We study non-smoothness of the fundamental solution for the Schrödinger equation with a spherically symmetric and super-quadratic potential in the sense that *V*(*x*) ≥ *C*|*x*|^{2+ɛ} at infinity with constants *C* > 0 and *ɛ* > 0. More precisely, we show the fundamental solution *E*(*t*, *x*, *y*) does not belong to *C*^{1} as a function of (*t*, *x*, *y*), which partially solves Yajima’s conjecture.

## I. INTRODUCTION

*V*(

*x*)

*d*≥ 3, $i=\u22121$, $\Delta =\u2211j=1d\u22022/\u2202xj2$, and $u:R\xd7Rd\u2192C$. The aim of this paper is to show non-smoothness of the fundamental solutions under the following assumption, which partially solves Yajima’s conjecture.

^{12}

*The potential*$V(x)\u2208C3(Rd)$

*is a real valued spherically symmetric function. Moreover, letting*$V(x)=V\u0303(|x|)$

*, there are constants*

*R*> 0

*and*

*c*> 1

*such that for*

*r*∈ [

*R*,

*∞*)

*C*> 0 such that

*V*(

*x*) ≥

*C*|

*x*|

^{2c}for |

*x*| ≫ 1, which implies

*H*= −Δ +

*V*is essentially self-adjoint on $C0\u221e(Rd)$. Thus the equation generates a unique unitary propagator

*U*(

*t*) =

*e*

^{−itH}on $L2(Rd)$ and

*u*(

*t*,

*x*) = (

*U*(

*t*)

*u*

_{0}) (

*x*) is the unique solution of (1.1). We denote the integral kernel of

*U*(

*t*) by

*E*(

*t*,

*x*,

*y*), that is,

*E*(

*t*,

*x*,

*y*) the fundamental solution of (1.1).

The main theorem of this paper is the following, which claims that *E*(*t*, *x*, *y*) is generically nowhere *C*^{1}.

*Suppose that* *d* ≥ 3 *and that* *V*(*x*) *satisfies Assumption 1.1. Then, for any* $t0\u2208R$ *and* *r*_{1}, *r*_{2} > 0*, there exist* $x0,y0\u2208Rd$*, satisfying* |*x*_{0}| = *r*_{1} *and* |*y*_{0}| = *r*_{2}*, such that the fundamental solution of (1.1) does not belong to* *C*^{1} *near* (*t*_{0}, *x*_{0}, *y*_{0}).

Moreover, we can obtain the following proposition which is (almost) stronger than Theorem 1.2.

*Suppose that*

*d*≥ 3

*and that*

*V*(

*x*)

*satisfies Assumption 1.1. Let*

*Y*

_{n}

*be a spherical harmonic of arbitrary degree*

*n*

*with*$\Vert Yn\Vert L2(Sd\u22121)=1$

*and let*

*Then, for any*$(t0,x0,y0)\u2208R\xd7Rd\xd7Rd$

*,*$EYn$

*is not in*

*C*

^{1}

*near*(

*t*

_{0},

*x*

_{0},

*y*

_{0}).

Here $(\u22c5,\u22c5)H$ denotes the inner product of a Hilbert space $H$, and we suppose $(\u22c5,\u22c5)H$ is linear with respect to the first entry and anti-linear with respect to the second entry.

*H*= −Δ +

*V*by the same symbol. Since

*V*(

*x*) →

*∞*as |

*x*| →

*∞*under Assumption 1.1, the spectrum

*σ*(

*H*) of

*H*is discrete. Thus we have, at least formally,

*u*

_{l}being an eigenfunction of

*H*:

*Hu*

_{l}=

*λ*

_{l}

*u*

_{l}.

*H*is decomposed by the partial wave expansion; let {

*Y*

_{nm}∣

*n*= 0, 1, 2, …,

*m*= 1, 2, …,

*d*

_{n}} be a complete orthonormal system of $L2(Sd\u22121)$ such that each

*Y*

_{nm}is a spherical harmonic of degree

*n*, where $dn=d+n\u22121d\u22121\u2212d+n\u22123d\u22121$ is the dimension of the space of the spherical harmonics of degree

*n*. Then we have

*f*

_{n,l}∈

*L*

^{2}((0,

*∞*)) is the normalized

*λ*

_{n,l}-eigenfunction of the Schrödinger operator on the half line (0,

*∞*):

*f*

_{n,l}(0) = 0. Hence, if

*Y*

_{n}is a spherical harmonic of degree

*n*, we have

^{12}

*E*(

*t*,

*x*,

*y*), the formal summation over spherical harmonics

*C*

^{1}, which is, however, still not clear. Instead, we can obtain Theorem 1.2.

*(i)*$EYn(t,x,y)$

*is actually defined as a distribution on*$R2d+1$

*by*

*The representation (1.5) implies the identity*

The smoothness of the fundamental solution is related to the growth rate of *V*. Fujiwara (Ref. 2, Theorem 1.1) has given the construction of the fundamental solution *E*(*t*, *x*, *y*) with the classical orbit if *V* is at most of quadratic growth, which shows that *E*(*t*, *x*, *y*) is smooth with respect to (*x*, *y*) for any *t* ≠ 0 small enough as a corollary.

*x*

_{0},

*ξ*

_{0}) is not in the wave front set of

*E*(

*t*

_{0}, ·) if and only if

*φ*is a Schwartz function,

*φ*

_{λ}(

*x*) =

*λ*

^{d/2}

*φ*(

*λ*

^{1/2}

*x*) and

*x*(

*t*;

*t*

_{0},

*x*

_{0},

*λξ*

_{0}) is the solution of $x\u0307(t)=\xi (t),\xi \u0307(t)=\u2212\u2207V(t,x),x(t0)=x0,\xi (t0)=\lambda \xi 0$, which implies the smoothness of the fundamental solutions for Schrödinger equations with sub-quadratic potentials.

On the other hand, Yajima (Ref. 12, Theorem 1.2) has studied that if *V* is super-quadratic and the spatial dimension is one, then *E*(*t*, *x*, *y*) is not smooth anywhere with respect to (*t*, *x*, *y*). The proof is given by the estimates of eigenvalues and eigenfunctions of *H*. Yajima has conjectured that the same result as in Yajima (Ref. 12, Theorem 1.2) is true even for higher dimensional cases.

As a first step for generalization of Yajima’s result, we treat the case that *V* is spherically symmetric with the dimension *d* ≥ 3. We use the unitary equivalence of *H* = −Δ + *V* to *⨁*_{n,m}*H*_{nm}, where the operator *H*_{nm} is defined on (0, *∞*) (see Lemma 2.3), and the estimates of eigenvalues and eigenfunctions which are shown by the same way as in Ref. 12. Our main theorem partially solves Yajima’s conjecture. In the case that *d* = 2, the projection of *H* onto the subspace of spherically symmetric functions is unitarily equivalent to $H0=\u2212d2dr2\u221214r2+V\u0303(r)$ on *L*^{2}((0, *∞*)) (see Sec. 2). The potential $\u221214r2+V\u0303(r)$ is not bounded from below near 0, which requires additional argument of self-adjoint extensions. We shall discuss the two-dimensional case in the forthcoming paper.

For a Schrödinger operator −Δ_{g} on a complete Riemannian manifold (*M*, *g*), the smoothness of the fundamental solution depends on whether *M* is compact or not. Kapitanski and Rodnianski (Ref. 3, Theorem I–III) has studied that *E*(*t*, *x*, *y*) is not smooth if *M* is the circle. Taylor (Ref. 8, Sec. 1) has mentioned that *E*(*t*, *x*, *y*) is not smooth if *M* is the sphere. Yajima (Ref. 12, Remark 4) has pointed out that *E*(*t*, *x*, *y*) is not smooth if *M* is the bounded interval [0, *π*] with the Dirichlet condition. Taira (Ref. 7, Remark 3.3) has studied that *E*(*t*, *x*, *y*) is not smooth if *M* is compact. When *M* is non-compact, Doi (Ref. 1, Theorem 1.5) has studied the smoothness of the fundamental solution in terms of the wave front set. Taira (Ref. 7, Theorems 1.1) has given a sufficient condition under which *E*(*t*, *x*, *y*) is smooth.

We introduce some notation. For sets *U* and *V*, we write *U* ⋐ *V* if *U* is relatively compact with respect to *V*. We write $C0\u221e(U)={f\u2208C\u221e(U)\u2223suppf\u22d0U}$, where supp*f* denotes the support of *f*. For an open interval *I*, we denote the Sobolev space on *I* of order $k\u2208N$ by *H*^{k}(*I*) = {*f* ∈ *L*^{2}(*I*)∣*f*^{(m)} ∈ *L*^{2}(*I*) for *m* = 1, …, *k*}. We denote by $H01(I)$ the closure of $C0\u221e(I)$ in *H*^{1}(*I*). We denote the Schwartz space by $S(Rd)$. For any $f\u2208S(R)$, the Fourier transform $f\u0302$ of *f* is defined by $f\u0302(\lambda )=\u222bRe\u2212it\lambda f(t)dt$.

## II. SCHRÖDINGER OPERATOR IN THE POLAR COORDINATES

In the following, space dimension *d* is larger than or equal to 3. We call a function *Y* on $Sd\u22121$ *spherical harmonic of degree n* if *Y* is the restriction to $Sd\u22121$ of a homogeneous harmonic polynomial of degree *n*. We denote by $Hn$ the space of spherical harmonics of degree *n*. The following lemma is well-known [for more details and proofs, see Stein and Weiss (Ref. 6, Sec. 2 in Chap. IV) or Yajima (Ref. 13, Sec. 13 in Chap. 4)].

*Let* $\Delta Sd\u22121$ *be the Laplace-Beltrami operator on* $Sd\u22121$*. Then:*

$\u2212\Delta Sd\u22121Y=n(n+d\u22122)Y$

*for*$Y\u2208Hn$.$L2(Sd\u22121)=\u2a01n=0\u221eHn$.

$dn=dimHn=d+n\u22121d\u22121\u2212d+n\u22123d\u22121$.

*Y*

_{nm}∣

*m*= 1, …,

*d*

_{n}} be an orthogonal basis of $Hn$, then {

*Y*

_{nm}∣

*n*= 0, 1, … and

*m*= 1, …,

*d*

_{n}} is a complete orthonormal system of $L2(Sd\u22121)$. If we define $Jnm:L2(Rd)\u2192L2((0,\u221e))$ by

*m*, we let

*If* (*d*, *n*) ≠ (3, 0)*,* *H*_{n} *is essentially self-adjoint on* $C0\u221e((0,\u221e))$*. In particular, the domain of its closure is the maximal domain* {*g* ∈ *L*^{2}((0, *∞*))∣*H*_{n}*g* ∈ *L*^{2}((0, *∞*))}*. If* (*d*, *n*) = (3, 0)*,* *H*_{0} *is essentially self-adjoint on* $J00C0\u221e(Rd)$*, and the domain of its closure is* ${g\u2208H01((0,\u221e))\u2223Hng\u2208L2((0,\u221e))}$.

Weyl’s limit point-limit circle criterion (Ref. 5, Theorem X.7) implies that *H*_{n} is essentially self-adjoint on $C0\u221e((0,\u221e))$ if and only if *H*_{n} is in the limit point case at both 0 and *∞*. It follows from Ref. 5 (Theorem X.8) that *H*_{n} is in the limit point case at *∞* for any (*d*, *n*), since the potential of *H*_{n} is bounded from below. Reference 5 (Theorem X.10) implies that *H*_{n} is in the limit point case at 0 if and only if the coefficient $(d\u22121)(d\u22123)4+n(n+d\u22122)$ of *r*^{−2} is not smaller than $34$, i.e., (*d*, *n*) ≠ (3, 0). Thus we have the assertion in the case (*d*, *n*) ≠ (3, 0).

*d*,

*n*) = (3, 0),

*H*

_{n}is in the limit point case at

*∞*and is in the limit circle case at 0. In this case the self-adjoint extensions of $H0=\u2212d2dr2+V\u0303(r)$ on $C0\u221e((0,\u221e))$ are those with the domain

*θ*∈ [0, 2

*π*). We can see that $C0\u221e((0,\u221e))\u2282J00C0\u221e(Rd)$ and that $J00C0\u221e(Rd)\u2282D\theta $ only if

*θ*= 0, since

*J*

_{00}

*u*(0) = 0 and (

*J*

_{00}

*u*)′(0) = 1 if $u\u2208C0\u221e(Rd)$ equals to $|S2|\u22121/2=(4\pi )\u22121/2$ near 0. Therefore, the self-adjoint extension of

*H*

_{0}on $J00C0\u221e(Rd)$ is unique and the domain of its closure is $D0$.□

We denote the above-mentioned self-adjoint extensions by the same symbols.

*One has*

*More precisely,*$J*({gnm}nm)\u2208D(H)$

*if and only if*

*g*

_{nm}∈

*D*(

*H*

_{n})

*for any*

*n*,

*m*

*and*$\u2211n,m\Vert Hngnm\Vert 2<\u221e$

*, and*

## III. ESTIMATES OF EIGENVALUES AND EIGENFUNCTIONS

*n*, and consider the eigenvalue problem

*H*

_{n}. The following argument is based on Ref. 12 (Sec. 3).

### A. Asymptotic behavior in bounded regions

*λ*→

*∞*. Let

*λ*

_{0}be sufficiently large so that Ω

_{λ}is an interval for

*λ*≥

*λ*

_{0}. We write

*y*=

*S*(

*r*) and

*f*(

*r*) =

*a*(

*r*)

*w*(

*y*). Then (3.1) implies

*w*satisfies the integral equation

*w*(0) and

*w*′(0) are given by

*f*(1),

*f*′(1),

*U*

_{d,n}(1), $Ud,n\u2032(1)$ and

*λ*; in fact,

*C*

_{λ}= (

*w*(0) −

*iw*′(0)) so that

*w*(0) cos

*y*+

*w*′(0) sin

*y*= Re(

*C*

_{λ}

*e*

^{iy}).

*Suppose that a real valued function*

*f*

*satisfies (3.1) on*Ω

_{λ}

*. Then, there exists a constant*

*C*> 0

*independent of*

*λ*≥

*λ*

_{0}

*and*

*r*∈ Ω

_{λ}

*such that*

*R*be the same as in Assumption 1.1. Since $Ud,n\u2032(r)\u2264CUd,n(r)\u2264C(\lambda \u2212Ud,n(r))$ on Ω

_{λ}∩ [

*R*,

*∞*), we have, for any

*r*

_{0}∈ Ω

_{λ}∩ (

*R*,

*∞*),

*r*

_{1}∈ Ω

_{λ}∩ (0, 1), we have

*a*(

*r*) =

*λ*

^{−1/4}(1 +

*O*(

*λ*

^{−1})) uniformly on any compact set of (0,

*∞*). Hence, for any compact set

*I*⊂ (0,

*∞*), we have

*I*⊂ Ω

_{λ}for sufficiently large

*λ*and

*r*∈

*I*.

The next lemma gives a lower bound of |*C*_{λ}| in the previous lemma. The assertion is proved by the same way as in Ref. 12 (Lemma 3.2) and we omit the proof.

*Suppose that a real valued function*

*f*∈

*L*

^{2}((0,

*∞*))

*satisfies (3.1) and*$\Vert f\Vert L2((0,\u221e))=1$

*, and let*

*C*

_{λ}

*be as in Lemma 3.1. Then there is a constant*

*C*> 0

*such that*

*where*

*c*> 1

*is the constant in Assumption 1.1*.

### B. Gaps of the eigenvalues

Let *λ*_{n,0} < *λ*_{n,1} < ⋯ be the eigenvalues of *H*_{n}. We show that the gap *λ*_{n,l+1} − *λ*_{n,l} of eigenvalues increases polynomially with respect to *λ*_{n,l}.

*There exists a constant*

*C*> 0

*which satisfies*

*for any n and large*

*l*.

The proof of the above lemma is found in Ref. 12 (Lemma 3.3), for which we use.

*Let*

*d*≥ 3

*and*

*n*

*be fixed. Then as*

*l*→

*∞*

*where*$V\u0303(Xn,l)=\lambda n,l$.

We show the Proof of Proposition 3.4 in the rest of this section. Note that it has been proved for the case *d* = 3, see Titchmarsh (Ref. 9, p. 151).

### C. Asymptotic behavior of eigenfunctions

*For sufficiently large*

*λ*

*, there is a real-valued function*

*ψ*

_{λ}∈

*H*

^{2}((1/2,

*∞*))

*such that*

*and*

*where*

*and*

*T*=

*T*

_{λ}> 1

*is the value satisfying*

*U*

_{d,n}(

*T*) =

*λ*.

*H*

_{n}is the limit point case at

*∞*, see the Proof of Lemma 2.2. We give an instant proof for the readers’ convenience. Suppose that there are linearly independent

*L*

^{2}-eigenfunctions

*f*

_{1}and

*f*

_{2}of

*H*

_{n}with eigenvalue

*λ*. Then the ellipticity of

*H*

_{n}implies $f1,f2\u2208Hloc2((0,\u221e))$, and the Wronskian $W(r)=f1(r)f2\u2032(r)\u2212f1\u2032(r)f2(r)\u22600$ is independent of

*r*. Thus we learn

*λ*be sufficiently large, and let

*T*=

*T*

_{λ}> 1 be as above. We set

*ζ*(

*r*) =

*π*/2 for

*r*>

*T*, and arg

*ζ*(

*r*) = −

*π*for

*r*<

*T*.

*ζ*= 0, i.e.,

*r*=

*T*. Changing variables

*r*→

*ζ*and

*η*=

*a*(

*r*)

^{−1}

*ψ*(

*r*), where $a(r)=\lambda \u2212Ud,n(r)\u22121/4$, imply, at least formally,

*J*

_{ν}is the Bessel function of the first kind and $H\nu (j)$ is the Hankel function, are the solutions of

*ψ*

_{j}(

*r*) =

*a*(

*r*)

*η*

_{j}(

*ζ*(

*r*)),

*j*= 1, 2, are real valued. We also remark that

*η*(

*ix*) ∼

*η*

_{2}(

*ix*) as

*x*→

*∞*, since

*ψ*

_{2}(

*r*) =

*a*(

*r*)

*η*

_{2}(

*ζ*(

*r*)) is

*L*

^{2}near

*∞*. Thanks to the standard ODE calculus, we only have to observe the solution of the integral equation

*ζ*, and the contour of the above first integral is taken over the path connecting

*ζ*and

*i∞*in (−

*∞*, 0] ∪

*i*[0,

*∞*). We note that (3.13) implies the integral in (3.15) maps the functions

*η*satisfying (3.13) into themselves, which guarantees that the solution of (3.15) satisfies (3.13) and hence

*ψ*

_{λ}(

*r*) =

*a*(

*r*)

*η*(

*ζ*(

*r*)) is real valued.

*F*

_{λ}

*η*(

*ζ*), and we define a function space

*X*and its norm by

*η*

_{1}(

*ζ*) =

*O*(

*e*

^{Imζ}),

*η*

_{2}(

*ζ*) =

*O*(

*e*

^{−Imζ}) [see Ref. 11 (Sec. 7.21) for the asymptotic behavior of $J13(\zeta )$ and $H13(1)(\zeta )$ as |

*ζ*| →

*∞*], and that

*η*

_{2}‖

_{X}<

*∞*and ‖

*F*

_{λ}‖

_{X→X}=

*O*(

*λ*

^{−1/2}

*T*

^{−1}), which implies

*λ*and $\zeta \u2208(\zeta (12),0]\u222ai[0,\u221e)$, and thus

*r*<

*T*, we have $H13(1)(\zeta )=23e\u221216\pi i{J13(z)+J\u221213(z)}$, where

*ζ*=

*e*

^{−iπ}

*z*[see Ref. 9 (Sec. 7.8)]. By using

*λ*, we obtain (3.8).

*ζ*on $(\zeta (12),0)\u222ai(0,\u221e)$ implies

*ψ*

_{λ}∈

*H*

^{2}((1/2,

*∞*)), it suffices to confirm that $\psi \lambda \u2032$ is continuous at

*r*=

*T*. We compute

*j*= 1, 2, are continuous at

*r*=

*T*. The same argument for the proof of continuity of $\phi \xb1\u2032$ in Appendix A, where $\phi \xb1(r)=a(r)\zeta 12J\xb113(\zeta )$, shows

*r*=

*T*, which implies

*ψ*

_{λ}is continuous at

*r*=

*T*.□

*For sufficiently large*

*λ*

*, there is*

*ϕ*

_{λ}∈

*H*

^{2}((0, 2))

*such that*

*and*

*d*,

*n*) ≠ (3, 0) for simplicity. It is easy to see that $\varphi +(r)=r12J(n+d\u221222)(r\lambda )$ and $\varphi \u2212(r)=r12Y(n+d\u221222)(r\lambda )$, where

*Y*

_{ν}is the Bessel function of the second kind, are solutions of

*ϕ*of (3.1) with

*ϕ*∼

*ϕ*

_{+}, regarding the term $V\u0303(r)\varphi (r)$ as a perturbation employing the norm $\Vert f\Vert =supr\u2208(0,2)min(1,(\lambda r)n+d\u221212)\u22121|f(r)|$. Finally we have (3.21) and (3.22) by the asymptotic behavior (3.18).□

### D. Proof of Proposition 3.4

Suppose that *λ*_{n,l} is the *l* + 1th eigenvalue of *H*_{n} and that *f*_{n,l} is the real valued eigenfunction for *λ*_{n,l}.

*Z*in Lemma 3.5 with

*λ*=

*λ*

_{n,l}. Let $mn,l\u2208Z$ and $\delta \u2208[\u2212\pi 2,\pi 2)$ be the constants satisfying

*Let*

*X*

_{n,l}> 1

*be the constant in Proposition 3.4 such that*$V\u0303(Xn,l)=\lambda n,l$

*. Then*

*A*and

*B*, we have

By Lemma 3.7, it suffices to show that *m*_{n,l} = *l* for sufficiently large *l*, which follows from counting the zeros of *f*_{n,l} in (0, *∞*) by two different methods.

We easily see [e.g. Ref. 9 (Sec. 5.4)] that *f*_{n,l} has *l* zeros in (0, *∞*), since *f*_{n,l} is the *l* + 1th eigenfunction of *H*_{n}. On the other hand, we can find that *m*_{n,l} is the number of the zeros of *f*_{n,l}, which concludes the proof. In the rest of the proof, we show that.

*Let* $p=\pi \u22121\lambda n,l\u22122n+d\u221214\u2208Z$ *and* $r0=(p+2n+d\u221214)\pi /\lambda n,l$ *(note that* $1\u22121\pi \lambda n,l<r0\u22641$*). Then*

*f*_{n,l}*has no zeros in*[*T*,*∞*).*f*_{n,l}*has**m*_{n,l}−*p**zeros in*[*r*_{0},*T*).*f*_{n,l}*has**p**zeros in*(0,*r*_{0}).

(i) is proved by contradiction. If there exists a zero of *f*_{n,l} in [*T*, *∞*), *f*_{n,l} and $fn,l\u2032$ are positive (or negative) on the right side of the zero. Since $fn,l\u2033(r)=(Ud,n(r)\u2212\lambda n,l)fn,l(r)$, *f*_{n,l} is convex (or concave). Hence *f*_{n,l} tends to *∞* (or −*∞*). This contradicts that *f*_{n,l} ∈ *L*^{2}((0, *∞*)).

The proof is divided into four steps.

We remark that the Proof of Lemma 3.5 implies $fn,l(r)=c\psi \lambda n,l(r)=ca(r)\eta (\zeta (r))$ with some $c\u2208R\{0}$, where $a(r)=(\lambda n,l\u2212Ud,n(r))\u221214$, $\zeta (r)=\u222bTr\lambda n,l\u2212Ud,n(s)ds$ and

*η*(*ζ*) is the solution of (3.15). Since*ζ*([*r*_{0},*T*)) = [*ζ*(*r*_{0}), 0), the number of zeros of*f*_{n,l}(*r*) in [*r*_{0},*T*) is same as that of zeros of*η*(*ζ*) in [*ζ*(*r*_{0}), 0).We show.

$\u2212\zeta (r0)=mn,l\u2212p+14\pi +o(1)$, and

*m*_{n,l}−*p*→*∞*as*l*→*∞*.$\eta 2(\zeta )=2\u2061cos\u2212\zeta \u221214\pi +O(|\zeta |\u22121)$.

Let

*ɛ*= inf{maximal values of |*η*_{2}(*ζ*)| in (−*∞*, 0)}. Then*ɛ*> 0 holds.

- Sincethe definitions of$\u2212\zeta (r0)\u2212Zn,l=\u222br01{\lambda n,l\u2212Ud,n(r)}12dr=(1\u2212r0)\lambda n,l+O\lambda n,l\u221212,$
*m*_{n,l},*p*,*r*_{0}show thatSince $\u2212\zeta (r0)\u2265\u222b1T\lambda n,l\u2212Ud,n(s)ds>c\lambda n,l\u2192\u221e$ as$\u2212\zeta (r0)=2n+d4+mn,l\pi \u2212r0\lambda n,l+\delta =mn,l\u2212p+14\pi +o(1).$*l*→*∞*,*m*_{n,l}−*p*→*∞*also holds. is proved by $H13(1)(\zeta )=23e\u221216\pi i{J13(ei\pi \zeta )+J\u221213(ei\pi \zeta )}$ in (−

*∞*, 0) and the asymptotic behavior (3.18) of $J\xb113$.*η*_{2}is concave (resp. convex) if*η*_{2}(*ζ*) > 0 (resp.*η*_{2}(*ζ*) < 0) since*η*_{2}solves $d2\eta d\zeta 2+1+536\zeta 2\eta =0$. Hence this and (2) imply*ɛ*> 0.Step 3. Let

*S*= {*ζ*∈ [*ζ*(*r*_{0}), 0)∣|*η*(*ζ*)| <*ɛ*/3}. Then*η*(*ζ*) is monotone on each segment of*S*. In fact, $S\u2282\eta 2\u22121((\u22122\epsilon /3,2\epsilon /3))$ for large*l*by (3.17), and thus (3.20) implies*η*′(*ζ*) ≠ 0 for*ζ*∈*S*. Moreover, we note that.- If
*ζ*is between*ζ*(*r*_{0}) and $\u2212mn,l\u2212p+14\pi $, Step 2 impliesfor sufficiently large$|\eta (\zeta )|\u2265|\eta 2(\zeta )|\u2212O\lambda n,l\u221212T\u22121\u2265\eta 2\u2212mn,l\u2212p+14\pi \u2212o(1)\u22652\u2212o(1)>\epsilon /2$*l*. - Since
*f*_{n,l}is bounded and*a*(*r*)^{−1}→ 0 as*r*→*T*,Therefore each segment of$\eta (\zeta )=c\u22121a(r)\u22121fn,l(r)\u21920as\u2009\zeta \u2192\u22120.$*S*, except the one of the form (−*a*, 0), has exactly one zero, and the number of segments of*S*\ (−*a*, 0) is same as that of zeros of*η*_{2}in $[\u2212mn,l\u2212p+14\pi ,0)$ thanks to (3.17).

Step 4. The number of the zeros of $J13(z)+J\u221213(z)$ in $(0,mn,l\u2212p+14\pi ]$ is

*m*_{n,l}−*p*for large*l*[Ref. 9 (Lemma 7.9b)]. Thus, since $H13(1)(\zeta )=23e\u221216\pi i{J13(ei\pi \zeta )+J\u221213(ei\pi \zeta )}$ in (−*∞*, 0), $\eta 2(\zeta )=12\pi \zeta 12H13(1)(\zeta )$ has exactly*m*_{n,l}−*p*zeros in $[\u2212mn,l\u2212p+14\pi ,0)$, which concludes the proof of (ii).

is proved by the same argument as above, noting that $\varphi +(r)=r12J(n+d\u221222)(r\lambda )$ has

*p*zeros in (0,*r*_{0}) [the number of the zeros of $J(n+d\u221222)(r)$ is*p*in $(0,(p+2n+d\u221214)\pi )$ by Ref. 9 (Lemma 7.9a)].□

## IV. PROOF OF THEOREM 1.2 AND PROPOSITION 1.3

*U*(

*t*) =

*e*

^{−itH}, the spectral decomposition theorem and Fubini’s theorem, where $\kappa \u0302(\lambda )=\u222bRe\u2212it\lambda \kappa (t)dt$. It follows from Lemma 2.3 that

*C*

^{1}on $(0,\u221e)\xd7{x\u2208Rd\u2223x\u22600}\xd7{y\u2208Rd\u2223y\u22600}$, which we show by contradiction. Suppose that $EYn$ is in

*C*

^{1}near (

*t*

_{0},

*x*

_{0},

*y*

_{0}) with

*t*

_{0}> 0,

*x*

_{0}≠ 0, and

*y*

_{0}≠ 0. The definition of $EYn$ and Remark 1.4 imply that

*I*⋐ (0,

*∞*) and 0 <

*r*

_{j}<

*R*

_{j},

*j*= 1, 2, where $Br,R={x\u2208Rd\u2223r<|x|<R}$. We may assume

*Y*

_{n}=

*Y*

_{n1}without loss of generality.

*ϕ*(

*r*) =

*J*

_{n1}Φ(

*r*) ≢ 0 and

*ψ*(

*r*) =

*J*

_{n1}Ψ(

*r*) ≢ 0 are non-negative. We set

*C*> 0 and a sequence (

*τ*

_{l},

*j*

_{l},

*k*

_{l}) such that $\tau l+jl+kl\u2192\u221e$, as

*l*→

*∞*such that

*τ*=

*λ*

_{n,l}and $jl=kl=\lambda n,l$.

*f*

_{n,l},

*l*= 0, 1, …, is the normalized eigenfunction of

*H*

_{n}with eigenvalue

*λ*

_{n,l}. Then it follows from Lemma 3.3 and $\kappa \u0302\u2208S(R)$ that for any $N\u2208N$

*ϕ*⋐ Ω

_{λ}for sufficiently large

*λ*, where Ω

_{λ}is as in Lemma 3.1. Thus, by using (3.4), we can compute

*l*with some

*c*

_{0}> 0, which concludes the assertion.□

*ϕ*

_{j}(

*r*) =

*e*

^{irj}

*ϕ*(

*r*) and

*ψ*

_{k}(

*r*) =

*e*

^{−irk}

*ψ*(

*r*). Thus the same argument as in the Proof of Theorem 1.3 implies $G(\tau l,jl,kl)\u2265c0\lambda n,l\u22121/2c$. Therefore we obtain $E(t,x,y)\u2209C1(I\xd7Br1,R1\xd7Br2,R2)$ for any interval

*I*⋐ (0,

*∞*) and 0 <

*r*

_{j}<

*R*

_{j},

*j*= 1, 2, which completes the proof.□

## ACKNOWLEDGMENTS

The authors would like to thank the referee for helpful comments. The authors would also like to thank Fumihito Abe, Ryo Muramatsu and Kouichi Taira for helpful discussions. K.K. was partially supported by JSPS KAKENHI Grant No. 22K03394. Y.T. was partially supported by JSPS KAKENHI Grant No. 23K12991. The authors appreciate useful comments made by the referee.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Keiichi Kato**: Writing – original draft (equal); Writing – review & editing (equal). **Wataru Nakahashi**: Writing – original draft (equal); Writing – review & editing (equal). **Yukihide Tadano**: Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX A: PROOF OF (3.14)

Since $H13(1)(\zeta )=23(e\pi 6iJ13(\zeta )+e\u2212\pi 2iJ\u221213(\zeta ))$, we show *φ*_{±}(*r*) = *a*(*r*)*η*_{±}(*ζ*(*r*)) ∈ *H*^{2}((1/2, *∞*)) with $\eta \xb1(\zeta )=\zeta 12J\xb113(\zeta )$. We easily see that *φ*_{±} are *H*^{2} away from *r* = *T*, since *η*_{±} solve (3.12) on {*ζ* ≠ 0}. Thus it suffices to show that *φ*_{±} and $\phi \xb1\u2032$ are continuous at *r* = *T* and that $\phi \xb1\u2033$ is *L*^{2} near *r* = *T*, which imply that *φ*_{±} are *H*^{2} near *r* = *T*.

*φ*

_{±}are continuous near

*r*=

*T*.

*r*=

*T*follows from (A1), while the continuity of $\phi \u2212\u2032$ at

*r*=

*T*is proved by, in addition to (A1),

*η*

_{±}solve (3.12) on {

*ζ*≠ 0}, we have

*L*

^{2}near

*r*=

*T*.

### APPENDIX B: PROOF OF (3.16)

*U*

_{d,n},

*g*, and $U$ are the following,

*U*(

*r*) instead of

*U*

_{d,n}(

*r*) until the end of the proof. We divide the integral as follows,

*ɛ*> 0 is a small constant which will be fixed later.

Estimate of *I*_{2}.

*ζ*, we have

*T*/2 ≤

*r*≤ 2

*T*,

*U*′(

*r*) is monotonically increasing. By using this property and (1.3), we have, for any

*T*≤

*r*≤ 2

*T*,

*T*/2 ≤

*r*≤

*T*,

*ɛ*> 0 so that

*ɛ*)

*T*≤

*r*≤ (1 +

*ɛ*)

*T*. In fact, $logU(ar)U(r)=\u222brarU\u2032(s)U(s)ds$ and (1.3), (1.4) imply for

*a*≥ 1

*ɛ*→ 0. By using (B1) and (B2), and Taylor’s expansion of (1 −

*x*)

^{−2}with $x=25U\u2033(r)(\lambda \u2212U(r))U\u2032(r)2\u2212S(r)\u2208(\u221212,12)$, we have

*I*

_{2}=

*O*(

*λ*

^{−1/2}

*T*

^{−1}).

Estimate of *I*_{1}.

*R*be the constant as in Assumption 1.1. By (1.3), there exists a constant

*C*> 0 such that

*λ*=

*U*(

*T*) ≥

*CT*

^{2c}, i.e., $\lambda \u22121\u2264\lambda \u221212c\u2264CT\u22121$. Then we have

*U*((1 −

*ɛ*)

*T*) ≤

*c*

_{ɛ}

*U*(

*T*) =

*c*

_{ɛ}

*λ*with some

*c*

_{ɛ}< 1,

*ζ*, we have

*I*

_{1}=

*O*(

*λ*

^{−1/2}

*T*

^{−1}).

Estimate of *I*_{3}.

*r*

^{−1}

*U*(

*r*) is monotonically increasing, we have

*ζ*′(

*s*) = (

*λ*−

*U*(

*s*)),

*U*((1 +

*ɛ*)

*T*) ≥

*c*

_{ɛ}

*U*(

*T*) =

*c*

_{ɛ}

*λ*with some

*c*

_{ɛ}> 1, we have $U(s)U(s)\u2212\lambda =O(1)$ and (

*U*(

*s*) −

*λ*)

^{−1/2}=

*O*(

*λ*

^{−1/2}) for

*s*≥ (1 +

*ɛ*)

*T*. By using these estimates and (1.4), we have

*I*

_{3}=

*O*(

*λ*

^{−1/2}

*T*

^{−1}).□

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