One-loop analysis with nonlocal boundary conditions

In the eighties, Schroder studied a quantum mechanical model where the stationary states of Schrodinger's equation obey nonlocal boundary conditions on a circle in the plane. For such a problem, we perform a detailed one-loop calculation for three choices of the kernel characterizing the nonlocal boundary conditions. In such cases, the zeta(0) value is found to coincide with the one resulting from Robin boundary conditions. The detailed technique here developed may be useful for studying one-loop properties of quantum field theory and quantum gravity if nonlocal boundary conditions are imposed.


I. INTRODUCTION
In the late eighties, motivated by physical models of Bose condensation and mathematical study of Schrödinger operators, the work in Ref. [1] studied spectral properties of the Laplace operator with nonlocal boundary conditions. Within this framework, on considering the region B R ≡ x, y : x 2 + y 2 ≤ R 2 , (1.1) one builds, out of a function q which is both Lebesgue summable and square-integrable on the real line, the periodic function which has period 2πR and approaches q if R → ∞. On using polar coordinates (r, ϕ), the nonlocal boundary-value problem studied in Ref. [1] reads as whereq is the Fourier transform of q, i.e. [1] We note from (1.6) thatq must have dimension length −1 , and hence q must have dimension length −2 .
In the present paper, we have tried to work out the one-loop properties pertaining to the problem defined by Eqs. (1.3) and (1.4). In the physics-oriented literature, one-loop calculations are more frequently performed in the case of quantum field theories, but the quantum mechanical framework is already of interest [2], and may provide valuable information on the behaviour of solutions of elliptic equations under a scale dilation. Such a property is neatly described by the ζ(0) value, where ζ is the spectral (or generalized) ζ-function of the elliptic operator A under consideration, defined by a heat kernel K(ξ, η; t), whose diagonal K(ξ, ξ; t) yields, upon integration over the whole region B R in (1.1), the integrated heat kernel (for gauge theories, the trace to be integrated is instead the fibre trace of the heat-kernel diagonal) which has, as t → 0 + , the asymptotic expansion [3][4][5] In our 2-dimensional region B R , the method used in Ref. [3] considers the so-called spectral ζ-function at large h, i.e. (λ n and h being dimensionless in (2.3)) which is related to the integrated heat kernel (2.1) by the identity If one now inserts into the left-hand side of (2.4) the asymptotic expansion (2.2), one finds On the other hand, on considering the equation (1.6), which is the equation obeyed by the eigenvalues E = h 2 R 2 by virtue of the boundary conditions, one has also the identity (see [3] and our appendix) (2.7) In the course of performing sums over all positive and negative values of l, it is helpful to exploit the identity as well as the even nature of β l as a function of l (see section 4). This implies that a real root of G l with positive l is also a real root of G l with negative l, because We can therefore limit ourselves to summing over positive values of l, writing that, in (2.6), − contribution of (l = 0). (2.10) Following Ref. [3], we use in (2.6) and (2.10) the uniform asymptotic expansion of J l and its first derivative J ′ l , which involve the polynomials u k and v k occurring below and in the appendix. On denoting by C a constant, and defining α l (ih) ≡ √ l 2 + h 2 , we obtain where, having defined the variable and exploiting the following among the many Debye-Olver polynomials [6]: which implies the simple but very helpful relations By virtue of (2.6) and (2.11) the three contributions independent of β l are obtained by applying twice the operator 1 2h d dh to the first line on the right-hand side of (2.11). For this purpose, we need the following identities: Thus, upon applying the split (2.10), the terms independent of β l are obtained by taking twice (from the factor 2 multiplying ∞ l=0 in (2.10)) the following sums: The sums (3.5)-(3.7) can be studied in a thorough way with the help of the Euler-Maclaurin summation formula [7]. This states that, if f is a real-or complex-valued function defined on [0, ∞), and if its derivatives of even order are absolutely integrable on (0, ∞), one has, for n = 1, 2, ..., where the Bernoulli numbersB s are defined by the expansion while the remainder R m (n) is majorized according to [7] |R As n approaches ∞, Eq. (3.8) provides a very useful asymptotic expansion for the desired sum of the series, i.e.
The integral on the right-hand side of (3.11) can be evaluated or studied in a qualitative way, while the derivatives of odd order at 0 and at ∞ can be obtained in a systematic way.
We refer the reader to the last chapter of the book by Hardy [8] for a thorough analysis of the Euler-Maclaurin formula.
For our purposes, after having re-expressed σ 1 in the form where we have set 14) we now take the limit as n → ∞ in Eq.
We then find that, in (3.11), only the first derivative of F at l = 0 gives a contribution proportional to h −4 , and indeed equal to B. Contributions of σ 2 and σ 3 We now rely again upon the limit as n → ∞ of Eq. (3.8). By virtue of (3.11), only half the value at l = 0 of 1 2 α −4 l contributes to the h −4 term in σ 2 , i.e.
whereas σ 3 gives a vanishing contribution to the term proprtional to h −4 We now aim at studying the contribution of the second line of the asymptotic expansion (2.11) to Eq. (2.6). For this purpose, on the one hand we denote by Ω all terms added to 1 within the square brackets in (2.11), and consider the asymptotic expansion where On the other hand, it is clear that no further progress can be made without explicit forms of the β l coefficients. For example, we find from (1.7) and (1.9) supplemented, in principle, by infinitely many other terms, i.e.
In the formula (4.10), it is helpful to use (4.3) where we re-express a 1 , b 1 and b 2 in the form where the numerical coefficients a 1r , b 1r and b 2r can be read off from (2.13), (2.15) and (2.16).
Thus, a patient calculation shows that (the superscript (l) denotes here dependence on l) where κ (l) and hence, by repeated application of (3.1), we obtain from (4.10) and (4.13) a result which agrees with the application of Eq. (3.11). Moreover, the asymptotic expansion (3.11) implies that the first sum on the right-hand side of (4.9) is equal to where, on defining Y ≡ y h , we find It is clear once more that h plays the role of regularizing parameter, since without it the integral (4.21), and many of the integrals below, would not exist. Last, the third sum on the right-hand side of (4.9) is again studied with the help of (3.11), and we find where and hence (4.24) does not contribute to h −4 either. Furthermore, the last two sums on the right-hand side of (4.17), which contain the effect of β l , with the particular choice (4.6) for this coefficient are found to involve where − 1 2 is the term denoted in (2.10) by minus the contribution of (l = 0), and arises from σ 2 in (3.6).

C. Other choices of β l
For a generic choice of β l coefficient, our Eq. (4.9) gets replaced by If β l is taken in the form (4.5), we find, by virtue of (3.11), the asymptotic expansion (K n being the standard notation for modified Bessel functions of second kind and order n) which has no term proportional to h −4 , while β l in the form (4.8) leads to and the integral on the right-hand side of (4.31) does not have a term proportional to h −4 at large h, either. Moreover, the last two sums in (4.17) suggest introducing the notation Now we find, for the two choices of β l according to (4.5) or (4.8),

34)
(4.37) Among the integrals occurring in (4.34)-(4.37), only those on the right-hand side of (4.34) might give rise to a contribution proportional to h −4 , because The resulting integrand ϕ(z) has a third-order pole at z = i h 2 , with residue which is one of the three terms occurring in (4.38).

V. CONCLUDING REMARKS
As far as we know, our paper has performed the first ζ(0) calculation with nonlocal boundary conditions in quantum mechanics. We have proved explicitly that at least three choices of kernel in the nonlocal boundary operator exist (see (4.5), (4.6) and ( It remains to be seen whether, for yet other choices of β l in (4.17) and (4.29), a contribution to ζ(0) can be found which is compatible with (1.7), (1.9), the assumption q ∈ L 1 (R) ∩ L 2 (R) and condition (A7) of the Appendix. This means having to study the sums The mere recourse to the formula [3] suggests a negative answer, because for example, upon requiring β l (β l − 1) = κ l, κ > 0, one finds the positive root which has a growth rate incompatible with (1.7) and (1.9), if one looks for functions q ∈ L 1 (R) ∩ L 2 (R). However, the general starting point should be the asymptotic expansion inspired by (3.11), i.e.
and the application of (5.3) to (5.1) deserves further work.
Furthermore, with the help of the experience gained from our calculation, it should be possible in the near future to investigate the one-loop properties of Euclidean quantum gravity with nonlocal boundary conditions, along the lines of Refs. [9,10], where it was suggested that the Universe might become classical, after a quantum origin, by virtue of a wave function that decays as it occurs in the case of quantum mechanical surface states [1] with nonlocal boundary conditions. In order to help the general reader and stress the relevance of our work, we find it also appropriate to write what follows.
The use of spectral ζ-functions has led to several important developments over the last decades, with application to partition functions of strings and membranes, Casimir effect, relation between the generalized Pauli-Villars and covariant regularizations, spontaneous compactification in two-dimensional quantum gravity, stability of the rigid membrane, topological symmetry breaking in self-interacting theories, functional determinants for bosonic and fermionic fields, ground-state energy under the influence of external fields, Bose-Einstein condensation of ideal Bose gases under external conditions [11][12][13]. Furthermore, the work in Ref. [14] obtained heat-kernel coefficients of the Laplace operator on the D-dimensional ball, Ref. [15] evaluated Casimir energies for massive fields in the bag, while the Casimir energy for a massive fermionic quantum field with a spherical boundary was obtained in Ref. [16].
The regularization used in our paper, which relies as we said on the pioneering work in Ref. [3], was applied successfully in Ref. [17] to the first correct calculation of one-loop conformal anomaly for a massless fermionic field with local boundary conditions, at a time when the powerful geometric formulas in Ref. [4] were not in final form. It is therefore encouraging that, after almost thirty years, such a regularization technique is still useful in opening yet new perspectives. For example, it would be interesting to apply it to the Casimir effect [13] in cosmological backgrounds, and to establish a correspondence with yet other models in Ref. [13]. For the gravitational field, one cannot generalize the Schröder scheme by simply studying the eigenvalue problem for an operator of Laplace type on metric perturbations h µν , subject to nonlocal boundary conditions. The reason is that boundary conditions invariant under infinitesimal diffeomorphisms on h µν take, in field-theoretic language, the form [18] (Π h) ij where Π is a projection operator that picks out only spatial components of h µν , while Φ µ (h) is the gauge-fixing functional. Thus, nonlocality in the boundary conditions can only result from the gauge-fixing term; but then the invertible operator on metric perturbations is no longer differential but it belongs to the broader class of pseudodifferential operators [9,10], for which a ζ(0) calculation is still a challenging task. Thus, new exciting goals are in sight in the area of physical applications of spectral ζ-functions, bearing also in mind the enlightening assessment in Ref. [19]. and hence