\input amstex \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \NoBlackBoxes \TagsOnRight \topmatter \title Higher Order Trace Relations for Schr\"odinger Operators \endtitle \rightheadtext{ Higher Order Trace Relations} \author F.~Gesztesy$^1$, H.~Holden$^2$, B.~Simon$^3$, and Z.~Zhao$^1$ \endauthor \leftheadtext{F.~Gesztesy, H.~Holden, B.~Simon, and Z.~Zhao} \dedicatory Dedicated to Henry P. McKean on the occasion of his sixtieth birthday \enddedicatory \thanks $^1$ Department of Mathematics, University of Missouri, Columbia, MO 65211. E-mail for F.G.: mathfg\linebreak\@mizzou1.missouri.edu; e-mail for Z.Z.: mathzz\@mizzou1.missouri.edu \endthanks \thanks $^2$ Department of Mathematical Sciences, The Norwegian Institute of Technology, University of Trondheim, N-7034 Trondheim, Norway. E-mail: holden\@imf.unit.no \endthanks \thanks $^3$ Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 253-37, Pasa-dena, CA 91125. This material is based upon work supported by the National Science Foundation under Grant No. DMS-9101715. The Government has certain rights in this material. \endthanks \thanks To be submitted to {\it{Commun.~Pure Appl.~Math.}} \endthanks \abstract We extend the trace formula recently proven for general one-dimensional Schr\"o-dinger operators which obtains the potential $V(x)$ from a function $\xi(x, \lambda)$ by deriving trace relations computing moments of $\xi(x, \lambda)\,d\lambda$ in terms of polynomials in the derivatives of $V$ at $x$. We describe the relation of those polynomials to KdV invariants. We also discuss trace formulae for analogs of $\xi$ associated with boundary conditions other than the Dirichlet boundary condition underlying $\xi$. \endabstract \endtopmatter \document \bigpagebreak \flushpar {\bf \S1. Introduction} This paper is one in a series [14--20] concerning a basic function, $\xi(\lambda, x)$, associated to any one-dimensional Schr\"odinger operator, $H=-\frac{d^2}{dx^2}+V$ in $L^{2}(\Bbb R)$ and its application to inverse spectral problems. A basic formula proven in [18] is that $$V(x)=E_{0}+\lim\limits_{\alpha\downarrow 0} \int\limits^{\infty}_{E_0} d\lambda \, e^{-\alpha\lambda} (1-2\xi(\lambda, x)), \tag 1.1$$ where $E_{o}=\inf\,\text{spec}(-\frac{d^2}{dx^2}+V)$. (1.1) was proven in [18] assuming $V$ is bounded below, continuous, and $|V(x)|\leq C_{1}e^{C_{2}x^{2}}$. Our definition of $\xi$ is $$\xi(\lambda, x):=\frac{1}{\pi}\,\text{Arg}((G(\lambda+i0, x, x)) \tag 1.2$$ (where $G(z, x, x')$ denotes the Green's function of $H$, that is, the integral kernel of $(H-z)^{-1}$), although we derived that from a basic definition as the Krein spectral shift in going from $H$ to $H_{D; x}$, the operator on $L^{2}(-\infty, x)\oplus L^{2}(x, \infty)$ with Dirichlet boundary condition at $x$. The key to (1.1) then was $$\text{Tr}(e^{-tH}-e^{-tH_{D; x}})=\frac12 \,(1-tV(x)+o(t)) \quad\text{as }t\downarrow 0. \tag 1.3$$ (1.3) is related to (1.1) because the Krein spectral shift [30] is a function $0\leq\xi(\lambda, x)\leq 1$ obeying $$\text{Tr}[f(H)-f(H_{D;x})]=-\int\limits^{\infty}_{E_o} d\lambda \, f'(\lambda)\xi(\lambda, x) \tag 1.4$$ for a rich set of $f$'s including exponentials (e.g., $f\in C^{2}(\Bbb R)$, $(1+\lambda^{2})f^{(j)}\in L^{2}((0, \infty))$, $j=1, 2$ and also $f(\lambda)= (\lambda -z)^{-1}$, $z\in\Bbb C\backslash [E_{o}, \infty$)) so that $$\text{Tr}(e^{-tH}-e^{-tH_{D; x}})=t\int\limits^{\infty}_{E_{o}} d\lambda\, e^{-t\lambda}\xi(\lambda, x).$$ One of our goals in the present paper is to prove (1.1) in greater generality; we only need $V$ bounded from below with no growth restriction at infinity. $V$ need not be continuous; a local $L^1$ condition suffices. (1.1) then holds at points of Lebesgue continuity of $V$. Our main goal though is to prove higher order trace formulas. In great generality (suppose $V$ has an asymptotic Taylor series at $x_0$), we'll extend (1.3) to $$\text{Tr}(e^{-tH}-e^{-tH_{D; x_{0}}})\operatornamewithlimits{\sim} \limits_{t\downarrow 0} -\sum^{\infty}_{j=0} s_{j}(x_{0})t^{j},$$ where $s_{j}(x)=(-1)^{j+1}(j!)^{-1}r_{j}(x)$ and the $r_j$ are KdV invariants defined recursively in Theorem 5.1 below. With more information one can relate this to a similar formula in terms of $\xi$ (for simplicity of notation we suppose that $E_{o}=0$): $$r_{j}(x_{0})=j\,\lim\limits_{\alpha\downarrow 0} \int\limits ^{\infty}_{0}d\lambda\,e^{-\lambda\alpha}\lambda^{j-1} \biggl( \frac12 -\xi(\lambda, x_{0})\biggr). \tag 1.5$$ The key to handling potentials with no growth condition at infinity is a path space representation for $\text{Tr}(e^{-tH}-e^{-tH_{D; x}})$. Properties of the paths needed are proven in \S2. Then in \S3, we prove (1.1) for general $V$. In \S4, we show that $\text{Tr} (e^{-tH}-e^{-tH_{D; x}})$ has an asymptotic expansion to all orders in $t$ at $t=0$ if $V$ is $C^\infty$. In \S5, we relate the coefficients of this expansion to the KdV invariants, and in \S6 we discuss what happens if boundary conditions other than Dirichlet are used. Historically, trace formulas for Schr\"odinger operators on a finite interval originated with a 1953 paper by Gel'fand and Levitan [11] with later contributions by Dikii [6], Gel'fand [9], Halberg-Kramer [22], and Gilbert-Kramer [21]. The case of periodic potentials was first studied by Hochstadt [25] who obtained a trace formula for $V(x)-V(0)$ in terms of appropriate Dirichlet eigenvalues in the special case of finite-gap potentials. The periodic trace formula (5.59) for finite-gap potentials $V(x)$ in terms of Dirichlet eigenvalues was first derived by Dubrovin [7]. The periodic trace formulas (5.59) for all higher order Korteweg-de Vries invariants $s_{j}(x)$ were first proven in 1975 by McKean-van Moerbeke [35] and independently by Flaschka [8], the trace formula for $s_{1}(x)=\frac{1}{2}V(x)$ for general periodic $C^3$ potentials by Trubowitz [40] in 1977. More recently, the trace formula (5.59) for $V(x)$ has been extended to certain classes of almost periodic potentials in Levitan [32,33], Kotani-Krishna [29], and Craig [2]. Analogous trace formulas for Schr\"odinger operators on the real line with potentials decaying sufficiently rapidly at infinity have been studied in 1979 by Deift and Trubowitz [5], and more recently by Venakides [41], Gesztesy-Holden [13], Gesztesy [12], and Gesztesy-Holden-Simon-Zhao [14]. These trace formulas are a key element of the solution of the inverse spectral problem for periodic potentials and the inverse scattering problem for potentials decaying sufficiently fast at infinity (see, e.g., [5], [7], [8], [25], [26], [32--36], [40], [41] and the references therein.) \bigpagebreak \flushpar {\bf \S2. The Xi Process} In [18], we introduced a probability measure on the set of paths on $[0, 1]$ as follows. Let $\alpha$ be the Brownian bridge, that is, the Gaussian process of $\{\alpha(s)\}_{0\leq s\leq 1}$ of mean zero and covariance $E_{\alpha}(\alpha (s)\alpha (t))=s(1-t)$ if $s\leq t$. In terms of Brownian motion, one can realize $\alpha$ as $\alpha (s)=b(s)-sb(1)$ (see [37] for discussion of Brownian motion, Gaussian processes, and the Brownian bridge). There is a Baire measure $\Cal D\alpha$ on $C([0, 1])$ induced by the process. Let $d\kappa$ be the measure on $\Bbb R\times C([0, 1])$ given by $dx\otimes (4\pi)^{-1/2}\Cal D\alpha$ where $dx$ is Lebesgue measure. Let $\omega(s)=x+\alpha(s)$ and let $\Omega_{0}\subset\Bbb R\times C([0, 1])$ be the set of paths given by $\{\omega\mid \omega(s)= 0 \text{ for some }s \in[0, 1]\}$. We claim that $$\int\limits_{\Omega_{0}} d\kappa=\frac12 \tag 2.1$$ for the free Feynman-Kac formula says \align e^{\Delta/2}(x, x) &= (4\pi)^{-1/2} \int\limits_{\{\omega(0)=x\}} \Cal D\alpha =(4\pi)^{-1/2}, \tag 2.2 \\ e^{\Delta_{D}/2} (x, x) &= (4\pi)^{-1/2} \int\limits_{\{\omega(0)=x; \omega(s)\neq 0 \text{\rom{ all }}s\in [0, 1]\}} \Cal D\alpha, \tag 2.3 \endalign where $\Delta_{D}=\Delta_{D; 0}$ has a Dirichlet boundary condition at $x=0$. Thus \align \int\limits_{\Omega_{0}} d\kappa &=\int\limits_{\Bbb R} dx\, \biggl[\exp\biggl(\frac12 \Delta\biggr)(x, x)-\exp \biggl( \frac12\Delta_{D}\biggr)(x,x)\biggr] \\ &= \int\limits_{\Bbb R} dx \exp \biggl(\frac12 \Delta\biggr)(x, -x) \\ &= \int\limits_{\Bbb R} dx \exp \biggl(\frac12 \Delta\biggr)(2x, 0) \\ &= \frac12 \int\limits_{\Bbb R} dy \exp \biggl(\frac12 \Delta\biggr) (y, 0)=\frac12. \endalign We define the xi process by placing the measure $\Cal D \omega\equiv 2\chi_{\Omega_{0}}\, d\kappa$ on $C([0, 1])$ with $\omega(s)=x+\alpha(s)$. The reason for the interest in $\Cal D \omega$ is that by writing (2.3) with a potential, one finds (see [37]): \proclaim{Proposition 2.1} Let $V$ be bounded below and continuous on $(-\infty, \infty)$, $H=-\frac{d^2}{dx^2} +V$ on $L^{2}(-\infty, \infty)$ and let $H_{D}=-\frac{d^2}{dx^2}+V$ on $L^{2}(-\infty, 0)\oplus L^{2}(0, \infty)$ with a Dirichlet boundary condition at $x=0$ \rom(i.e., $H_{D}=H_{D;0}$\rom). Then $$\text{\rom{Tr}}(e^{-tH}-e^{-tH_{D}})=\frac12 E_{\omega}\biggl(\exp\biggl(-t \int\limits^{1}_{0}ds\, V(\sqrt{2t}\, \omega(s))\biggr)\biggr). \tag 2.4$$ \endproclaim The Feynman-Kac formula (2.4) will be critical for the proof of our higher order trace relations. We'll need the following technical result (we use the notation employed in [37], i.e., $E(f)=\int\limits_{\Omega}f\, d\mu$, $E(A)=\int\limits_{A}d\mu=\mu(A)$, $E(f; A)=\int\limits_{A}f\,d\mu$, etc., where $(\Omega, \Cal F, \mu)$ denotes a probability space, $A\in\Cal F$, $f:\Omega\to\Bbb R$ is $\Cal F$-measurable): \proclaim{Theorem 2.2} $E_{\omega}\bigl(\{\omega\mid\sup\limits_{0\leq s\leq 1} |\omega(s)|\geq a\}\bigr)\leq C_{1}\exp(-C_{2}a^{2})$ for some $C_{1}, C_{2}>0$. \endproclaim \demo{Proof} Look at sets on $\Bbb R\times C([0, 1])$ with measure $d\kappa$. Let $T_{a}=\bigl\{\omega\in \Omega_{0}\mid\sup\limits_{0 \leq s\leq 1} |\omega(s)|\geq a\bigr\}$. Then $$\split T_{a} &\subset\biggl\{\omega\in \Omega_{0}\mid |\omega(0)|>\frac{a}{2}\biggr\} \cup\biggl\{\omega\mid |\omega(0)|<\frac{a}{2}, \sup\limits_{0\leq s\leq 1} \omega(s)\geq a\biggr\} \\ &\qquad\cup\biggr\{\omega\mid |\omega(0)|<\frac{a}{2}, \inf\limits_{0\leq s\leq 1} \omega(s)\leq -a\biggr\} \\ &\equiv T^{(1)}_{a}\cup T^{(2)}_{a}\cup T^{(3)}_{a}. \endsplit$$ Notice that we have dropped the $\omega\in \Omega_{0}$ condition from $T^{(i)}_{a}, i=2, 3$. In each case, we have a single condition on a value that we must take, for example: $$T^{(2)}_{a}=\biggr\{\omega\mid |\omega(0)|<\frac{a}{2}, \omega(s)=a \text{ for some }s\in [0, 1]\biggr\}.$$ Thus, each $\int\limits_{T^{(i)}_{a}} dx$ can be written in terms of a Dirichlet boundary condition (at $0$, $a$, $-a$, respectively) and then by the method of images in terms of the free heat kernel of $e^{\Delta/2}$. Explicitly, \align \int\limits_{T^{(1)}_{a}}dx &=\int\limits_{|x|>a/2}dx\, e^{\Delta/2} (x, - x), \\ \int\limits_{T^{(2)}_{a}}dx &=\int\limits_{|x|0, we can integrate in (3.4) over |x|\frac12 t^{1/4}, \delta_2 the region where |w+z|>\frac12 t^{1/4}, and \delta_3 the region where |w|0 along the positive real axis. Explicitly, one infers from (5.8), (5.9), and (5.35) that\gather r_{0}(x)=\frac12, \quad r_{1}(x)=\frac 12\, V(x), \quad r_{2}(x)=\frac12\, V(x)^{2}-\frac14\, V''(x), \quad \text{etc.} \tag 5.37 \\ \phi_{1}(x)=V(x), \quad \phi_{2}(x)=-V'(x), \quad \phi_{3}(x)=V''(x)- V(x)^{2}, \\ \phi_{4}(x)=4V(x)V'(x)-V'''(x), \quad \text{etc.}, \tag 5.38 \endgather $$and$$ \omega_{0}(x)=1, \quad \omega_{2}(x)=-2V(x), \quad \omega_{4}(x)=6V(x)^{2}-2V''(x), \quad \text{etc.} \tag 5.39 $$\endremark Next we relate (5.7) and (5.1). \proclaim{Theorem 5.3} Suppose V\in C^{\infty}(\Bbb R), V real-valued and bounded from below. Then for each N\in\Bbb N,$$ \text{\rom{Tr}}(e^{-tH_{D; x}}-e^{-tH}) \operatornamewithlimits{\sim}\limits_{t\downarrow 0} \sum^{N}_{j=0} s_{j}(x)t^{j}+O(t^{N+1}), \quad x\in\Bbb R, \tag 5.40 $$where s_{j}(x) are the KdV invariants$$ s_{j}(x)=(-1)^{j+1}(j!)^{-1}r_{j}(x), \quad j\in\Bbb N_{0} \tag 5.41 $$with r_{j}(x) given by {\rom(5.8)}. \endproclaim \demo{Proof} Since the existence of the asymptotic expansion (5.40) has been proven in \S4 we only need to identify the coefficients s_{j}(x) as in (5.41). Without loss of generality we may assume in addition that V\in C^{\infty}_{0}(\Bbb R). Let E_{o}=\inf\,\text{spec}(H), then one obtains from (1.4) and Fubini's theorem that$$\align \text{Tr}[(H_{D; x}-z)^{-1}-(H-z)^{-1}]&=-\int\limits^{\infty}_{E_{o}} \frac{d\lambda\,\xi(\lambda, x)}{(\lambda-z)^{2}} \\ &=\int\limits^{\infty}_{E_o}d\lambda\,\xi(\lambda, x) \int\limits^{\infty}_{0}dt\, (-t)e^{(z-\lambda)t} \\ &=\int\limits^{\infty}_{0}dt\, e^{zt}\int\limits^{\infty}_{E_o} d\lambda \, (-t)e^{-\lambda t}\xi(\lambda, x) \\ &=\int\limits^{\infty}_{0}dt \, e^{zt}\text{Tr} (e^{-tH_{D; x}}-e^{-tH}), \quad z0,\, x\in\Bbb R. \tag 5.43 $$Then$$ F(x, \cdot)\in C^{\infty}([0, \infty)), \quad \text{for each } x\in\Bbb R \tag 5.44 $$is proven at the end of \S4 and Theorem 4.1 yields for each N\in\Bbb N,$$ F(x, t)\operatornamewithlimits{\sim}\limits_{t\downarrow 0} \sum^{N}_{j=0} s_{j}(x)t^{j}+O(t^{N+1}). \tag 5.45 $$In particular,$$ \biggl|F(x, t)-\sum^{N}_{j=0}s_{j}(x)t^{j}\biggr|\leq C_{N}(x)t^{N+1}, \quad 0\leq t\leq 1 \tag 5.46 $$by estimating the remainder in the Taylor expansion for F(x, \cdot). Thus$$ z\text{Tr}[(H_{D; x}-z)^{-1}-(H-z)^{-1}]=z\int\limits^{1}_{0}dt\, e^{zt}F(x, t)+z\int\limits^{\infty}_{1}dt \, e^{zt}F(x, t):=G_{1}(x, z) +G_{2}(x, z). \tag 5.47 $$Clearly,$$ |G_{2}(x, z)|=\biggl|z\int\limits^{\infty}_{1}dt\, e^{zt}F(x, t)\biggr| \leq (-z)\int\limits^{\infty}_{1} dt\, e^{zt}|F(x, t)|\leq C_{0}e^{z}, \quad z<\min(0, E_{o}) \tag 5.48 $$since |F(x, t)|\leq e^{-tE_{o}} (because of 0\leq\xi(\lambda, x)\leq 1). Moreover,$$\multline G_{1}(x, z)=z\int\limits^{1}_{0}dt\, e^{zt}\biggl[F(x, t) -\sum^{N}_{j=0}s_{j}(x)t^{j}+\sum^{N}_{j=0}s_{j}(x)t^{j}\biggr]\\ \shoveleft{\operatornamewithlimits{\sim}\limits_{z\downarrow -\infty} \sum^{N}_{j=0} s_{j}(x)\biggl[z\int\limits^{\infty}_{0}dt\, e^{zt}t^{j}+O(e^{\epsilon z})\biggr] + z\int\limits^{1}_{0} dt\, e^{zt}\biggl[F(x, t)-\sum^{N}_{j=0} s_{j}(x)t^{j}\biggr]} \\ \shoveleft{\operatornamewithlimits{\sim}\limits_{z\downarrow -\infty} \sum^{N}_{j=0} s_{j}(x)(-1)^{j+1}(j!)[z^{-j}+O(e^{\epsilon z})] + z \int\limits^{1}_{0}dt\, e^{zt}\biggl[F(x, t)-\sum^{N}_{j=0} s_{j}t^{j}\biggr], \quad z<\min(0, E_{o})} \\ \endmultline \tag 5.49 $$for some 0<\epsilon<1. Thus$$ z\text{Tr}[(H_{D; x}-z)^{-1}-(H-z)^{-1}]\operatornamewithlimits{\sim} \limits_{z\downarrow -\infty}\sum^{N}_{j=0} s_{j}(x)(-1)^{j+1}(j!) z^{-j}+O(z^{-N-1}) \tag 5.50 $$using the estimate (5.46). A comparison of (5.7) and (5.50) then yields (5.41). \qed \enddemo Relations (5.37) and (5.41) then yield explicitly$$ s_{0}(x)=-\frac12, \quad s_{1}(x)=\frac12 V(x), \quad s_{2}(x)=\frac18 V''(s)-\frac14 V(x)^{2}, \quad\text{etc.} \tag 5.51 $$Finally we epxress the KdV invariants s_{j}(x) in terms of \xi(\lambda, x) according to Theorem 4.3. \proclaim{Theorem 5.4} Suppose V\in C^{\infty}(\Bbb R), V real-valued and bounded from below. Assume that \rom{(4.2)} holds and denote E_{o}=\inf\,\text{spec}(H). Then$$\aligned s_{0}(x) &=-\frac12 , \\ s_{j}(x) &=\frac{(-1)^{j+1}}{j!} \left\{\frac{E^{j}_{o}}{2}\right. + j \, \lim_{t\downarrow 0} \int\limits^{\infty}_{E_o}d\lambda\, e^{-t\lambda}\lambda^{j-1} \biggl[\frac12 -\xi(\lambda, x)\biggr] \quad j\in\Bbb N,\, x\in\Bbb R. \endaligned \tag 5.52 $$Explicitly, one has$$\align s_{1}(x) &=\frac12 V(x) \\ &=\frac{E_o}{2}+\lim_{t\downarrow 0}\int\limits^{\infty}_{E_o} d\lambda\, e^{-t\lambda}\biggl[\frac12 -\xi(\lambda, x)\biggr], \tag 5.53 \\ s_{2}(x) &=\frac18 V''(x)-\frac14 V(x)^{2} \\ &=-\frac{E^{2}_{o}}{4}-\lim_{t\downarrow 0}\int\limits^{\infty}_{E_o} d\lambda\, e^{-t\lambda}\lambda\biggl[\frac12 -\xi(\lambda, x)\biggr], \quad\text{\rom{etc.}} \tag 5.54 \endalign $$\endproclaim We will illustrate these results in the special case where V(x) is periodic. \example{Example 5.5} Assume V\in C^{\infty}(\Bbb R), V real-valued, for some a>0, V(x+a)=V(x) for all x\in\Bbb R. In this case the spectrum of H is given by$$ \text{spec}(H)=\bigcup^{\infty}_{n=1}[E_{2(n-1)}, E_{2n-1}]. \tag 5.55 $$Then for each x\in\Bbb R, \xi(\lambda, x) is real-valued for \lambda\in (E_{2n-1}, E_{2n}) and purely imaginary for \lambda\in (E_{2(n-1)}, E_{2n-1}) (see, e.g., [4], [28]). More precisely,$$ \xi(\lambda, x)=\cases 0, &\quad\lambda0\quad\text{ for Im}(z)>0} \\ \endmultline \tag 6.8 $$by Cauchy's inequality. Equation (5.25) then turns into$$ \text{Tr}[(H_{\beta; x}-z)^{-1}-(H-z)^{-1}]=-\frac{d}{dz} \ln[(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x,)]\quad \beta\in\Bbb R,\, x\in\Bbb R. \tag 6.9 $$In order to introduce \xi_{\beta}(\lambda, x), Krein's spectral shift function associated with the pair (H_{\beta; x}, H) (in analogy to \xi(\lambda, x)\equiv\xi_{\infty}(\lambda, x) associated with (H_{D; x}\equiv H_{\infty; x}, H)), we next investigate [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)] a bit further. First of all we notice that$$ H_{\beta; x}\leq H, \quad \beta\in\Bbb R, \, x\in\Bbb R \tag 6.10 $$as opposed to$$ H_{D; x}=H_{\infty; x}\geq H, \quad x\in\Bbb R. \tag 6.11 $$One way of understanding (6.10) is in terms of quadratic forms. Let Q(H_{\beta=0})=N_{y} be the form domain of the Neumann boundary condition object. Then \varphi's in N_y are continuous on \Bbb R\backslash\langle y\rangle and have continuous boundary values \varphi(y\pm 0). Q(H_{\beta})=N_y with$$ (\varphi, H_{\beta}\varphi)=(\varphi, H_{\beta=0}\varphi)-\beta [|\varphi(y+)|^{2}-|\varphi(y-)|^{2}]. $$Let N^{0}_{y}=\{\varphi\in N\mid\varphi(y+)=\varphi(y-)\}. Thus H is just the form H_\beta restricted to N^{0}_{y}, so H_{\beta, y}\leq H. Moroever, one easily verifies the identity$$\multline [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)]=\beta^{2}G(z, x, x) \\ +\beta\biggl[\frac{d}{dx}\, G(z, x, x)\biggr] +H(z, x, x), \quad z\in\Bbb C\backslash\Bbb R,\, \beta\in\Bbb R, \, x\in\Bbb R, \endmultline \tag 6.12 $$where$$ H(z, x, x)=\frac{f'_{+}(z, x)f'_{-}(z, x)}{W(f_{+}(z), f_{-}(z))} \tag 6.13 $$and$$ \frac{d}{dx}\,H(z, x, x)=[V(x)-z]\frac{d}{dx}\, G(z, x, x). \tag 6.14 $$From$$ G(z, x, x)\operatornamewithlimits{=}\limits_{z\downarrow -\infty} \frac{1}{2|z|^{1/2}} + o(|z|^{-1/2}), \tag 6.15 $$in accordance with$$ G(z, x, x)>0\quad\text{ for } z<\inf\,\text{spec}(H), \tag 6.16 $$and from (6.14) one infers$$\align H(z, x, x) &=H(z, x_{o}, x_{o})+\int\limits^{x}_{x_o}dx'\, [V(x')-z]\biggl[\frac{d}{dx'} G(z, x', x')\biggr] \\ &\operatornamewithlimits{=}\limits_{z\downarrow -\infty} H(z, x_{o}, x_{o})+o(|z|^{1/2}) \tag 6.17 \endalign $$upon integration by parts. In particular, the leading asymptotic behavior of H(z, x, x) as z\downarrow -\infty is independent of x and can be obtained from the free case V^{(o)}(x)\equiv 0. Since for V^{(o)}(x)=0,$$ G^{(o)}(z, x, x)=\frac{i}{2z^{1/2}}, \quad H^{(o)}(z, x, x)=\frac{iz^{1/2}}{2}, \tag 6.18 $$one infers$$ H(z, x, x)\operatornamewithlimits{=}\limits_{z\downarrow -\infty} -\frac{|z|^{1/2}}{2}+ o(|z|^{1/2}) \tag 6.19 $$and hence$$ [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)]<0\quad\text{for $- z>0$ large enough}. \tag 6.20 $$Thus the exponential Herglotz representation [1] for [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)] yields$$ [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)]=\exp\biggl\{c +\int\limits_{\Bbb R}\biggl[\frac{1}{\lambda-z}- \frac{\lambda}{1+\lambda^{2}}\biggr]\,[\xi_{\beta}(\lambda, x)+1]\biggr\}d\lambda \tag 6.21 $$for some c\in\Bbb R, where for each x\in\Bbb R and a.e.~\lambda\in\Bbb R$$ \xi_{\beta}(\lambda, x)=\frac{1}{\pi}\,\lim_{\epsilon\downarrow 0} \,\text{Im}\left\{\ln[(\beta+\partial_{1})(\beta+\partial_{2}) G(\lambda+i\epsilon, x, x)]\right\}-1 \tag 6.22 $$and$$ -1\leq\xi_{\beta}(\lambda, x)\leq 0, \quad\text{a.e.~$\lambda\in\Bbb R$}, \quad \xi_{\beta}(\lambda, x)=0 \quad \lambda<\inf\,\text{spec}(H_{\beta; x}) \tag 6.23 $$in agreement with (6.10) and (6.20). Hence$$ \text{Tr}[f(H_{\beta; x})-f(H)]=\int\limits_{\Bbb R}d\lambda\, f'(\lambda)\xi_{\beta}(\lambda, x) \tag 6.24 $$for any f\in C^{2}(\Bbb R) with (1+\lambda^{2})f^{(j)}\in L^{2}((0, \infty)), j=1, 2 and for f(\lambda)=(\lambda-z)^{-1}, z\in\Bbb C\backslash[\inf\,\text{spec}(H_{\beta; x}), \infty). The following example in the free case V^{(o)}(x)\equiv 0 illustrates these facts. \example{Example 6.1} V^{(o)}(x)\equiv 0. Then G^{(o)}(z, x, x')=\frac{i}{2z^{1/2}}\cdot e^{iz^{1/2}|x-x'|}, \text{Im}(z^{1/2})\geq 0 yields$$ [(\beta+\partial_{1})(\beta+\partial_{2})G^{(o)}(z, x, x)]= (i/2)[\beta^{2}z^{-1/2}+z^{1/2}], \quad \beta\in\Bbb R \tag 6.25 $$and$$ \xi^{(o)}_{\beta}(\lambda, x)=\cases 0, &\lambda<-\beta^{2} \\ -1, &-\beta^{2}<\lambda<0, \quad \beta\in\Bbb R\backslash\{0\}, \\ -\frac12, &\lambda >0 \endcases\tag 6.26  \xi^{(o)}_{o}(\lambda, x)= \cases 0, &\lambda<0 \\ -\frac12, &\lambda>0. \endcases \tag 6.27 $$Thus$$ \text{Tr}[(H^{(o)}_{\beta; x}-z)^{-1}-(H^{(o)}-z)^{-1}]=\frac{\beta^{2}-z} {2z(z+\beta^{2})}, \quad \beta\in\Bbb R,\, z\in\Bbb C\backslash\{ \{-\beta^{2}\}\cup[0, \infty)\}, \tag 6.28  \text{Tr}[e^{-tH^{(o)}_{\beta; x}}-e^{-tH^{(o)}}]=-\frac12 + e^{t\beta^{2}}, \quad \beta\in\Bbb R,\, t >0, \tag 6.29 $$where H^{(o)}=-\frac{d^2}{dx^2}, \Cal D(H^{(o)})=H^{2,2}(\Bbb R). One has$$ \text{spec}(H^{(o)}_{\beta; x})=\{-\beta^{2}\}\cup[0, \infty), \quad \beta\in\Bbb R. \tag 6.30 $$Next we recall the well-known fact that the Weyl m-functions \phi_{\pm}(z, x) associated with H_{D,\pm; x} in L^{2}((x, \pm\infty)) (see the paragraph following (5.23)) have the asymptotic expansion (5.18) as z\to i\infty whenever V satisfies (6.2), see [3], [23], [24]. (Actually the l.p.~property of h at \pm\infty is irrelevant in this context and the asymptotic expansion (5.18) is valid outside any cone |\tan\theta|<\epsilon for \epsilon>0 arbitrarily small.) Hence (5.23), (5.31), and (6.14) imply the existence of asymptotic expansions for G(z, x, x), \frac{d}{dx}G(z, x, x), H(z, x, x)=[\partial_{1}\partial_{2}G(z, x, x)], and \frac{d}{dx}H(z, x, x) as z\to i\infty to all orders in z. In the following we derive recursion relations for the coefficients in the expansion for [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)] by reducing it to those of G(z, x, x) and H(z, x, x) under the assumptions (6.2) on V. The ansatz$$ G(z, x, x)\operatornamewithlimits{\sim}\limits_{z\to i\infty} \frac{i}{2}\sum^{\infty}_{j=0}g_{j}(x)z^{-j-1/2} \tag 6.31 $$inserted into the well-known differential equation for G(z, x, x) (essentially equivalent to (5.19))$$ 4[V(x)-z]G(z, x, x)^{2}+\biggl[\frac{d}{dx}\, G(z, x, x)\biggr]^{2}- 2G(z, x, x)\biggl[\frac{d^2}{dx^2}\, G(z, x, x)\biggr]=1 \tag 6.32 $$then yields the recursion relation [10]$$\gathered g_{0}(x)=1, \quad g_{1}(x)=\frac12 \,V(x), \\ g_{j+1}(x)=-\frac12 \sum^{j}_{\ell=1}g_{\ell}(x)g_{j+1-\ell}(x) +\frac12 \,V(x)\sum^{j}_{\ell=0} g_{\ell}(x)g_{j-\ell}(x) \\ +\frac18 \sum^{j}_{\ell=0}g'_{\ell}(x)g'_{j-\ell}(x) - \frac14 \sum^{j}_{\ell=0}g''_{\ell}(x)g_{j-\ell}(x), \quad j\in\Bbb N. \endgathered \tag 6.33 $$Equivalently, one could have used the linear third order equation$$ \biggl[\frac{d^3}{dx^3}\,G(z, x, x)\biggr]-4[V(x)-z]\biggl [\frac{d}{dx}\, G(z, x, x)\biggr]+V'(x)G(z, x, x)=0 \tag 6.34 $$to obtain$$\aligned g_{0}(x) &= 1,\\ g'_{j}(x)&=-\frac14 \,g'''_{j-1}(x)+V(x)g'_{j-1}(x)+\frac12 \,V'(x)g_{j- 1}(x), \quad j\in\Bbb N \endaligned \tag 6.35 $$which yields g_{j}(x) upon (homogeneous) integration. Here g_j are homogeneous differential polynomials in V of degree$$ \deg(g_{j})=2j, \quad j\in\Bbb N_{0} \tag 6.36 $$assuming \deg(V^{(m)})=m+2, m\in\Bbb N_{0}. Explicitly, one obtains$$\multline g_{0}=1, \quad g_{1}(x)=\frac12 V(x), \quad g_{2}(x)=\frac38 V(x)^{2}- \frac18 V''(x), \\ g_{3}(x)=\frac{1}{32}V''''(x)-\frac{5}{16}V(x)V''(x)- \frac{5}{32}V'(x)^{2}+\frac{5}{16} V(x)^{3}, \quad\text{etc.} \endmultline \tag 6.37 $$Equation (6.14) then yields$$ \frac{d}{dx}\, H(z, x, x)\operatornamewithlimits{\sim}\limits_{z\to i\infty} \frac{i}{2}\sum^{\infty}_{j=0} [V(x)g'_{j}(x)-g'_{j+1}(x)] z^{-j-1/2} \tag 6.38 $$and hence$$ H(z, x, x)\operatornamewithlimits{\sim}\limits_{z\to i\infty} \frac{i}{2}\sum^{\infty}_{j=0}\biggl[\int\limits^{x}dx'\, V(x') g'_{j}(x')-g_{j+1}(x)\biggr] z^{-j-1/2} +C(z). \tag 6.39 $$Here \int\limits^{x}dx'\,V(x')g'_{j}(x') denotes homogeneous integration, that is, all integration constants are put zero. Moreover, as proven in [10],$$ g_{\ell}(x)g'_{j}(x)=\frac{d}{dx}\, h_{\ell, j}(x) \tag 6.40 $$for some homogeneous differential polynomial h_{\ell, j} in V and hence Vg'_{j}=2g_{1}g'_{j} is a total derivative (see (6.39)). The x-independent constant C(z) in (6.39) can be obtained from the free case V(x)\equiv 0 and one gets (cf.~(6.18), (6.19))$$ C(z)=iz^{1/2}/2. \tag 6.41 $$Alternatively, one could have used$$ H(z, x, x)^{-1}=\phi_{+}(z, x)^{-1}-\phi(z, x)^{-1} \tag 6.42 $$and the asymptotic expansions (5.18) for \phi_{\pm}(z, x). Combining (6.12), (6.31), (6.39), and (6.41) then yields$$\multline [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)]=(iz^{1/2}/2) +(i/2)\sum^{\infty}_{j=0}\biggl[\beta^{2}g_{j}(x)+\beta g'_{j}(x) \\ +\int\limits^{x}dx'\, V(x')g'_{j}(x')-g_{j+1}(x)\biggr] z^{-j-1/2} =(iz^{1/2}/2)\sum^{\infty}_{j=0} c_{\beta, j}(x) z^{-j}, \endmultline \tag 6.43 $$where$$\aligned c_{\beta, 0}(x)&=1, \\ c_{\beta, j}(x)&=\beta^{2}g_{j-1}(x)+\beta g'_{j-1}(x)+\int\limits^{x} dx'\, V(x')g'_{j-1}(x')-g_{j}(x), \quad j\in\Bbb N. \endaligned \tag 6.44 $$Explicitly, one gets$$\gathered c_{\beta, 0}(x)=1, \quad c_{\beta, 1}(x)=\beta^{2}-\frac12 \,V(x), \\ c_{\beta, 2}(x)=\frac12 \,\beta^{2}V(x)+\frac12 \,\beta V'(x)-\frac18 \,V(x)^{2}+\frac18 V''(x), \quad\text{etc}. \endgathered \tag 6.45 $$Hence, applying (5.27)--(5.29) again, one infers$$ \ln[(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)]=\ln(iz^{1/2}/2) + \sum^{\infty}_{j=1}d_{\beta, j}(x)z^{-j}, \tag 6.46 $$where$$\aligned d_{\beta, 1}(x) &=c_{\beta, 1}(x)=\beta^{2}-\frac12 \,V(x), \\ d_{\beta, j}(x) &=c_{\beta, j}(x)-\frac1j \sum^{j-1}_{\ell=1}\ell c_{\beta, j-\ell}(x) d_{\beta, \ell}(x), \quad j\geq 2. \endaligned \tag 6.47 $$Explicitly,$$\aligned d_{\beta, 1}(x)&=\beta^{2}-\frac12 \,V(x), \\ d_{\beta, 2}(x)&=-\frac12 \,\beta^{4}+\beta^{2}V(x)+\frac{\beta}{2} \,V'(x) -\frac14 \,V(x)^{2}+\frac18 \,V''(x), \quad\text{etc.} \endaligned \tag 6.48 $$\endexample This finally leads to the following theorem. \proclaim{Theorem 6.2} Suppose V\in C^{\infty}(\Bbb R), V real-valued and bounded from below. Then for each N\in\Bbb N,$$\multline \text{\rom{Tr}}[(H_{\beta; x}-z)^{-1}-(H-z)^{-1}]=-\frac{d}{dz}\ln [(\beta+\partial_{1})(\beta+\partial_{2})G(z, x, x)] \\ \operatornamewithlimits{\sim}\limits_{z\to i\infty} \sum^{N}_{j=0} r_{\beta, j}(z) z^{-j-1}+O(z^{-N-1}), \quad \beta\in\Bbb R,\, x\in\Bbb R, \endmultline \tag 6.49 $$where$$\aligned r_{\beta, 0}(x)&=-\frac 12,\\ r_{\beta, j}(x)&=jc_{\beta, j}(x)-\sum^{j-1}_{\ell=1} c_{\beta, j- \ell}(x) r_{\beta, \ell}(x), \quad j\in\Bbb N \endaligned \tag 6.50 $$with c_{\beta, j}(x) computed from {\rom{(6.44)}}. \endproclaim \demo{Proof} It suffices to note that$$ r_{\beta, 0}(x)=-\frac12, \quad r_{\beta, j}(x)=jd_{\beta, j}(x), \quad j\in\Bbb N \tag 6.51 $$upon differentiating (6.46) with respect to z. \qed \enddemo Explicitly, one obtains from (6.50), (6.44),$$\aligned r_{\beta, 0}(x) &=-\frac12, \quad r_{\beta, 1}(x)=\beta^{2}-\frac12 \,V(x), \\ r_{\beta, 2}(x) &=-\beta^{4}+2\beta^{2}V(x)+\beta V'(x)-\frac12 \,V(x)^{2}+\frac14 \,V''(x), \quad\text{etc.} \endaligned \tag 6.52 $$It remains to express r_{\beta, j}(x) in terms of \xi_{\beta}(\lambda, x) in analogy to the resolvent regularization procedure sketched in Remark 5.6. By exactly the same procedure one proves the following result. \proclaim{Theorem 6.3} Suppose V\in C^{\infty}(\Bbb R), V real-valued and bounded from below. Assume that \rom{(4.2)} holds and denote E_{\beta, o}(x)=\inf\,\text{\rom{spec}}(H_{\beta; x}). Then$$\align r_{\beta, 0}(x) &=-\frac12 , \tag 6.53 \\ r_{\beta, 1}(x) &=\beta^{2}-\frac12 \, V(x) \\ &=\frac{E_{\beta, o}(x)}{2} + \lim_{z\to i\infty} \int\limits^{\infty}_ {E_{\beta, o}(x)} d\lambda\, \frac{z^2}{(\lambda -z)^{2}} \biggl[\frac12 -\xi(\lambda, x)\biggr], \tag 6.54 \\ r_{\beta, j}(x) &=\frac{E_{\beta, o}(x)^{j}}{2} + \lim_{z\to i\infty} \int\limits^{\infty}_{E_{\beta, o}(x)} d\lambda\, \frac{z^{j+1}}{(\lambda -z)^{j+1}}\, j(-\lambda)^{j-1} \biggl[\frac12 -\xi(\lambda, x)\biggr], \quad j\in\Bbb N, \, x\in\Bbb R. \tag 6.55 \endalign $$\endproclaim Finally, the analog of Example 5.5 in the case where V(x) is periodic reads as follows. \example{Example 6.4} Assume V\in C^{\infty}(\Bbb R), V real-valued, for some a>0, V(x+a)=V(x) for all x\in\Bbb R. Then the spectrum of H is given by (5.55) while the spectrum of H_{\beta; x} is of the type$$\gathered \text{spec}(H_{\beta; x})=\{\lambda_{\beta, n}(x)\}_{n\in\Bbb N_{0}}\cup \bigcup^{\infty}_{n=1}[E_{2(n-1)}, E_{2n-1}], \\ \lambda_{\beta, 0}(x)\leq E_{0}, \quad E_{2n-1}\leq \lambda_{\beta, n}(x)\leq E_{2n}, \quad n\in\Bbb N. \endgathered \tag 6.56 $$The analog of (5.56) then reads$$ \xi_{\beta}(\lambda, x)=\cases 0, &\lambda<\lambda_{\beta, 0}(x), \quad E_{2n-1}<\lambda<\lambda_{\beta, n}(x), \quad n\in\Bbb N \\ -1, &\lambda_{\beta, 0}(x)<\lambda