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On metric graphs the trace of the heat kernel was first studied by Roth, [50], deriving an exact (Selberg-like) trace formula for it. After that, the (one-particle) heat kernel has experienced an accelerated attention and analogous questions has been asked to the manifold case, [21, 44, 28, 35, 42, 48, 43, 7, 15]. The motivation of studying contact interactions in many-particle physics for one-dimensional systems dates back to the famous Lieb-Liniger model, [38], used to test Bogoliubov’s perturbation theory for Bose gases. Since then many- particle contact interactions are well established in one dimensions and graphs for testing and modeling famous phenomenons such as superconductivity or Bose-Einstein condensation, [20, 51, 11, 10, 30, 12, 18]. For the importance of one-dimensional contact interactions in physics and recent experimental implementations we refer to [14]. Those exciting results inspired us to consider as a very first example the small-t asymptotics of the trace of the heat kernel for a simple but non-standard Lieb-Liniger type system of a singular but non-constant contact interaction on the real line, R. More precisely, the operator which we consider is the one-dimensional two-particle Schr¨odinger operator given by the formal expression (1) ∆ := ∂2 ∂2 + ρ(x ,x )δ(x x ) , − ρ − x1 − x2 1 2 1 − 2 with L2(R2) as the two-particle Hilbert space and a δ-potential modulated by a potential ρ. We assume that the modulating potential, ρ, is smooth and possesses compact support. The heat semi-group, et∆ρ , satisfies ∂ et∆ρ = ∆ in a strong sense, and the heat kernel, t |t=0 ρ kρ(t)( , ), t R, is the integral kernel generating et∆ρ by · · ∈ (2) (et∆ρ ψ)(x)= kρ(t)(x, y)ψ(y)dy .
RZ2 However, since the underlying configuration space is non-compact the Schr¨odinger operator (1) possesses an essential spectrum, and we have to regularize the heat semi-group in order to make it a trace-class operator. In this paper, we follow the method of [7] and we don’t derive a parametrix expansion for the heat kernel. Instead, we exploit that the resolvent and the heat semi-group are related via a Laplace transformation. This approach is similar to [32], where the authors used the so-called Agmon-Kannai method to derive an asymptotic expansion for the resolvent kernel of the corresponding Schr¨odinger operator. The Agmon-Kannai method, in turn, is a tool to obtain a series representation of the resolvent with operator-valued coefficients and is based on a recursive construction involving commutators of the free and the full resolvent of the considered operator, [46]. We don’t use the Agmon-Kannai method here, but we exploit that the resolvent of our system allows a rather explicit representation in terms of suitable integral operators which is based on a more-general formula of Kre˘ın. We start by first establishing an asymptotic expansion for large but negative energies of the regularized trace of the resolvent. Then, we use a suitable version of the converse Watson lemma to deduce the small-t asymptotics of the regularized trace of the heat semi-group (and heat kernel). In general, our method is taking advantage of the symmetry properties of the underlying system, and the convenience of our method is that it allows a compact, explicit and quick formulation of the heat-kernel coefficients. Finally, we refer to Appendix A, where we recall standard notations and definitions used in this paper. HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 3
2. Preliminaries We begin our investigations by implementing a rigorous version of the formal Schr¨odinger operator (1). To obtain a well-defined operator in L2(R2) it is convenient to associate with (1) a quadratic form being complete and semi-bounded from below. Then, there is a unique self-adjoint operator corresponding to the quadratic form, [55, Section 4.2], which may be regarded as a rigorous version of (1). To get the quadratic form of the operator (1) we first observe that the modulating potential ρ only needs to be known on the diagonal (one-dimensional submanifold) (3) D := (x,x) : x R R2 { ∈ }⊂ due to the δ-potential interaction. Now, using the identification ρ(x,x) = ρ(x) we restrict R ourselves to compactly supported and smooth potentials, i.e., ρ C0∞( ). To identify the associated quadratic form and operator we proceed as in [9, Section∈ 3.1], replacing the interval [0, 1] with R to obtain our case related to (1). We follow the steps of [9, Section 3.1] and to 2 do so, we use D and partition R into the two disjoint open sets D+ and D by − 2 (4) R = D ˙ D ˙ D+ , D+ = (x1,x2) : x1
1 1 (5) bv : H (D ) H 2 (D). ± ± → Both above maps are continuous linear maps. In the same way, the gradients (6) : H2(D ) H1(D ) H1(D ) ∇ ± → ± ⊕ ± are well-defined and continuous maps. Therefore, the inward normal derivatives, ∂n, w.r.t. the boundary D of the domains D act as ± 2 2 1 1 (7) ∂n : H (D+) H (D ) H 2 (D) H 2 (D) , ⊕ − → ⊕ and are well-defined by
(8) ∂n(ψ+ ψ ) := ψn,+ ψn, , ⊕ − ⊕ − and
(9) ψn, := (∂x1 ∂x2 )ψ . ± ∓ − ± With these technical tools at hand, we may now use [9, p. 6] allowing a proper identification of the two-particle and one-dimensional Schr¨odinger operator in (1) with a one-particle and two-dimensional operator acting on R2. For the readers convenience we denote this operator by ∆ρ as well. The functions of the corresponding operator domain, D(∆ρ), obey the − 2 2 following regularity and boundary conditions. If ψ D(∆ρ) then ψ H (D+) H (D ) L2(R2) and the following boundary conditions are satisfied∈ in a L2-sense,∈ [9, p.⊕ 6]: − ⊂
(10) bv+ ψ = bv ψ =: Ψ , ψn,+ + ψn, = ρΨ . − − 1 2 Moreover, by [9, p. 6] the operator ∆ρ is associated with the quadratic form (qρ,H (R )) defined by −
(11) q (ψ) := ψ, ψ R2 dx + ρΨdx , ρ h∇ ∇ i RZ2 DZ 4 SEBASTIAN EGGER
1 2 where we used (4). On the other hand, given (qρ,H (R )) then the associated operator is ∆ρ and is self-adjoint and bounded from below, [13, Theorem 4.2]. We denote by λmin,ρ := inf− λ R : λ σ( ∆ ) the bottom of the spectrum σ( ∆ ) of the Schr¨odinger operator, { ∈ ∈ − ρ } − ρ and we recall the well-known fact λmin,0 = 0. 3. The resolvent kernel At the beginning of this section, we derive an explicit expression of the resolvent, λ / σ( ∆ ), ∈ − ρ 1 (12) R (λ) := ( ∆ λ)− . ρ − ρ − To do so, we follow [13], and in the following, we choose √λ = k C such that Im k> 0. We introduce the integral kernels ∈ 1 (13) G (k)(x ,x ,y ,y ) := K ( ik (x y )2 + (x y )2) , 0 1 2 1 2 2π 0 − 1 − 1 2 − 2 and p 1 (14) g(k)(x,y) := K ( i√2k x y ) . 2π 0 − | − | 2 2 The integral kernel G0(k) corresponds for Im k> 0 to a bounded operator R0(k) : L (R ) L2(R2) and g(k) to a bounded operator g(k) : L2(R) L2(R), respectively. That may be→ → deduced from the asymptotic behavior of the K0-Bessel function for large arguments, [39, p. 139]. It is worth mentioning that the logarithmic singularity of the K0-Bessel function at the origin, [39, p, 65], doesn’t affect the boundedness of R0 and g due to Young’s inequality. 2 Note that R0(k) is the (free) resolvent of the pure Laplacian on R . To write down the resolvent explicitly we need one more integral operator connecting L2(R2) and L2(R). We introduce 1 (15) b(k)(x,y ,y ) := K ( ik (x y )2 + (x y )2) , 1 2 2π 0 − − 1 − 2 and those integral kernel generates for Im k > 0p a bounded operator b(k) : L2(R2) L2(R). R → R Finally, we make the simple observation that any potential ρ C0∞( ), or only ρ L∞( ), generates a bounded multiplication operator ρ : L2(R) L2(R∈). ∈ With these operators at hand, we now invoke [13, Corollary→ 2.1] saying that the resolvent Rρ(λ) may be written as 1 (16) R (λ)= R (k) b k ∗ (1 + ρg(k))− ρb(k) , ρ 0 − √ π where we put k = λ. In the following, it is convenient to denote by Cα, α < 2 , the cone around the positive imaginary axis iR+ with opening angle 2α, i.e., π (17) C := z C : arg(z) < α . α ∈ | − 2 | Then, for k C , α< π , and k largen enough one has, [13, Corollaryo 2.2], ∈ α 2 | | (18) ρg(k) < 1 . k k reg π Now, we introduce the regularized resolvent Rρ (k), k Cα, α < 2 , and k sufficiently reg 2 2 ∈ | | large, defined as Rρ (k) := Rρ(k ) R0(k ). Due to the semi-boundedness of qρ (and q0) reg − π the operator Rρ (k) exists for k Cα, α < 2 and k sufficiently large. We will show that reg ∈ | | Rρ (k) is also a trace class operator. For this, it is advantageous to ’shift’ a square root of the potential in (16) from right to left. This will eventually reveal that the supports of the HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 5
integral kernels are compact w.r.t. appropriate variables, and will be exploited to estimate reg the integrals from above. Specifically, the following rearrangement of Rρ (k) is possible. Lemma 3.1. For k C , α< π , and k sufficiently large, we have ∈ α 2 | | reg ∗ 1 (19) R (k)= ρ b k (1 + sgn ρ ρ g(k) ρ )− sgn ρ ρ b(k) . ρ − | | | | | | | | p 1 p p p Proof. Using (18) we may write (1 + ρg(k))− as a Neumann series. Now, (20) (ρg(k))nρ = ρ (sgn ρ ρ g(k) ρ )n sgn ρ ρ | | | | | | | | for every n N proves the claim. p p p p ∈ 0 3.1. Trace class property of the regularized resolvent. Now, we are in the position to prove that the regularized resolvent is actually a trace class operator.
π reg Proposition 3.2. For k Cα, α < 2 , and k sufficiently large, Rρ (k) is a trace class operator. ∈ | | reg Proof. We want to employ [55, Satz 3.23] saying that it’s enough to show that Rρ (k) can be factorized as reg (21) Rρ (k)= AB by two Hilbert-Schmidt operators A : L2(R) L2(R2) and B : L2(R2) L2(R). Looking at (19) we are tempted to identify → →
(22) A = ρ b k ∗ , − | | and p 1 (23) B = (1 + sgn ρ ρ g(k) ρ )− sgn ρ ρ b(k) . | | | | | | Indeed, by [55, Satz 3.18] the Hilbert-Schmidtp p property isp closed under taking the adjoint. Moreover, by [55, Satz 3.20] we may neglect the operator (1 + sgn ρ ρ g(k) ρ ) 1 sgn ρ, | | | | − and it remains to show that ρ b (k) is of Hilbert-Schmidt class. We proceed to show that | | p p the integral kernel of ρ b (k) is in L2(R R2) which then proves by [55, Satz 3.19] the p claim. We recall (15) and| calculate| × p (2π)2 ρ(x) b (k) (x,y ,y ) 2dxdy dy | || 1 2 | 1 2 ZR RZ2
= ρ(x) K ik (x y )2 + (x y )2 2dxdy dy (24) | || 0 − − 1 − 2 | 1 2 Z Z0 supp ρ BR( ) p + ρ(x) K ik (x y )2 + (x y )2 2dxdy dy , | || 0 − − 1 − 2 | 1 2 2 suppZ ρ R BZR(0) \ p
where BR(0) is a ball with sufficient large radius R such that (25) x supp ρ y = x or y = x ∈ ⇒ 1 6 2 6 6 SEBASTIAN EGGER
2 holds in R BR(0). Such a radius, R, exists as the support of ρ is bounded. By the same reason and due\ to the asymptotic behavior (80) we have that
ρ(x) K ik (x y )2 + (x y )2 2dxdy dy | || 0 − − 1 − 2 | 1 2 Z Z0 supp ρ BR( ) p (26) with some C > 0. By (81) a similarly estimates to (26) also holds for the second integral on the r.h.s. in (24) proving the claim. In order to evaluate the trace of the heat kernel, we want to employ a generalized version of Mercer’s theorem. To achieve this, we first have to show that the heat kernel exists and then to work out suitable continuity and decay properties satisfied by that integral kernel. π reg Lemma 3.3. Given k Cα, α< 2 , and k sufficiently large, Rρ (k) is an integral operator with an continuous and∈ exponentially decaying| | kernel for large arguments. Proof. First, we choose for our convenience an appropriate ik R+, and the remaining cases may then be similarly proven. With a similar argument as− in∈ the proof of Proposition 3.2, using the (absolute) integrability of the logarithm, we may regard ρ b(k) as a map | | p 2 2 (27) ˜b : (R , R2 ) L (R) , k · k → defined by (28) (˜b(y ,y ))(x) := ρ(x) b(k)(x,y ,y ) . 1 2 | | 1 2 p To see that ˜b is continuous we first take into account that the support of ρ is finite and therefore we only have to investigate singular points of (28) w.r.t. the argument. Looking at (15) we see that the singularity is in the logarithm of (80) where x = y1 = y2. We may choose y1 = y2 = 0, y1′ = y2′ =: y′ > 0 and the other cases are similar. We are going to use that for every ǫ> 0 there is a δ(ǫ) such that (0, 0) (y′,y′) R2 δ(ǫ), and (x,x) / B2δ(ǫ)(0, 0) (29) k − k ≤ ∈ ln (x,x) R2 ln (y′ x,y′ x) R2 ǫ . ⇒ | k k − k − − k |≤ HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 7 Hence, choosing ǫ suitable small gives 2 ln (x,x) R2 ln (y′ x,y′ x) R2 dx | k k − k − − k | suppZ ρ 2 = ln (x,x) R2 ln (y′ x,y′ x) R2 dx | k k − k − − k | (x,x)/B Z(0,0) supp ρ { ∈ 2δ(ǫ) }∩ 2 + ln (x,x) R2 ln (y′ x,y′ x) R2 dx | k k − k − − k | (30) (x,x) B Z(0,0) supp ρ { ∈ 2δ(ǫ) }∩ 2δ(ǫ) y′ 2 Cǫ + C′ ln 1 dx ≤ | | − x || Z0 ∞ y′ 2 , Cǫ + C′ ln 1 x dx = Cǫ + O(y′), ≤ x2 | | − || y′(2δZ(ǫ))−1 where in the last line we performed the substitution x y , and we used [49, pp. 240,241]. → x Observing that C, C′ > 0 only depend on the size of supp ρ 1 and choosing y′ sufficiently close 0 proves the claim. | | 2 In the same way as above, we view ( ρ b(k))∗ as a continuous map from (R , R2 ) to 2 | | + k · k L (R). This gives the same integral kernel ˜b as in (28) since ik R and then the K0- p2 − ∈ Bessel function is real valued, but the R -variables are now indicated by (x1,x2). We denote for convenience, see (19), 1 2 2 (31) q = (1 + sgn ρ ρ g(k) ρ )− : L (R) L (R) , | | | | → reg and since q is continuous we observep that thep integral kernel rρ(k)((x1,x2), (y1,y2)) of Rρ (k) is given by (32) r (k)((x ,x ), (y ,y )) = ˜b(x ,x ),q˜b(y ,y ) 2 R . ρ 1 2 1 2 h 1 2 1 2 iL ( ) Now, the algebraic identity r (k)((x ,x ), (y ,y )) r (k)((x′ ,x′ ), (y′ ,y )′) ρ 1 2 1 2 − ρ 1 2 1 2 (33) = ˜b(x ,x ),q˜b(y ,y ) 2 R ˜b(x′ ,x′ ),q˜b(y′ ,y′ ) 2 R h 1 2 1 2 iL ( ) − h 1 2 1 2 iL ( ) = (˜b(x ,x ) ˜b(x′ ,x′ )),q˜b(y ,y ) 2 R + ˜b(x′ ,x′ ),q(˜b(y ,y ) ˜b(y′ ,y′ )) 2 R , h 1 2 − 1 2 1 2 iL ( ) h 1 2 1 2 − 1 2 iL ( ) and a suitable application of H¨older’s inequality prove the first part of the claim. To see the exponential decay we recall that the operators ˜b in (32) involve the kernel ρ(x) b(k)(x,y ,y ) given in (15). As before, it’s enough to assume x supp ρ, and since | | 1 2 ∈ the support of ρ is finite we have for sufficiently large (y ,y ) R2 the inequality p k 1 2 k 1 (34) (y x,y x) R2 > (y ,y ) R2 . k 1 − 2 − k 2k 1 2 k We obtain, using H¨older’s inequality, ˜ ˜ rρ(k)((x1,x2), (y1,y2)) = b(x1,x2),qb(y1,y2) L2(R) (35) | | |h i | ˜b(x ,x ) 2 R q ˜b(y ,y ) 2 R , ≤ k 1 2 kL ( )k kk 1 2 kL ( ) 8 SEBASTIAN EGGER 2 where q is the L -operator norm of q. For sufficiently large (y1,y2) R2 we may use (81) for (15).k k Together with (34) we then obtain k k 2 2 k√(x y1) +(x y2) ˜b(y ,y ) 2 R C e− − − dx k 1 2 kL ( ) ≤ (36) suppZ ρ k 1 √y2+y2 C′e− 2 1 2 ≤ with some C, C′ > 0. Plugging (36) in (35) proves the second part of the claim. Having established the existence of a (continuous) integral kernel we take over the notation π of the above proof. We denote for k Cα, α< 2 , and k sufficiently large, the integral kernel reg ∈ | | of Rρ (k) by rρ(k)( , ). The following proposition· · tells us how we may calculate the trace of the regularized resol- vent. π Proposition 3.4. The trace of the regularized resolvent may be calculated by, k Cα, α< 2 , and k sufficiently large, ∈ | | reg ρ (37) Tr Rρ (k)= rreg(k)(x, x)dx . RZ2 reg Proof. The claim follows by an application of [24, p. 117] saying that the properties of Rρ (k) ρ and of its kernel rreg(k) derived in Proposition 3.2 and Lemma 3.3 are sufficient to deduce the claim. The above (trace-class) result tempts us to define the regularized trace of the resolvent as the trace of the regularized resolvent Definition 3.5. The regularized trace of the resolvent Rρ(λ) is defined as reg √ (38) Trreg Rρ(λ) := Tr Rρ ( λ) , with λ such that k = √λ satisfies the assumption of Proposition 3.4. 3.2. An asymptotic analysis of the trace of the regularized resolvent. To determine the asymptotic expansion of the trace of the regularized resolvent we have to introduce a couple of auxiliary objects and notations permitting a closed presentation. We start with defining a diffeomorphism, n N , ∈ 0 (39) φ : Rn+1 Rn+1 , → by wl = φ(y0,...,yn)l = yl yl+1 , l 0,...,n 1 , (40) − ∈ { − } wn = φ(y0,...,yn)n = yn + y0 . Note that for n = 0 we have w0 = 2y0. The inverse map (diffeomorphism) 1 n+1 n+1 (41) φ− : R R → reads as n l 1 1 1 − (42) y = φ− (w , . . . , w ) = [ w w ], l 0,...,n , l 0 n l 2 m − m ∈ { } mX=l Xl=0 HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 9 1 and, in particular, for n = 0 we have y0 = 2 w0. Moreover, we are going to utilize the following (combinatorial) set of maps, n N , ∈ 0 s : s : 0,...,n 1 1, 1 , n N , (43) Sn := { { − } → { − }} ∈ , n = 0 . (∅ In addition, we employ the multy-index notation n (44) α := (α0,...,αn 1) , αl N0, l 0,...,n 1 . − ∈ ∈ { − } together with n 1 − (45) αn := α . | | l Xl=0 We remark here that αn is only defined for n = 0. R 1 6 N For any ρ C0∞( ) the maps φ− in (41) and s Sn, n 0, in (43) are employed to gen- ∈ ∈ Rn∈+1 R Rn+1 R erate a smooth and compactly supported map ρn,s from to , i.e., ρn,s C0∞( , ), defined by, (w , . . . , w ) Rn+1, ∈ 0 n ∈ n 1 ρ((φ− (s(0)w0,...,s(n 1)wn 1, wn))l) , n N , (46) ρ (w , . . . , w ) := − − ∈ n,s 0 n l=0 Qw0 ρ( 2 ) , n = 0 . Our asymptotic analysis also deploys the following notations of partial derivatives of ρn,s, using (46) and (44), αn ∂| | n (47) ∂α ρn,s(w0, . . . , wn 1, wn) := ( α0 αn−1 ρn,s)(w0, . . . , wn 1, wn) . − ∂w0 ...∂wn−1 − R Finally, fixing ρ C0∞( ), the following functions will turn out of particular interest (48) ∈ cαn,s,l dξdt0...dtn−1 N 3 n αn +1 , n , 2 α +1 n−1 +n R (1+ξ ) 2 (√2((cosh(t0)+iξs(n 1))) 0 ...(√2(cosh(tn−1)+iξs(n 1))) ∈ R − − R Rdξ = 3 , n = 0 , l = 0 , R (1+ξ2) 2 R0 , else , and, l N0, ∈ cαn,s,l ∂αn ρn,s(0,..., 0,y)dy, n N , |αn|=l, R ∈ sP∈Sn R (49) bn,l := 2ρ( y )dy, n = 0, l = 0 , 2 R R0, else , where we incorporated [25, 3.251 11.] and S = , (43). We remark that by [25, 3.252 11.] 0 ∅ (50) cα0,s,0 = 2 . Equipped with the above identities, we are now able to determine the large-λ asymptotic expansion of the regularized trace of the resolvent. For this, we remind that Cα is a sector with opening angle α around the positive imaginary axis, (17). 10 SEBASTIAN EGGER Theorem 3.6. The regularized trace of the resolvent Trreg Rρ(λ) possesses for λ and √ π | |→∞ k := λ Cα with α < 2 a complete asymptotic expansion in integer powers of k of the form − ∈ ∞ ( m +1) (51) Tr R ( λ) b λ− 2 , reg ρ − ∼ m mX=0 where the coefficients bm are given by 1 (n+1) (52) b = ( 2π)− b . m 8 n,l n,l, − l+Xn=m The first two coefficients read as 1 √2 b = ρ(y)dx , b = ρ(y)2dy . (53) 0 −4π 1 32 ZR ZR Remark 3.7. The condition on λ implies that λ has to be in a cone around the positive axis R+ with opening angle smaller than 2π. Moreover, we point out that for m 2 integrals of ≥ derivatives of ρ appear in (52) for bm. Proof. For our convenience we choose k = √ λ and consider only the case k˜ := ik R+, k − − ∈ | | sufficiently large. The general case k Cα may be treated analogously. We also remark that in the following every interchange of the∈ order of integration is justified by Fubini’s theorem, [5, 23.7 Corollary]. 1 We are going to use the resolvent representation (16). First, we expand (1 + ρg(k))− into a Neumann series and attain 1 Tr R ( λ)= Tr(b k ∗ (1 + ρg(k))− ρb(k)) reg ρ − − (54) ∞ = ( 1)n+1 Tr(b k ∗ (ρg(k))nρb(k)) . − n=0 X It is possible to get a large-k asymptotic expansion of Tr(b k ∗ (ρg(k))nρb(k)) for every n N , and then we rearrange the terms w.r.t. powers of k˜ in (54). To see the first part of 0 the∈ afore mentioned, we use the integral kernels (14) and (15) for b(k) and g(k), and we use the notation Y = (y ,y ,...,y ) Rn+1, giving 0 1 n ∈ Tr(b k ∗ (ρg(k))nρb(k)) (n+2) 2 2 = (2 π)− K (k˜ (x y ) + (x y ) )ρ(y ) . . . 0 1 − 0 2 − 0 0 ∗ (55) RZ2 RnZ+1 p . . . K0(√2k˜ y0 y1 )ρ(yn 1)K0(√2k˜ yn 1 yn ) . . . ∗ | − | − | − − | ∗ . . . ρ(y )K (k˜ (y x )2 + (y x )2)dY dx dx . ∗ n 0 n − 1 n − 2 1 2 Now, by slight abuse of notation,p we apply the (orthogonal) coordinate transformation 1 (56) (x1,x2) (x1 + x2,x1 x2) , → √2 − HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 11 +n followed by an insertion of (82) in (55), using the notation T = (t0,...,tn 1) R , and (92) in (55) which yields − ∈ Tr(b k ∗ (ρg(k))nρb(k)) (n+2) 2 2 = (2 π)− K (k˜ (x √2y ) + x )ρ(y ) . . . 0 1 − 0 2 0 ∗ RZn RnZ+1 RZ2 q √2k˜ y0 y1 cosh(t) √2k˜ yn−1 yn cosh(t) . . . e− | − | ρ(yn 1)e− | − | . . . ∗ − ∗ (57) . . . ρ(y )K (k˜ (√2y x )2 + x2)dx dx dY dT ∗ n 0 n − 1 2 1 2 q i√2kξ˜ (y0 yn) π (n+2) e− − √2k˜ y0 y1 cosh(t0) = (2π)− ρ(y0)e− | − | . . . 2 3 2k˜ (1 + ξ2) 2 ∗ RZ+n ZR RnZ+1 √2k˜ yn−1 yn cosh(tn−1) . . . ρ(yn 1)e− | − | ρ(yn)dY dξdT . ∗ − For n = 0 we directly calculate the trace and obtain 1 1 Tr(b k ∗ ρb(k)) = ρ(y0)dξdy0 2 3 8πk˜ (1 + ξ2) 2 ZR ZR (58) 1 = ρ(y0)dy0 , 4πk˜2 ZR where in the last line we used [25, 3.252 11.]. For n = 0 we want to invoke for our asymptotic analysis the integration by parts method. For this,6 it is expedient to first use (41) as an appropriate substitution of variables. This transformation implies n 1 − (59) w = y y , l 0 − n Xl=0 1 and the determinant of the Jacobian det J(φ− ) of this coordinate transformation is constant, 1 1 and given by det J(φ− ) = 2 . Hence, changing the variables in (57), using (40), (59) and 12 SEBASTIAN EGGER n W = (w0, . . . , wn 1) R , gives − ∈ ikξ˜ (y0 yn) e− − √2k˜ y0 y1 cosh(t0) 3 ρ(y0)e− | − | . . . (1 + ξ2) 2 ∗ RZ+n ZR RnZ+1 √2k˜ yn−1 yn cosh(tn−1) . . . ρ(yn 1)e− | − | ρ(yn)dY dξdT ∗ − 1 1 1 √2k˜( w0 cosh(t0)+iξw0) = 3 ρ(φ− (w0, . . . , wn)0)e− | | . . . 2 (1 + ξ2) 2 ∗ RZ+n ZR ZR RnZ+1 1 √2k˜( wn−1 cosh(tn−1)+iξwn−1) . . . ρ(φ− (w0, . . . , wn)n 1)e− | | ∗ − ∗ 1 ρ(φ− (w , . . . , w ) )dW dw dξdT ∗ 0 n n n (60) 1 1 = 3 . . . 2 2 2 ∗ s Sn n n (1 + ξ ) X∈ RZ+ ZR ZR RZ √2k˜(sgn(w0) cosh(t0)+iξ)w0 √2k˜(sgn(wn−1) cosh(tn−1)+iξ)wn−1 . . . e− . . . e− ∗ ∗ n 1 ρ((φ− (w0, . . . , wn 1, wn))l)dW dwndξdT ∗ − Yl=0 1 1 = 3 ρn,s(w0, . . . , wn) 2 (1 + ξ2) 2 ∗ s Sn +n R R +n X∈ RZ Z Z RZ √2k˜(cosh(t0)+iξs(0))w0 √2k˜(cosh(tn−1)+iξs(n 1))wn−1 e− . . . e− − dW dw dξdT . ∗ n It is for the following integration by parts method important that in the last line of (60) only the variables wl with l 0,...,n 1 appear in the exponential function. We perform an integration by parts w.r.t.∈ { the W variables.− } The obtained terms which don’t possess any W integrals anymore may then be ordered w.r.t. powers of k˜. Using our notation (45) and (47), we obtain, l N , ∈ 0 1 3 ρn,s(w0, . . . , wn) (1 + ξ2) 2 ∗ RZ+n ZR ZR RZ+n √2k˜(cosh(t0)+iξρ(0))w0 √2k˜(cosh(tn−1)+iξs(n 1))wn−1 e− . . . e− − dW dw dξdT ∗ n αn n 1 ˜ n (61) = k−| |− 3 ∂α ρn,s(0,..., 0, wn) 2 αn, (1 + ξ ) 2 ∗ RZ+n ZR RZ+ |αXn|≤l dwndξdT αn+1 αn +1 ∗ (√2(cosh(t0) + iξs(n 1))) 0 . . . (√2(cosh(tn 1) + iξs(n 1))) n−1 − − − (l+n+1) + O(k˜− ) . The last line follows by integration by parts and the observation that every partial derivative evaluated at (0,..., 0, wn) is generated exactly once. Moreover, we used that the T and ξ (l n+1) integration don’t affect the order estimate O(k˜− − ). HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 13 Now, sorting (61) w.r.t. powers of k˜ by making use of our definitions (48) and (49) we get l n 1 1 1 (l+n+1) (62) Tr(b k ∗ (ρg(k)) ρb(k)) = bn,l′ + O(k˜− ) . ˜2 (2π)n+1 ˜l′+n 8k l′=0 k X Note that (62) is conform with (58) for l = n = 0. Finally, we use (54) plugging in there the asymptotic expansion (62), and we sort the obtained sum again w.r.t. powers of k˜. This then yields the asymptotic expansion w.r.t. powers of ik˜ = k = √ λ, (52). The calculations of the first two coefficients are as follows − 1 1 1 (63) b = b , b = b + b , 0 −16π 0,0 1 −16π 0,1 32π2 1,0 and it remains to calculate the three coefficients in (63) by (49). The first two ones are simple and given by, (49), (64) b0,0 = 4 ρ(y)dy , b0,1 = 0 , ZR where we used (50). For b1,0 we get by (49) n n (65) b1,0 = cα ,s+,0 ρ0,s+ (0,y)dy + cα ,s−,0 ρ0,s− (0,y)dy , ZR ZR 1 where s+(0) := 1 and s (0) := 1. We have for α = 0 the simple relations − − | | 1 (66) ρ (0,y) = (ρ( y))2 , 0,s± 2 and, α1 = 0, | | 1 1 cosh(t0) iξ 1 (67) cα ,s±,0 = 3 2 ∓ 2 dt0dξ . √2 (ξ2 + 1) 2 cosh(t0) + ξ RZ+ ZR Taking into account that the imaginary part cancels out in (67) we are only interested on the real part of (67) given by, [49, 2.5.49. 3.], π 1 π2 (68) cα1,s ,0 = Re cα1,s ,0 = dξ = . ± ± √2 (1 + ξ2)2 2√2 ZR Hence, inserting (68) and (66) into (65) gives 2 π 1 2 2 2 (69) b1,0 = ρ( x) dx = √2π ρ(x) dx . √2 2 ZR ZR Now, plugging (69) and (64) into (63) proves the claim. Regarding Theorem 3.6, we make the following remark. Remark 3.8. On the r.h.s. of formula (48) only the real part is essential as the imaginary part vanishes. 14 SEBASTIAN EGGER 4. The asymptotic expansion of the regularized trace of the heat kernel Armed with all the results inferred in our paper so far we are now in the position to deduce the existence of the heat kernel and to conclude the small-t asymptotic expansion of the regularized trace of the heat semi-group et∆ρ (and heat kernel). We are going to exploit that the resolvent of a contraction semi-group admits a represen- tation as a Laplace transformation of the heat semi-group, [56, Satz VII.4.10]. As for the resolvent, the heat semi-group et∆ρ isn’t a trace-class operator due to the presence of an essen- tial spectrum of ∆ρ. Again, we may regularize the trace of the heat semi-group analogously − t∆ρ t∆ρ t∆0 to Definition 3.5 by subtracting the free heat semi-group, i.e., e reg := e e , t> 0. t∆ρ { } − To see that e reg is trace class as well, and how we may calculate its trace, we prove the following lemma.{ } t∆ρ Lemma 4.1. The operator e reg is for t > 0 a trace-class integral operator with kernel ρ 2 2 { } 2 2 kreg(t)( , ) C (R R ) L (R R ). Its trace is given by · · ∈ ∞ × ∩ ∞ × (70) Tr et∆ρ = kρ (t)(x, x)dx . { }reg reg RZ2 Proof. With the same Dunford-Pettis argument as in [36, Lemma 6.1] we may infer that t∆ρ e reg is an integral operator possessing a smooth and bounded kernel for t> 0. We shall {use the} Dunford-Taylor integral identity, [33, Section IX.1.6], t∆ρ i λt reg (71) e = e− R (√λ)dλ , { }reg 2π ρ Zγ where γ is a suitable contour encircling the spectrum of ∆ρ in a positively orientated way. − reg With a similar method as in the proof of Lemma 3.3 we may infer that Rρ (√λ) is continuous in trace norm for suitable γ’s. Furthermore, due to the asymptotics (51) we conclude that the integral converges in trace norm. Now, [55, Satz 3.22] proves the first part of the claim. The second part my be proven analogously to Proposition 3.4 incorporating the above properties ρ of the integral kernel kreg(t). It is reasonable to define the regularized trace of the heat semi-group (and heat kernel) analogously to (38). Definition 4.2. The regularized trace of the heat semi-group (heat kernel) is defined as (72) Tr et∆ρ := Tr et∆ρ = kρ (t)(x, x)dx , t> 0 . reg { }reg reg RZ2 We are now ready to present the result concerning our desired small-t asymptotic expansion of the regularized trace of the heat semi-group (heat kernel). R Theorem 4.3. Let ρ C0∞( ). Then, the regularized trace of the heat semi-group resp. heat kernel possesses a complete∈ asymptotic expansion in powers of t given by, t 0, → t∆ ∞ n (73) Tr e ρ a t 2 , reg ∼ n nX=0 HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 15 where b n!22n+1b (74) a = 2n , n N , a = 2n+1 , n N , 2n n! ∈ 0 2n+1 (2n + 1)!√π ∈ 0 and the bn’s are given in (52). Proof. In view of Lemma 4.1, we may utilize the well-known identity, [56, Satz VII.4.10], λt t∆ρ (75) Tr R ( λ)= e− Tr e dt , reg ρ − reg RZ+ with Re λ > 0 and λ > λmin,ρ (sufficiently large). Now, we want to apply the converse | | π Watson lemma, Lemma B.5. For this, we observe that the condition arg( λmin,σ λ ) 2 π | | |− | ≤ is equivalent to k Cα with α 4 . Hence, we may apply Lemma 4.1 and use the asymptotic ∈ ≤ n expansion of Trreg Rσ( λ) in Theorem 3.6. Comparing (51) with (98) we infer that λn = 2 +1. Finally, we use [39, pp.− 2,3] to calculate 3 1 1 (2n + 1)√π (76) Γ(n + ) = (n + )Γ(n + )= , 2 2 2 n!22n+1 and we plug (76) together with the bn’s, (52), in (98). That reveals the identity (74). We end this paper with a comparison of our heat kernel asymptotics with known results for a Schr¨odinger operator on R2, however, with a smooth potential, V . Using the Theo- rems 4.3 and 3.6 we obtain for our system the leading asymptotic estimate 1 √2√t (77) Tr et∆σ = σ(x)dx + σ2(x)dx + O(t) , t 0 . reg −4π 16√π → ZR ZR R2 On the other hand, for a Schr¨odinger operator of the form ∆+ V ( ) with V C0∞( ) the result on [32, p, 405] is (using an analogous notation), t −0+, · ∈ → t(∆ V ( )) 1 t 2 (78) Tr e − · = V (x)dx + (3V (x) ∆V (x))dx . reg −4π 24π − RZ2 RZ2 We see that the first coefficients and the power of t agree, but not the second coefficient and the corresponding power of t owing to the fact that our potential is supported only on a codimension one submanifold. Acknowledgment. The author is very grateful to Ram Band for helpful discussions and comments and he is indebted to Frank Steiner for pointing out various useful relations of Bessel functions. The work has been supported by ISF (Grant No. 494/14). References [1] W. Arendt, R. Nittka, W. Peter, and F. Steiner. 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Notations First, we introduce some notations and denote by , R2 and R2 the standard inner product and the standard Euclidean norm on R2, respectively.h· ·i Tok ·ease k notation we denote points in R2 by bold letters, e.g., x = (x ,x ). We put 0 arg z < 2π as the range of the 1 2 ≤ argument for z C, and set the brunch cut of the square root √ on R+ such that √z = i z for z R+. By∈ we denote the one-dimensional Hausdorff· measure. The spectrum| of| 1 p an operator− ∈ O is denoted| · | by σ(O) and by O we refer to the standard operator norm, [55, k k 18 SEBASTIAN EGGER Sections 2.1, 5.1]. Moreover, we use standard notations for the set of n-times continuously n Rm n Rm Rm differentiable functions (possessing compact support), C ( ) (C0 ( )), on , and for the set of square Lebesgue-integrable functions, L2(Rm), on Rm. We put the Fourier transform Ff of a suitable function f on R as 1 iξx (79) (Ff)(ξ) := e− f(x)dx . √2π ZR We also remind that by Plancherel’s theorem the Fourier transform generates a unitary map on L2(R), [26, Section 2.2.3]. In addition, by Γ( ) we identify the Gamma function and by K ( ) the K -Bessel function (Mcdonald function)),· [39, pp. 1,66]. 0 · 0 Appendix B. Integral identities for the K0-Macdonald function First, we remind the asymptotic behavior of the K0-Macdonald function for large and small arguments, γ Euler-Mascheroni constant, [39, p. 69], z (80) K (z)= (ln( )+ γ)(1 + O(z)) , z / R+ , 0 − 2 − ∈ and [39, p. ,139], δ > 0, π z 1 3 (81) K (z)= e− (1 + O(z− )) , z , arg z < π δ . 0 2z | |→∞ | | 2 − r In addition, we invoke for our analysis the following integral identity, [39, p. 85], x, ξ R+, ∈ ∞ 2 2 x cosh(t) (82) K0( x + ξ )= e− cos(ξ sinh(t))dt . p Z0 The following simple lemma will be used. Lemma B.1. For κ> 0 and η R we have, ξ R, ∈ ∈ κ η 2 2κ iµξ (83) (F e− |·− |)(ξ)= 2 2 e− . rπ κ + ξ Proof. We first treat the case η = 0: κ 1 ixξ κ x 1 ixξ ixξ κ x (F e− |·|)(ξ)= e e− | |dx = (e + e− )e− | |dx √2π √2π ZR Z+ (84) R 2 κ = 2 2 . rπ κ + ξ Now, for η = 0 we use the identity (Ff( µ))(ξ) = (Ff)(ξ)e iµξ giving (83). 6 ·− − Proposition B.2. For y, y , x R we have ′ 2 ∈ K (k (x y)2 + x2)K (k (x y )2 + x2)dx 0 1 − 2 0 1 − ′ 2 1 ZR q q (85) 2 cosh(t) cosh(t′) cos(x2 sinh(t)) cos(x2 sinh(t′)) iξ(y y′) = 2 2 2 2 e− − dtdt′dξ . πk ((cosh(t)) + ξ )((cosh(t′)) + ξ ) ZR RZ+ RZ+ HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 19 Proof. We use Plancherel’s theorem and Lemma B.1 to calculate ′ ′ k x1 (y y ) cosh(t) k x1 cosh(t ) e− | − − | e− | | dx1 ZR (86) 2 2 k cosh(t) cosh(t′) iξ(y y′) = 2 2 2 2 e− − dξ . π ((k cosh(t)) + ξ )((k cosh(t′)) + ξ ) ZR Now, we use (82) and then Fubini’s theorem, [5, 23.7 Corollary], for an interchange of the x1, t and t′ integration. Moreover, we are going to use the coordinate transformation ξ kξ giving → K (k (x y)2 + x2)K (k (x y )2 + x2)dx 0 1 − 2 0 1 − ′ 2 1 ZR q q ′ ′ k x1 (y y ) cosh(t) k x1 cosh(t ) (87) = e− | − − | cos(kx2 sinh(t))e− | | cos(kx2 sinh(t′))dtdtdx1 ZR RZ+ RZ+ 2 cosh(t) cosh(t′) cos(kx2 sinh(t)) cos(kx2 sinh(t′)) ikξ(y y′) = 2 2 2 2 e− − dtdt′dξ . πk ((cosh(t)) + ξ )((cosh(t′)) + ξ ) ZR RZ+ RZ+ We investigate the decay properties of the integral in (85) for large x2. To ease notation we introduce cosh(t) cosh(t′) (88) F (t,t′,ξ) := 2 2 2 2 , ((cosh(t)) + ξ )((cosh(t′)) + ξ ) and we obtain Lemma B.3. The following estimate ′ F (t,t ,ξ) cos(kx sinh(t)) cos(kx sinh(t ))e iξ(y y )dtdt dξ ′ 2 2 ′ − − ′ ZR Z+ Z+ (89) R R 1 (∂t,t′ F (t,t′,ξ)) 2 dtdt′dξ ≤ (kx2) cosh(t) cosh(t′) RZ+ RZ+ holds. Proof. We first observe that F , ∂lF , l = t, t′, and ∂t,t′ F are integrable. Then, we treat the t integration with the integration by parts method and arrive at F (t,t′,ξ) cosh(t) F (t,t′,ξ) cos(kx sinh(t))dt = cos(kx sinh(t))dt 2 cosh(t) 2 RZ+ RZ+ (90) 1 (∂tF (t,t′,ξ)) = sin(kx2 sinh(t))dt . kx2 cosh(t) RZ+ 20 SEBASTIAN EGGER iτ ξ(y y′) Doing the same w.r.t. the t′ integration and exploiting that sin(τ) e = e− − = 1, τ R, gives | | ≤ | | | | ∈ ikξ(y y′) F (t,t′,ξ) cos(kx sinh(t)) cos(kx sinh(t′))e− − dtdt′dξ | 2 2 | RZ+ 1 ∂ ′ F (t,t ,ξ) ′ = t,t ′ sin(kx sinh(t)) sin(kx sinh(t ))e ikξ(y y )dtdt dξ (91) 2 2 2 ′ − − ′ (kx2) | cosh(t) cosh(t′) | RZ+ 1 ∂t,t′ F (t,t′,ξ) 2 dtdt′dξ . ≤ (kx2) cosh( t) cosh(t′ ) RZ+ We are now able to perform an x2-integration in (85), and we remark that in the following appearing integrals the order of integration is crucial. Proposition B.4. For y, y R we have ′ ∈ K (k (x y)2 + x2)K (k (x y )2 + x2)dx dx 0 1 − 2 0 1 − ′ 2 1 2 ZR ZR q q (92) π ikξ(y y′) = 3 e− − dξ . 2k2(1 + ξ2) 2 ZR Proof. Due to Lemma B.3 the x2-integral exists. We perform in (85) the substitution t,t′ d 1 √ 2 1 → arsinh(t), arsinh(t′), use ( dt sinh(t)) = (cosh(arsinh(t)))− = ( 1+ t )− , cos(x) cos(y) = 1 (cos(x + y) + cos(x y)) and make the substitution x x2 . This gives 2 − 2 → k cosh(t) cosh(t′) cos(kx2 sinh(t)) cos(kx2 sinh(t′)) ikξ(y y′) 2 2 2 2 e− − dξdtdt′dx2 ((cosh(t)) + ξ )((cosh(t′)) + ξ ) ZR RZ+2 ZR (93) 1 cos(x2(t + t′)) + cos(x2(t t′)) ikξ(y y′) = 2 2 2 −2 e− − dξdtdt′dx2 . k 2(t +1+ ξ )(t′ +1+ ξ ) ZR RZ+2 ZR Due to decay property proven in Lemma B.3 it is not hard to see that we may extend the setting on [6, p. 36] to our case. Hence, we may use the δ-identity, [6, pp. 33,34], 1 (94) cos(ξ(x x ))dξ = δ(x x ) . 2π − 0 − 0 ZR We arrive at K (k (x y)2 + x2)K (k (x y )2 + x2)dx dx 0 1 − 2 0 1 − ′ 2 1 2 ZR q q 2 ikξ(y y′) = e− − dξdt (95) k2(t2 +1+ ξ2)2 RZ+ ZR π ikξ(y y′) = 3 e− − dξ , 2k2(1 + ξ2) 2 ZR HEAT-KERNEL TRACE OF A SINGULAR TWO-PARTICLE 1-D CONTACT INTERACTION 21 where in the last line we used [25, 3.241 4.]. B.1. A converse Watson lemma. To connect the large-λ (or large-k) asymptotics of the resolvent with the small-t asymptotics of the heat kernel the following lemma will be used, which may be obtained by replacing a by an on [57, p. 31]. n Γ(λn) Lemma B.5 (Converse Watson Lemma). Let f( ) be a continuous function in (0, ), f(t)= 0 for t< 0, and e c f( ) L1(0, ). Let F be the· Laplace transform of f, i.e., ∞ − · · ∈ ∞ ∞ zt (96) F (z) := f(t)e− dt . Z0 If F possesses the uniform asymptotic expansion ∞ λn π (97) F (z) a z− , z , arg(z c) , ∼ n | |→∞ − ≤ 2 n=0 X and λ monotonously as n , then n →∞ →∞ ∞ an λn 1 + (98) f(t) t − , t 0 . ∼ Γ(λ ) → n=0 n X (S. Egger) Department of Mathematics, Technion-Israel Institute of Technology 629 Amado Building, Haifa 32000, Israel E-mail address: [email protected]