirichlet boundary conditions at a and b ( i.e., α = β = 0) . Then, φ ( z, x, a ) = ( z − V ) − / sin (cid:0) ( z − V ) / ( x − a ) (cid:1) ,θ ( z, x, a ) = cos (cid:0) ( z − V ) / ( x − a ) (cid:1) , x ∈ ( a, b ) , z ∈ C . (5.44) PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 35 Furthermore, the eigenvalues and eigenfunctions for T , with q ( x ) = V > , x ∈ ( a, b ) , are given by λ k = k π / ( b − a ) − + V ,y k ( x ) = ( λ k − V ) − / sin (cid:0) ( λ k − V ) / ( x − a ) (cid:1) , k ∈ N (5.45)( in particular, z = 0 is not an eigenvalue of T , ) , and F , ( z ) = ( z − V ) − / sin (cid:0) ( z − V ) / ( b − a ) (cid:1) , z ∈ C . (5.46) Applying Corollary with m = 0 one finds for n = 1 , , the expression for n = 4 is significantly longer and hence is omitted here ) , ζ (1; T , ) = ∞ X k =1 (cid:20) k π ( b − a ) + V (cid:21) − = tr L r (( a,b )) (cid:0) T − , (cid:1) = (cid:2) V / ( b − a ) coth (cid:0) V / ( b − a ) (cid:1) − (cid:3)(cid:14) (2 V ) , ζ (2; T , ) = V / ( b − a ) sinh (cid:0) V / ( b − a ) (cid:1) − (cid:0) V / ( b − a ) (cid:1) + 2 V ( b − a ) + 28 V sinh (cid:0) V / ( b − a ) (cid:1) , ζ (3; T , ) = (cid:0) V sinh (cid:0) V / ( b − a ) (cid:1)(cid:1) − (cid:2) V ( b − a ) −
16 cosh (cid:0) V / ( b − a ) (cid:1) + 16 + V / ( b − a ) (cid:0) a V − abV + 8 b V − (cid:1) coth (cid:0) V / ( b − a ) (cid:1) − aV / cosh (cid:0) V / ( b − a ) (cid:1)(cid:0) sinh (cid:0) V / ( b − a ) (cid:1)(cid:1) − + 3 bV / cosh (cid:0) V / ( b − a ) (cid:1)(cid:0) sinh (cid:0) V / ( b − a ) (cid:1)(cid:1) − (cid:3) . (5.47)Taking the limit V ↓ Remark . One can also verify the expressions found in Example 5.6 by meansof the one-dimensional Epstein ζ -function given by ζ E ( s ; m ) = ∞ X k = −∞ (cid:0) k + m (cid:1) − s , m = 0 , s > / ζ ( s ; T , ) in Example 5.6 can be written ζ ( s ; T , ) = ∞ X k =1 (cid:20) k π ( b − a ) + V (cid:21) − s = ( b − a ) s π − s ∞ X k =1 (cid:2) k + m (cid:3) − s = 2 − ( b − a ) s π − s (cid:2) ζ E ( s ; m ) − m − s (cid:3) , s > / , (5.49)where m = ( b − a ) V π − > . (5.50)Then the following formula for the analytic continuation of ζ E ( s ; m ) in s for m =0 , − , − , . . . (see [21, Sect. 4.1.1]) ζ E ( s ; m ) = π / Γ( s − )Γ( s ) m − s + 4 π s Γ( s ) m / − s ∞ X n =1 n s − / K s − / (2 πmn ) ,s = (1 / − ℓ, ℓ ∈ N , s ∈ C , (5.51) where K µ ( · ) is the modified Bessel function of the second kind (see for example [1,Chs. 9-10]), can be used to explicitly verify the expressions found in Example 5.6.We verify the expressions for ζ (1; T , ) and ζ (2; T , ) next. From (5.51) one has,using the fact that K / ( z ) = π / (2 z ) − / e − z , ζ E (1; m ) = πm − + 4 πm − / ∞ X n =1 n / π / (4 πmn ) − / e − πmn = πm − + 2 πm − ∞ X n =1 e − πmn = πm − + 2 πm − e πm − πm coth( πm ) . (5.52)Thus, from (5.49) and (5.50) one obtains ζ (1; T , ) = ( b − a ) π (cid:0) ζ E (1; m ) − m − (cid:1) = ( b − a ) π (cid:18) πm coth( πm ) − m (cid:19) = (cid:2) V / ( b − a ) coth (cid:0) V / ( b − a ) (cid:1) − (cid:3)(cid:14) (2 V ) , (5.53)in accordance with Example 5.6.Next we verify the expression for ζ (2; T , ) by first noting that ddm (cid:0) ζ E ( s ; m ) (cid:1) = − smζ E ( s + 1; m ) , (5.54)which implies the functional equation ζ E ( s + 1; m ) = − sm ddm (cid:0) ζ E ( s ; m ) (cid:1) . (5.55)From (5.52) and (5.55) with s = 1 one has ζ E (2; m ) = − π m ddm (cid:18) coth( πm ) m (cid:19) = π sinh(2 πm ) + 2 π m m sinh ( πm ) . (5.56)Thus from (5.49) and (5.50) one obtains ζ (2; T , ) = ( b − a ) π (cid:0) ζ E (2; m ) − m − (cid:1) = ( b − a ) π (cid:18) π sinh(2 πm ) + 2 π m m sinh ( πm ) − m (cid:19) = V / ( b − a ) sinh (cid:0) V / ( b − a ) (cid:1) − (cid:0) V / ( b − a ) (cid:1) + 2 V ( b − a ) + 28 V sinh (cid:0) V / ( b − a ) (cid:1) , (5.57)again in accordance with Example 5.6. All other positive integer values can befound recursively by means of (5.52) and the functional equation (5.55). ⋄ Next, we turn to the case of a nonnegative piecewise constant potential (a po-tential well):
PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 37 Example 5.8.
Let c, d ∈ ( a, b ) , c < d , V ∈ (0 , ∞ ) , consider q ( x ) = x ∈ ( a, c ) ,V x ∈ ( c, d ) , x ∈ ( d, b ) , (5.58) and denote by T , the associated Schr¨odinger operator with Dirichlet boundaryconditions at a and b . Then, for z ∈ C , φ ( z, x, a ) = z − / sin (cid:0) z / ( x − a ) (cid:1) , x ∈ ( a, c ) ,θ ( z, x, a ) = cos (cid:0) z / ( x − a ) (cid:1) , x ∈ ( a, c ) ,φ ( z, x, a ) = cos (cid:0) z / ( c − a ) (cid:1) ( z − V ) − / sin (cid:0) ( z − V ) / ( x − c ) (cid:1) + z − / sin (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( x − c ) (cid:1) , x ∈ ( c, d ) ,θ ( z, x, a ) = − z / sin (cid:0) z / ( c − a ) (cid:1) ( z − V ) − / sin (cid:0) ( z − V ) / ( x − c ) (cid:1) + cos (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( x − c ) (cid:1) , x ∈ ( c, d ) ,φ ( z, x, a ) = (cid:20) cos (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1) − ( z − V ) / z − / sin (cid:0) z / ( c − a ) (cid:1) sin (cid:0) ( z − V ) / ( d − c ) (cid:1)(cid:21) × z − / sin (cid:0) z / ( x − d ) (cid:1) (5.59)+ (cid:20) cos (cid:0) z / ( c − a ) (cid:1) ( z − V ) − / sin (cid:0) ( z − V ) / ( d − c ) (cid:1) + z − / sin (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1)(cid:21) cos (cid:0) z / ( x − d ) (cid:1) ,x ∈ ( d, b ) ,θ ( z, x, a ) = − (cid:20) z / sin (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1) + ( z − V ) / cos (cid:0) z / ( c − a ) (cid:1) sin (cid:0) ( z − V ) / ( d − c ) (cid:1)(cid:21) × z − / sin (cid:0) z / ( x − d ) (cid:1) + (cid:20) − z / sin (cid:0) z / ( c − a ) (cid:1) ( z − V ) − / sin (cid:0) ( z − V ) / ( d − c ) (cid:1) + cos (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1)(cid:21) cos (cid:0) z / ( x − d ) (cid:1) ,x ∈ ( d, b ) . In particular, φ ( z, b, a ) = ∞ X m =0 z m φ m ( b ) , z ∈ C , (5.60) where φ ( b ) = h cosh (cid:0) V / ( d − c ) (cid:1) + V / ( c − a ) sinh (cid:0) V / ( d − c ) (cid:1)i ( b − d )+ V − / sinh (cid:0) V / ( d − c ) (cid:1) + ( c − a ) cosh (cid:0) V / ( d − c ) (cid:1) , φ ( b ) = (cid:0) V / (cid:1) − (cid:8) (cid:2)(cid:0) aV ( c − d ) − c V + cdV − (cid:1) sinh (cid:0) V / ( c − d ) (cid:1) (5.61)+ V / ( c − d ) cosh (cid:0) V / ( c − d ) (cid:1)(cid:3) + V (cid:2) sinh (cid:0) V / ( d − c ) (cid:1) ( aV ( b − d ) − bcV + cdV −
3) + V / (3 a − b − c + d ) cosh (cid:0) V / ( d − c ) (cid:1)(cid:3) × ( b − d ) (cid:2) V / sinh (cid:0) V / ( d − c ) (cid:1) + cosh (cid:0) V / ( d − c ) (cid:1)(cid:3) + V / ( a − c ) (cid:9) , etc.By construction, φ ( z, a, a ) = 0 , so eigenvalues are given by solving φ ( z, b, a ) = 0 ,or, equivalently, by solving tan (cid:0) z / ( b − d ) (cid:1) (5.62) = − z cos (cid:0) z / ( c − a ) (cid:1) sin (cid:0) ( z − V ) / ( d − c ) (cid:1) − p z ( z − V ) sin (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1)p z ( z − V ) cos (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1) − ( z − V ) sin (cid:0) z / ( c − a ) (cid:1) sin (cid:0) ( z − V ) / ( d − c ) (cid:1) . From (2.16) , one has F , ( z ) = (cid:20) cos (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1) − ( z − V ) / sin (cid:0) z / ( c − a ) (cid:1) z / sin (cid:0) ( z − V ) / ( d − c ) (cid:1)(cid:21) sin (cid:0) z / ( b − d ) (cid:1) z / + (cid:20) cos (cid:0) z / ( c − a ) (cid:1) ( z − V ) − / sin (cid:0) ( z − V ) / ( d − c ) (cid:1) (5.63)+ z − / sin (cid:0) z / ( c − a ) (cid:1) cos (cid:0) ( z − V ) / ( d − c ) (cid:1)(cid:21) cos (cid:0) z / ( b − d ) (cid:1) ,z ∈ C . Hence, applying Corollary with m = 0 one explicitly finds the sum of theinverse of these eigenvalues, namely ζ (1; T , ) = tr L r (( a,b )) (cid:0) T − , (cid:1) = − φ ( b ) /φ ( b ) (5.64) = − (cid:8) V (cid:2) ( V ( c − a )( b − d ) + 1) sinh (cid:0) V / ( d − c ) (cid:1) − V / ( a − b − c + d ) cosh (cid:0) V / ( d − c ) (cid:1)(cid:3)(cid:9) − × (cid:8) (cid:2)(cid:0) aV ( c − d ) − c V + cdV − (cid:1) sinh (cid:0) V / ( c − d ) (cid:1) + V / ( c − d ) cosh (cid:0) V / ( c − d ) (cid:1)(cid:3) + V (cid:2) sinh (cid:0) V / ( d − c ) (cid:1) ( aV ( b − d ) − bcV + cdV −
3) + V / (3 a − b − c + d ) cosh (cid:0) V / ( d − c ) (cid:1)(cid:3) × ( b − d ) (cid:2) V / sinh (cid:0) V / ( d − c ) (cid:1) + cosh (cid:0) V / ( d − c ) (cid:1)(cid:3) + V / ( a − c ) (cid:9) . Taking the limits c ↓ a and d ↑ b of (5.64) recovers the expression in Example 5.6.Furthermore, taking the limit V ↓ n = 2 is significantly longer and hence it is omitted here.5.3. Example of a Negative Constant Potential.
Next, we derive spectral ζ -function values for the case of a negative constantpotential. This case is dealt with separately since the question as to whether z = 0is an eigenvalue of T , depends on the actual constant value of the potential. Example 5.9.
Let V ∈ (0 , ∞ ) , consider q ( x ) = − V , x ∈ ( a, b ) , and denote by T , the associated Schr¨odinger operator with Dirichlet boundary conditions at a PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 39 and b . Then, φ ( z, x, a ) = ( z + V ) − / sin (cid:0) ( z + V ) / ( x − a ) (cid:1) ,θ ( z, x, a ) = cos (cid:0) ( z + V ) / ( x − a ) (cid:1) , z ∈ C . (5.65) Furthermore, eigenvalues and eigenfunctions for T , with q ( x ) = − V < , x ∈ ( a, b ) , are given by λ k = k π ( b − a ) − V , y k ( x ) = ( λ k + V ) − / sin (cid:0) ( λ k + V ) / ( x − a ) (cid:1) , k ∈ N , (5.66) where one notes that due to q ( x ) = − V < , z = 0 is an eigenvalue of T , forcertain values of V . Specifically, if one has V = k π / ( b − a ) , for some k ∈ N , (5.67) then z = 0 is a simple eigenvalue of T , . Otherwise, z = 0 is not an eigenvalue of T , . Moreover, F , ( z ) = ( z + V ) − / sin (cid:0) ( z + V ) / ( b − a ) (cid:1) , z ∈ C . (5.68) Applying Corollary with m = 0 when V = k π / ( b − a ) , k ∈ N , one findsfor n = 1 , , the expression for n = 4 is significantly longer and hence is omittedhere ) , ζ (1; T , ) = ∞ X k =1 (cid:20) k π ( b − a ) − V (cid:21) − = tr L r (( a,b )) (cid:0) T − , (cid:1) = (cid:2) V / ( a − b ) cot (cid:0) V / ( b − a ) (cid:1) + 1 (cid:3)(cid:14) (2 V ) , ζ (2; T , ) = V / ( b − a ) sin (cid:0) V / ( b − a ) (cid:1) + 2 cos (cid:0) V / ( b − a ) (cid:1) + 2 V ( b − a ) − V sin (cid:0) V / ( b − a ) (cid:1) , ζ (3; T , ) = (cid:0) V sin (cid:0) V / ( b − a ) (cid:1)(cid:1) − (cid:2) − V ( b − a ) −
16 cos (cid:0) V / ( b − a ) (cid:1) + 16 − V / ( b − a ) (cid:0) a V − abV + 8 b V − (cid:1) cot (cid:0) V / ( b − a ) (cid:1) − aV / cos (cid:0) V / ( b − a ) (cid:1)(cid:0) sin (cid:0) V / ( b − a ) (cid:1)(cid:1) − + 3 bV / cos (cid:0) V / ( b − a ) (cid:1)(cid:0) sin (cid:0) V / ( b − a ) (cid:1)(cid:1) − (cid:3) . (5.69) When V = k π / ( b − a ) for some k ∈ N , applying Corollary with m = 1 one finds for n = 1 , the expressions for n = 3 , are significantly longer and henceare omitted here ) , ζ (1; T , ) = ∞ X k =1 k = k (cid:20) k π ( b − a ) − V (cid:21) − = π ( b − a ) ∞ X k =1 k = k (cid:2) k − k (cid:3) − = (cid:0) V ( b − a ) − (cid:1) sin (cid:0) V / ( a − b ) (cid:1) + 3 V / ( a − b ) cos (cid:0) V / ( a − b ) (cid:1) V (cid:0) sin (cid:0) V / ( b − a ) (cid:1) + V / ( a − b ) cos (cid:0) V / ( b − a ) (cid:1)(cid:1) ,ζ (2; T , ) = 124 V (cid:0) sin (cid:0) V / ( b − a ) (cid:1) + V / ( a − b ) cos (cid:0) V / ( b − a ) (cid:1)(cid:1) × (cid:8) (cid:2) (cid:0) − V ( b − a ) (cid:1) sin (cid:0) V / ( a − b ) (cid:1) − V / ( b − a )( V ( b − a ) −
15) cos (cid:0) V / ( a − b ) (cid:1)(cid:3) + 3 (cid:0) sin (cid:0) V / ( b − a ) (cid:1)(cid:1) − (cid:2)(cid:0) V ( b − a ) −
3) sin (cid:0) V / ( a − b ) (cid:1) − V / ( b − a ) cos (cid:0) V / ( a − b ) (cid:1)(cid:3) × (cid:2) sin (cid:0) V / ( b − a ) (cid:1) − V / ( b − a ) cos (cid:0) V / ( b − a ) (cid:1)(cid:3)(cid:9) . (5.70)Taking the limit V ↓ Remark . In the case z = 0 is not an eigenvalue, one can verify these resultsvia the method outlined in Remark 5.7. Namely, letting m = − ( b − a ) V π − < m = ( i/π )( b − a ) V / (5.72)in (5.53) and (5.57), one verifies the expressions for n = 1 , ⋄ Example of a Linear Potential.
We finish with an example for calculating spectral ζ -function values for the linearpotential, q ( x ) = x , x ∈ ( a, b ). Example 5.11.
Consider q ( x ) = x , x ∈ ( a, b ) , and denote by T , the associatedSchr¨odinger operator with Dirichlet boundary conditions at a and b . Then, notingthat W (Ai , Bi)( x ) = π − ( cf. [1, Eq. 10.4.10]) , one finds φ ( z, x, a ) = π (Ai( a − z ) Bi( x − z ) − Bi( a − z ) Ai( x − z )) , (5.73) θ ( z, x, a ) = − π (Ai ′ ( a − z ) Bi( x − z ) − Bi ′ ( a − z ) Ai( x − z )) , z ∈ C , (5.74) where Ai( · ) and Bi( · ) represent the Airy functions of the first and second kind,respectively ( cf. [1, Sect. 10.4]) . In particular, substituting z = 0 in (5.73) yields φ ( x ) = π (Ai( a ) Bi( x ) − Bi( a ) Ai( x )) , θ ( x ) = − π (Ai ′ ( a ) Bi( x ) − Bi ′ ( a ) Ai( x )) , (5.75) and thus the Volterra Green’s function becomes g (0 , x, x ′ ) = π (Ai( x ) Bi( x ′ ) − Ai( x ′ ) Bi( x )) . (5.76) Hence, φ ( z, b, a ) = ∞ X m =0 z m φ m ( b ) , z ∈ C , (5.77) where φ ( b ) = π (Ai( a ) Bi( b ) − Bi( a ) Ai( b )) ,φ ( b ) = π ˆ ba dx (Ai( b ) Bi( x ) − Ai( x ) Bi( b ))(Ai( a ) Bi( x ) − Bi( a ) Ai( x ))= π (cid:2) Ai( a ) Ai( b ) (cid:0) Bi ′ ( a ) − Bi ′ ( b ) (cid:1) + Bi( a ) Bi( b ) (cid:0) Ai ′ ( a ) − Ai ′ ( b ) (cid:1) (5.78)+ (Ai ′ ( b ) Bi ′ ( b ) − Ai ′ ( a ) Bi ′ ( a ))(Bi( a ) Ai( b ) + Ai( a ) Bi( b )) (cid:3) , etc. PECTRAL ζ -FUNCTION FOR REGULAR STURM–LIOUVILLE OPERATORS 41 Furthermore, one has by construction, φ ( z, a, a ) = 0 , so eigenvalues are givenby solving φ ( z, b, a ) = 0 , or, equivalently, by solving Ai( a − z ) Bi( b − z ) = Bi( a − z ) Ai( b − z ) . In particular, the characteristic function is given by F , ( z ) = π (Ai( a − z ) Bi( b − z ) − Bi( a − z ) Ai( b − z )) , z ∈ C . (5.79) If zero is not an eigenvalue, applying Corollary with m = 0 one does find thesum of the inverse of these eigenvalues, namely ζ (1; T , ) = tr L r (( a,b )) (cid:0) T − , (cid:1) = − φ ( b ) /φ ( b )= π (Bi( a ) Ai( b ) − Ai( a ) Bi( b )) − (cid:2) Ai( a ) Ai( b ) (cid:0) Bi ′ ( a ) − Bi ′ ( b ) (cid:1) + Bi( a ) Bi( b ) (cid:0) Ai ′ ( a ) − Ai ′ ( b ) (cid:1) (5.80)+ (Ai ′ ( b ) Bi ′ ( b ) − Ai ′ ( a ) Bi ′ ( a ))(Bi( a ) Ai( b ) + Ai( a ) Bi( b )) (cid:3) . Acknowledgments.
We are indebted to Angelo Mingarelli for very helpful dis-cussions.
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