5.1 A Prime Example: Classical Laplacian and Brownian Motion

First we present the case of the classical Schrödinger operator with a potential well, for which not only estimates can be obtained but a full reconstruction of the ground state is possible by using the martingale \((M_t)_\) in (4.1). Alternatively, this can be done by an explicit solution of the Schrödinger eigenvalue equation, which in this case is a textbook example; however, our point here is that while the eigenvalue problem cannot in general be solved for non-local cases, the probabilistic approach is a useful alternative and this example shows best how this can be done by using occupation times.

Proposition 5.1

Let

$$\begin H = -\frac\frac^2}x^2} - v }}_} \end$$

be given on \(L^2(\mathbb )\). Then, the normalized ground state of H is

$$\begin \varphi _0(x) = A_0 e^|x|} }_} + B_0\cos \left( \sqrt \,x\right) }_}, \end$$

with

$$\begin A_0 = \sqrt}}} e^} \cos \left( a\sqrt\right) , \quad B_0 = \sqrt}}}. \end$$

Proof

Consider for any \(b,c \in \mathbb \) with \(b<0<c\), the first hitting times

$$\begin T_ = \inf \, \quad T_ = \inf \, \quad \text \quad T_ = T_ \wedge T_c, \end$$

for Brownian motion \((B_t)_\) starting at zero, and recall the general formula by Lévy [45]

$$\begin \mathbb ^x[e^}] = \frac}}} + e^ }} + \frac }}} + e^}}, \end$$

with \(b< x < c\), and

$$\begin \mathbb [e^] = e^|b|} \quad \text \quad \mathbb [e^}] = \frac\,\frac\right) }\, \frac\right) }, \quad u \ge 0. \end$$

(5.1)

It is well known that all these hitting times are almost surely finite stopping times with respect to the natural filtration. From (2.4) we have

$$\begin \varphi _0(x) = \mathbb [e^(a)} \varphi _0(B_t + x)], \end$$

where we denote

$$\begin U^x_(a) = \int _0^t }_} \hbox s = \int _0^t }_} \hbox s. \end$$

Then, \(U^x_}(a) = T_\) whenever \(|x| < a\), and is zero otherwise. Using Proposition 4.1, we obtain

$$\begin \varphi _0(x)= & \left[ e^ + vU^x_}(a)} \varphi _0(B_}+x)\right] . \end$$

Now suppose \(x > a\). By path continuity \(T_ = T_\) and thus

$$\begin \varphi _0(x)= & \ \mathbb [e^} \varphi _0(B_}+x)] = \varphi _0(a) \mathbb [e^}] \\= & \ \varphi _0(a) e^(x-a)}. \end$$

We obtain similarly for \(x < -a\) that \(T_ = T_\) and

$$\begin \varphi _0(x)= & \ \mathbb [e^} \varphi _0(B_}+x)] = \varphi _0(a) \mathbb [e^}] \\= & \ \varphi _0(a) e^(x-a)}. \end$$

using \(\varphi _0(-a)=\varphi _0(a)\). When \(-a< x < a\), the two-barrier formula in (5.1) gives

$$\begin \varphi _0(x)= & \ \mathbb [e^} \varphi _0(B_}+x)] \\= & \ \mathbb [e^} \varphi _0(B_}+x) }}_ < T_\}}] \\ & + \, \mathbb [e^} \varphi _0(B_}+x) }}_ > T_\}}] \\= & \ \varphi _0(a) \fracx)}a)}. \end$$

The constant \(\varphi _0(a)\) can be determined by the normalization condition \(\Vert \varphi _0\Vert _2=1\), which then yields the claimed expression of the ground state.

Remark 5.1

The argument can also be extended to higher dimensions. For instance, for \(d \ge 3\), denote by \(\mathcal _r(z)\) a ball of radius r centred in z, write \(\mathcal _r = \mathcal _r(0)\), and define the stopping times

$$\begin T_ = \inf \}_r \} \quad \text \quad \tau _ = \inf \_r \}. \end$$

Using the Ciesielski–Taylor formulae (see e.g. [18, eq. (2.15)] and [12, formula 2.0.1])

$$\begin & \mathbb ^x[e^}] = \left( \frac\right) ^} \frac}(|x|\sqrt)}}(r\sqrt)} \quad \text \\ & \mathbb ^x[e^}] = \left( \frac\right) ^} \frac}(|x|\sqrt)}}(r\sqrt)}, \end$$

and the properties of the Bessel function \(J_\) and modified Bessel functions \(I_\) and \(K_\) in standard notation (for properties of the Bessel functions, we refer to [62]), by a similar argument as above for the potential well \(-v }__a}\) we obtain

$$\begin \varphi _0(x)= & \ A_0 \left( \frac\right) ^} K_}\big (\sqrt\, |x|\big ) }}_} \\ & + B_0 \left( \frac\right) ^}J_}\big (\sqrt\,|x|\big ) }}_}, \end$$

where the constants \(A_0, B_0\) can be determined from \(L^2\)-normalization as before. The details are left to the reader.

5.2 Local Behaviour of the Ground State

To come to our main point in this section, we need some scaling estimates on the Lévy measure \(\nu _\) of the exterior of a ball.

Lemma 5.1

For every \(R>0\) there exists a constant \(C_ > 1\) such that

$$\begin \int __R}^c}j_(|x-y|)\textrmy\le \frac\int __^c}j_(|x-y|)\textrmy. \end$$

Moreover, if \(m=0\), then \(C_\) does not depend on R.

Proof

Since \(j_\) is non-increasing, for every \(\theta >0\) the set \(\(|x|)\ge \theta \}\) is a ball and then \(\nu _(\hbox x)\) is unimodal. As a consequence of Anderson’s inequality [2, Th. 1] we get \(\int __R^c}j_(|x-y|)\hbox y \ge \int __R^c}j_(|y|)\hbox y\), for every \(R>0\) and \(x \in \mathcal _R\). Taking \(R>0\), \(x \in \mathcal _R\) and \(k>2\), we obtain

$$\begin \int _^c_}j_(|x-y|)\hbox y&\le \int _^c_(x)}j_(|x-y|)\hbox y\\ &= \int _^c_}j_((k-1)|y|)\hbox y\\&=(k-1)^d\int _^c_}j_\left( (k-1)|y|\right) \hbox y. \end$$

First consider \(m=0\). We have

$$\begin \int _^c_}j_((k-1)|y|)\hbox y=\frac}\int _^c_}j_(|y|)\hbox y, \end$$

and thus

$$\begin \int _^c_}j_(|x-y|)\hbox y&\le \frac}\int _^c_}j_\left( |y|\right) \hbox y\\&\le \frac}\int _^c_}j_\left( |x-y|\right) \hbox y. \end$$

We can then set \(C_=1+2^\) to complete the proof.

Next consider \(m>0\). Using that \(j_(r)\sim C^_r^}e^r}\) as \(r \rightarrow \infty \), we have

$$\begin j_((k-1)|y|)&\le C_^ C^_(k-1)^}|y|^} \frac(k\!-\!1)|y|}}|y|}}e^|y|}\\&\le (C_^)^2 (k-1)^}e^kR}j_(|y|), \end$$

with some \(C^_>1\), and hence

$$\begin \int _^c_}j_(|y|) \hbox y&\le (C_^)^2(k-1)^}e^kR}\int __R^c}j_(|y|)\hbox y\\&\le (C_^)^2(k-1)^}e^kR}\int __R^c}j_(|x-y|)\hbox y. \end$$

Choosing \(C_\!>\!2\) such that \((C_^)^2(C_-1)^} e^C_R}\le \frac\) and using it instead of k, the claim follows.

Combining the last estimate with the Ikeda–Watanabe formula, we obtain the following result.

Lemma 5.2

For every \(R>0\) there exists a constant \(C_>0\) such that

$$\begin \mathbb ^x\big [g(\tau _); R\le |X_|\le C_R\big ] \ge \frac\mathbb ^x[g(\tau _R)] \end$$

for every non-negative function g and all \(x \in \mathcal _R\).

Proof

First consider \(g \in L^\infty (\mathbb ^d)\) and let \(C_>0\) be defined as in Lemma 5.1. By the Ikeda–Watanabe formula

$$\begin & \mathbb ^x[g(\tau _R); |X_|>C_R]\\ & \quad =\int _0^\int __R}g(t)p__R}(t,x,y) \int __R}^c}j_(|y-z|)\hbox z\hbox y\hbox t. \end$$

Using Lemma 5.1, we thus have

$$\begin & \mathbb ^x[g(\tau _R); |X_|>C_R]\\ & \quad \le \frac\int _0^\int __R}g(t)p__R}(t,x,y)\int __^c}j_(|y-z|)\hbox z\hbox y\hbox t =\mathbb ^x[g(\tau _R)]. \end$$

Next suppose that g is unbounded and let \(g_N(t)=g(t)\wedge N\) for \(N \in \mathbb \). Then, \(g_N \uparrow g\) pointwise, moreover

$$\begin \mathbb ^x[g_N(\tau _R); R\le |X_|\le C_R]\ge \frac\mathbb ^x[g_N(\tau _R)], \quad N \in \mathbb . \end$$

As \(N\rightarrow \infty \), by monotone convergence we then have

$$\begin \mathbb ^x[g(\tau _R); R\le |X_|\le C_R]\ge \frac\mathbb ^x[g(\tau _R)]. \end$$

Now we can turn to local estimates of the ground state. Consider the spherical potential well supported in \(\mathcal =\mathcal _a\) with some \(a>0\).

Theorem 5.1

Let \(\varphi _0\) be the ground state of \(H_\) with \(V=-v\textbf__a}\) and denote \(\textbf=(a,0,\ldots ,0)\). Then, the estimates

$$\begin \varphi _0(x)\asymp \varphi _0(\textbf) \times \mathbb ^x[e^] & \text\quad \; |x| \le a\\ \mathbb ^x[e^] & \text\quad \; |x| \ge a \end\right. } \end$$

hold, where the comparability constant depends on \(d,m,\alpha ,a,v,\lambda _0\).

Proof

Note that \(\varphi _0\) is rotationally symmetric by Theorem 4.1 and non-increasing by Proposition 4.2. We first prove the bound inside and next outside the well.

Step 1: First consider \(|x| \le a\). Using Proposition 4.1 with the almost surely finite stopping time \(\tau _a\), and that \(X_\in \mathcal _a^c\) and \(\varphi _0(X_)\le \varphi _0(\textbf)\), we have

$$\begin \varphi _0(x) = \mathbb ^x\big [e^\varphi _0(X_)\big ] \le \varphi _0(\textbf)\mathbb ^x\big [e^\big ]. \end$$

(5.2)

On the other hand, using that \(|X_|\le C^_a\), where \(C^_\) is defined in Lemma 5.2, we furthermore obtain

$$\begin \varphi _0(x)\ge & \ \mathbb ^x\big [e^\varphi _0(X_); a \le |X_|\le C^_a\big ] \\\ge & \ \varphi _0(C^_ })\mathbb ^x\big [e^; a \le |X_|\le C^_a\big ].\end$$

Recall that \(C^_>1\). Consider \(T_\) and \(T_M=T_a \wedge M\) for any positive integer \(M \in \mathbb \). By Proposition 4.1 applied to the almost surely finite stopping time \(T_M\), note that

$$\begin \varphi _0(C^_\textbf)=\mathbb ^_\textbf}[e^\varphi _0(X_)]\le \varphi _0(0)\mathbb ^_\textbf}[e^]. \end$$

By dominated convergence, in the limit \(M \rightarrow \infty \) we then get

$$\begin 0<\varphi _0(C^_\textbf)\le \varphi _0(0)\mathbb ^_\textbf}[e^], \end$$

implying \(C^_:=\mathbb ^_\textbf}(T_a=\infty )<1\). In particular, there exists a constant \(C^_>0\) such that \(\mathbb ^_\textbf}(T_a>C^_)<C^_\). Furthermore, by using Proposition 4.1 again, we get

$$\begin \varphi _0(C^_ })= & \ \mathbb ^_ }}[e^}\varphi _0(X_)]\\\ge & \ \mathbb ^_ }}[e^}\varphi _0(X_)]\\\ge & \ \mathbb ^_ }}[e^}\varphi _0(X_); T_\le C^_]. \end$$

Since on the set \(\_\}\) the random time \(T_M\) is almost surely constant as \(M \rightarrow \infty \), in the limit

$$\begin \varphi _0(C^_ })\ge & \ \mathbb ^_ }}[e^}\varphi _0(X_); T_\le C^_]\nonumber \\\ge & \ (1-C^_)e^_}\varphi _0(}) \end$$

(5.3)

follows, where we also used Proposition 4.2. On the other hand, by Lemma 5.2, we have

$$\begin \mathbb ^x\big [e^ ; a \le |X_|\le C^_a\big ] \ge \frac\mathbb ^x[e^]. \end$$

(5.4)

Combining (5.3)–(5.4) and choosing \(C^_= (1-C^_)e^_}\) we obtain

$$\begin \varphi _0(x)\ge \frac_}\varphi _0(\textbf)\mathbb ^x\big [e^\big ], \end$$

thus

$$\begin \varphi _0(x) \asymp \varphi _0(\textbf)\mathbb ^x\big [e^\big ], \quad |x|\le a, \end$$

where the comparability constant depends on \(d,m,\alpha ,a,|\lambda _0|\).

Step 2: Next consider \(|x|>a\), and let \(T_a\) and \(T_M\) be defined as before. By Proposition 4.1, we have

$$\begin \begin \varphi _0(x)&=\mathbb ^x[e^}\varphi _0(X_)] \ge \mathbb ^x[e^}\varphi _0(X_)]\\&\ge \mathbb ^x[e^}\varphi _0(X_); T_a<\infty ], \end \end$$

due to \(T_M \le T_a\). Taking the limit \(M \rightarrow \infty \) and observing that \(T_M\) is a definite constant if \(T_a<\infty \), we get

$$\begin \begin \varphi _0(x)&\ge \mathbb ^x[e^}\varphi _0(X_); T_a<\infty ]\ge \varphi _0(\textbf)\mathbb ^x[e^;T_a<\infty ]\\&=\varphi _0(\textbf)\mathbb ^x[e^]. \end \end$$

(5.5)

On the other hand,

$$\begin \varphi _0(x)\le \varphi _0(0)\mathbb ^x[e^}] \rightarrow \varphi _0(0)\mathbb ^x[e^}], \end$$

as \(M \rightarrow \infty \), by using dominated convergence. By Step 1, Theorem 3.1 and (5.7) we find a constant \(C^_\) such that

$$\begin \varphi _0(0) \le C^_\varphi _0(\textbf)\left( 1+\frac\right) =:C^_\varphi _0(\textbf). \end$$

and thus

$$\begin \varphi _0(x)\le C^_\varphi _0(\textbf)\mathbb ^x[e^}]. \end$$

(5.6)

This leads to

$$\begin \varphi _0(x)\asymp \varphi _0(\textbf)\mathbb ^x[e^}], \quad |x|\ge a, \end$$

where the comparability constants depend on \(d,m,\alpha ,a,v,|\lambda _0|\).

Remark 5.2 1.

In fact, along the way we also proved that

$$\begin & C^_\varphi _0(\textbf)e^_|\lambda _0|}\mathbb ^x[e^]\\ & \quad \le \varphi _0(x)\le C^_\varphi _0(\textbf)\mathbb ^x[e^], \end$$

for every \(|x|\le a\), with constants dependent only on \(d,m,\alpha ,a\) (and independent of v and \(\lambda _0\)).

2.

We point out that we have shown in particular that

$$\begin \mathbb ^x[e^]\le \frac_} \frac)} <\infty . \end$$

However, from (3.8) we know that \(\mathbb ^x[e^]\) is finite if and only if \(\lambda <\lambda _a\). Thus we have also shown that

$$\begin v-|\lambda _0|<\lambda _a. \end$$

(5.7)

We note that to prove this only monotonicity of \(\varphi _0\) outside the potential well is a required input, which has been proven in [4] without using (5.7) (which is, on the other hand, indispensable to obtain monotonicity inside the well). Thus this argument provides an alternative, purely probabilistic, proof of [4, Lem. 4.5].

Using the following estimate in conjunction with the estimates in Sect. 3, we can derive explicit local estimates for the ground states of the massless and massive relativistic operators.

Corollary 5.1

With the same notations as in Theorem 5.1, we have

$$\begin \varphi _0(x) \asymp \; \varphi _0(\textbf) \times \left\ 1+\frac\left( \frac\right) ^ & \text\quad \; |x|\le a \\ j_(|x|) & \text\quad \;|x|\ge a, \end\right. \end$$

where the comparability constant depends on \(d,m,\alpha ,a,v,|\lambda _0|\).

Proof

For \(|x|\le a\) the result is immediate by a combination of Theorems 5.1 and 3.1, using (5.7). For \(|x|\ge a\) we distinguish two cases. First, if \(m=0\), by [38, Cor. 4.1] there exists \(R_\) such that

$$\begin \varphi _0(x)\ge C^_|x|^\ge C^_j_(|x|), \quad |x|\ge R_, \end$$

where \(C^_\) is defined in the quoted result and \(C^_=C^_\frac\left| \Gamma \left( -\frac\right) \right| }\right) }\). Secondly, when \(m>0\) we use [38, Cor. 4.3(1)] to find that there exists \(R_\) such that

$$\begin \varphi _0(x)\ge C^_|x|^}e^|x|}, \quad |x| \ge R_. \end$$

Moreover, we know that \(j_(x)\sim |x|^}e^|x|}\) as \(|x| \rightarrow \infty \), hence there exists a constant \(C^_\) such that \(\varphi _0(x)\ge C^_j_(|x|)\) for \(|x|\ge R_\). Thus, by (5.6)

$$\begin \mathbb ^x[e^]\ge C^_j_(|x|), \quad |x|\ge R_. \end$$

Combining this with Corollary 3.3 and Theorem 3.3, we obtain

$$\begin \mathbb ^x[e^] \asymp j_(|x|), \quad |x|\ge a, \end$$

where the comparability constants depend on \(d,\alpha ,m,a,v,|\lambda _0|\).

Remark 5.3

By Remark 5.2 we have similarly

$$\begin & C^_\varphi _0(\textbf)e^_|\lambda _0|} \left( 1+\frac\left( \frac\right) ^\right) \\ & \quad \le \varphi _0(x)\le C^_\varphi _0(\textbf)\left( 1+\frac \left( \frac\right) ^\right) , \end$$

for \(|x|\le a\) it holds and with constants which depend only on \(d,m,\alpha ,a\) (and not on v and \(\lambda _0\)).

The local estimates on \(\varphi _0\) can further be improved to see the behaviour as \(|x| \rightarrow a\).

Proposition 5.2

There exist \(\varepsilon =\varepsilon _,C_>0\) such that for every \(x \in \mathcal _ \mathcal _\)

$$\begin \left| \frac)}-1\right| \le C_\big | |x|-a \big |^ \end$$

holds.

Proof

The estimate is clear once \(x \in \partial \mathcal _a\). Consider first the case \(x \in \mathcal _a\). By (5.2), we have

$$\begin \frac)}-1\le \mathbb ^x[e^-1]\le C_(a-|x|)^, \end$$

where we used Theorem 3.1. Taking \(x \in \mathcal _a^c\), we have by (5.5),

$$\begin 1-\frac)}\le \mathbb ^x[1-e^]. \end$$

Choosing \(R^_\) as in Proposition 3.4 and defining \(\varepsilon = (R^_-a)\wedge a\) the result follows.

By using the normalization condition \(\left\| \varphi _0 \right\| _=1\), we are able to provide a two-sided bound on \(\varphi _0(\textbf)\).

Proposition 5.3

Denote \(\mathcal I = \int _1^r^j^2_\left( ar\right) \hbox r\) and by B(x, y) the usual Beta-function. Then, with the same comparability constant as in Corollary 5.1,

$$\begin & \varphi _0(}) \asymp \left( a^d d\omega _d\left( \frac+2\fracB\left( d,1+\frac\right) \right. \right. \nonumber \\& \qquad \qquad \left. \left. +\left( \frac\right) ^2B\left( d,1+\alpha \right) +\mathcal I \right) \right) ^}, \end$$

Proof

We write \(\kappa = \frac\) for a shorthand. Consider \(|x| \le a\). By Corollary 5.1 we have

$$\begin & \frac}\varphi _0(\textbf)\left( 1+\kappa \left( \frac\right) ^}\right) \\ & \qquad \le \varphi _0(x) \le C_\varphi _0(\textbf) \left( 1+\kappa \right) \left( \frac\right) ^, \end$$

which gives

$$\begin \frac}\varphi _0(x) \le \varphi _0(\textbf)\left( 1+\kappa \left( \frac\right) ^\right) \le C_\varphi _0(x). \end$$

Taking the square on both sides and integrating over \(\mathcal _a\) we get

$$\begin \frac)^2}\int __a}\varphi ^2_0(x)\textx&\le \ \varphi _0^2(}) \int __a}\left( 1+\kappa \left( \frac\right) ^\right) ^2\textx \nonumber \\ &\le (C_)^2\int __a}\varphi ^2_0(x) \textx \end$$

(5.8)

Consider next \(|x|>a\). Proceeding similarly, we have

$$\begin & \frac)^2}\int _^c_a}\varphi ^2_0(x)\hbox x\nonumber \\ & \quad \le \varphi _0^2(\textbf)\int _^c_a}j^2_(|x|)\hbox x \le (C_)^2\int _^c_a}\varphi ^2_0(x) \hbox x. \end$$

(5.9)

Adding up (5.8)–(5.9) and using that \(\left\| \varphi _0 \right\| _=1\), we get

$$\begin \frac)^2}\le & \ \varphi _0^2(})\left( \int __a}\left( 1+\kappa \left( \frac\right) ^\right) ^2\textx +\int _^c_a}j^2_(|x|)\textx\right) \\\le & \ (C_)^2. \end$$

By evaluating the integrals and taking the square root, we obtain the required result.

As a direct consequence, we can rewrite Corollary 5.1 as follows.

Corollary 5.2

With the same notations as in Theorem 5.1, we have

$$\begin \varphi _0(x) \asymp \; \left\ 1+\frac\left( \frac\right) ^ & \text \; |x|\le a \\ j_(|x|) & \text \; |x|\ge a, \end\right. \end$$

where the comparability constant depends on \(d,m,\alpha ,a,v,|\lambda _0|\) and is independent of \(\varphi _0\).

5.3 Lack of Regularity at the Boundary of the Potential Well

From a quick asymptotic analysis of the profile functions appearing in the estimates in Corollary 5.1, the difference of the leading terms suggests that while the regime change around the boundary of the potential well is continuous, it cannot be smooth beyond a degree. To describe this quantitatively, we show next a lack of regularity of the ground state arbitrarily close to the boundary. For a result on Hölder regularity of solutions of related non-local Schrödinger equations, see [44].

Lemma 5.3

Consider the operator \(L_\) and the following two cases:

(1)

\(\alpha \in (0,1)\) and \(f \in C^_\textrm(\mathbb ^d) \cap L^\infty (\mathbb ^d)\) for some \(\delta \in (0,1-\alpha )\)

(2)

\(\alpha \in [1,2)\) and \(f \in C^_\textrm(\mathbb ^d) \cap L^\infty (\mathbb ^d)\) for some \(\delta \in (0,2-\alpha )\).

In either case above, the function \(\mathbb ^d \ni x \mapsto L_f(x)\) is continuous.

Proof

Note that under the assumptions above, \(L_f\) is well-defined pointwise via the integral representation (2.1). We show the statement for \(m=0\) only, for \(m>0\) the proof is similar by using the asymptotic behaviour of \(j_(r)\) around zero and at infinity.

To prove (1), we use the integral representation (2.1) and claim that in this case

$$\begin L_f(x)=& -C^_\lim _\left( \int _+\int _\right) \frac}\texty,\end$$

with the constant \(C^_\) entering the definition of the massless operator. Indeed, note that the second integral in the split is independent of \(\varepsilon \), while for the first integral we can use the Hölder inequality giving

$$\begin & \int _\frac}\hbox y\le C^\int _ \frac}\\ & \quad \le dC^\omega _d\int _^1\frac}\hbox \rho =\frac\omega _d}. \end$$

The claimed right hand side follows then by dominated convergence. Next choosing \(h \in \mathbb ^d\), \(|h|<1\), we show that \(\lim _L_f(x+h)=L_f(x)\). We write

$$\begin L_f(x+h)&=-C^_\int _^d}\frac}\hbox y\\&=-C^_\left( \int __3(x+h)}+\int _^c_3(x+h)}\right) \frac}\hbox y. \end$$

To estimate the first integral, note that \(\mathcal _3(x+h)\subseteq \mathcal _4(x)\) for every \(h \in \mathcal _1\). Let \(C^\) be the Hölder constant associated with \(\overline}_4(x)\) and observe that

$$\begin&\int __3(x+h)}\frac}\texty=\int __3}\frac}\texty\\&\le C^\int __3}\fracy}}= \fracd\omega _d}. \end$$

For the second integral, observe that if \(y \in \mathcal _2(x)\), then \(|x+h-y|\le |x-y|+|h|<3\) so that \(y \in \mathcal _3(x+h)\) for any \(h \in \mathcal _1\). This means that \(\mathcal ^c_3(x+h)\subseteq \mathcal _2^c(x)\) for all h and then

$$\begin \int _^c_3(x+h)}\frac}\texty&\le \int _^c_2(x)}\frac}\texty\\ &\le 2\left\| f \right\| _\int _^c_2(x)}\fracy}}\\&\le 2\left\| f \right\| _d\omega _d \int _^\frac}}\text\rho <\infty . \end$$

Thus, again we can use dominated convergence to prove the claim.

Next consider (2). Fix \(x \in \mathbb ^d\) and define the function

$$\begin \mathcal _1 \ni h \mapsto D_hf(x):=f(x+h)-2f(x)+f(x-h). \end$$

By Lagrange’s theorem there exist \(\xi _\pm (h) \in [x,x \pm h]\), where [x, y] denotes the segment with endpoints x, y, such that

$$\begin f(x+h)-2f(x)+f(x-h)=\langle \nabla f(\xi _(h))-\nabla f(\xi _(h)), h\rangle \end$$

and thus \(|D_hf(x)|\le |\nabla f(\xi _(h))-\nabla f(\xi _(h))||h|\). Since \(\xi _\pm (h) \in [x,x \pm h]\), in particular \(\xi _\pm (h) \in \mathcal _1(x)\), and we can use the Hölder property of the gradient to conclude that

$$\begin |\nabla f(\xi _(h))-\nabla f(\xi _(h))|\le C^(x)|\xi _(h)-\xi _(h)||h|^. \end$$

Moreover, \(|\xi _(h)-\xi _(h)|\le 2\), and thus \(|D_hf(x)|\le 2C^(x)|h|^\). Using that \(\int _^\frac}\hbox \rho =\frac\), by an application of [3, Prop. 2.6, Rem. 2.4] we then obtain

$$\begin L_f(x)=-\frac_}\int _^d}\frac}\hbox h, \quad x \in \mathbb ^d. \end$$

Taking \(k \in \mathcal _1\), we show that \(\lim _L_f(x+k)=L_f(x)\). Write

$$\begin L_f(x+k) =-\frac_}\int __3}\frac}\hbox h- \frac_}\int __3^c}\frac}\hbox h. \end$$

In the first integral, we have \(x+k \pm h \in \mathcal _4(x)\) for every \(k \in \mathcal _1\) and \(h \in \mathcal _3\), hence \(|D_hf(x+k)|\le 8C^(x)|h|^\), similarly to in the previous case, where \(C^(x)\) is the Hölder constant of \(\nabla f\) in \(\overline}_4(x)\). Thus, we obtain

$$\begin \int __3}\frac}\texth \le \fracd\omega 3^\delta } \int __3}\frach}}. \end$$

For the second integral, using that \(f \in L^\infty (\mathbb ^d)\) we get

$$\begin \int __3^c}\frac}\hbox h\le 4\left\| f \right\| _\int __3^c}\frach}}<\infty . \end$$

The proof is then completed by dominated convergence.

Theorem 5.2

Let \(\varphi _0\) be the ground state of \(H_\). The following hold:

(1)

If \(\alpha \in (0,1)\), then \(\varphi _0 \not \in C^_\textrm(\mathbb ^d)\) for every \(\delta \in (0,1-\alpha )\).

(2)

If \(\alpha \in [1,2)\), then \(\varphi _0 \not \in C^_\textrm(\mathbb ^d)\) for every \(\delta \in (0,2-\alpha )\).

Proof

We rewrite the eigenvalue equation like

$$\begin L_\varphi _0=(v\textbf__a}+\lambda _0)\varphi _0. \end$$

(5.10)

Suppose that \(\alpha \in (0,1)\) and \(\varphi _0 \in C^_\textrm(\mathbb ^d)\) for some \(\delta \in (0,1-\alpha )\). Then, by (1) of Lemma 5.3 we have that the left-hand side of (5.10) is continuous. On the other hand, take \(\textbf_1=(1,0,\dots ,0)\) and notice that

$$\begin&\lim _(v\textbf__a}((a+\varepsilon )\textbf_1) +\lambda _0)\varphi _0((a+\varepsilon )\textbf_1)=\lambda _0 \varphi _0(a\textbf_1)\\&\lim _(v\textbf__a}((a-\varepsilon )\textbf_1) +\lambda _0)\varphi _0((a-\varepsilon )\textbf_1)=(v+\lambda _0) \varphi _0(a\textbf_1), \end$$

thus the right-hand side is continuous in \(a\textbf_1\) if and only if \(\varphi _0(a\textbf_1)=0\), which is in contradiction with the fact that \(\varphi _0\) is positive. In particular, the same argument holds for any point \(x \in \partial \mathcal _a\); thus, the right-hand side of (5.10) has a jump discontinuity on \(\partial \mathcal _a\), which is impossible since the left-hand side is continuous. The same arguments hold for \(\alpha \in [1,2)\) by using part (2) of Lemma 5.3.

Remark 5.4 (1)

Instead of using \(C_\textrm^(\mathbb ^d)\) we also can prove part (1) of Lemma 5.3 with \(f \in C^(\overline}_r(x))\) for some \(x \in \mathbb ^d\), implying that \(L_f\) is continuous in x. With this localization argument, we obtain for \(\alpha \in (0,1)\) that \(\varphi _0 \not \in C_\textrm^(\mathcal _ \overline}_)\), for all \(\varepsilon \in (0,a)\) and \(\delta \in (0,1-\alpha )\). In particular, this implies that \(\varphi _0\) cannot be \(C^1\) on \(\partial \mathcal _a\). The same arguments apply to part (2) of Lemma 5.3 and the case \(\alpha \ge 1\), implying that \(\varphi _0\) cannot be \(C^2\) on \(\partial \mathcal _a\). We note that for the classical case the ground state is \(C^1\) but fails to be \(C^2\) at the boundary of the potential well.

(2)

It is reasonable to expect that \(\varphi _0\) has at least a \(C^\)-regularity, for all \(\varepsilon >0\) small enough, both inside and outside the potential well (away from the boundary). However, this needs different tools and we do not pursue this point here.

5.4 Moment Estimates of the Position in the Ground State

As an application of the local estimates of ground states, we consider now the behaviour of the following functional. Note that when the ground state is chosen to satisfy \(\Vert \varphi _0\Vert _2=1\), the expression \(\varphi _0^2(x)\hbox x\) defines a probability measure on \(\mathbb ^d\). Let \(p>0\) and define

$$\begin \Lambda _p(\varphi _0) = \left( \int _^d} |x|^p \varphi _0^2(x)\hbox x\right) ^, \end$$

which can then be interpreted for \(p \ge 1\) as the pth moment of an \(\mathbb ^d\)-valued random variable under this probability distribution. In the physics literature, the ground state expectation for \(p=2\) is called the size of the ground state.

Let \(m \ge 0\), \(\alpha \in (0,2)\), and define

$$\begin p_*(m,\alpha ):= d+2\alpha & \text\quad m=0\\ \infty & \text\quad m>0. \end\right. } \end$$

Also, we write for a shorthand

$$\begin \mathcal J_a = \frac = \frac}, \end$$

(5.11)

which is a constant related to the ratio between the energy gap separating the ground state eigenvalue from the bottom value of the potential and the energy needed to climb the potential well.

Lemma 5.4

The following cases occur:

1.

If \(0<p<p_*(m,\alpha )\), then \(\Lambda _p(\varphi _0)<\infty \).

2.

If \(p \ge p_*(m,\alpha )\), then \(\Lambda _p(\varphi _0)=\infty \).

Proof

It is a direct consequence of Corollary 5.1, using that \(j_(r)=C_ r^\), and \(j_(r)\approx r^e^r}\) as \(r \rightarrow \infty \) if \(m>0\). Indeed, while for \(m>0\) it is immediate, for \(m=0\) we have \(\rho ^j^2_(\rho )=C_\rho ^\), so that it is integrable at infinity if and only if \(d+2\alpha >p\).

Proposition 5.4

Let \(0<p<p_*(m,\alpha )\). There exist constants \(C^_, C^_>0\) such that

$$\begin \Lambda _p(\varphi _0) \ge C^_ \mathcal J_a^e^C^_|\lambda _0|}\varphi ^_0(\textbf). \end$$

(5.12)

Proof

By Remark 5.3, we get

$$\begin \varphi _0^2(x)&\ge \varphi ^2_0(\textbf)(C^_)^2\left( 1+2\frac \left( \frac\right) ^\right. \\&\left. \quad +\left( \frac\right) ^2 \left( \frac\right) ^\right) e^_|\lambda _0|}\\&\ge \varphi ^2_0(\textbf)(C^_)^2\mathcal J_a^2 \left( \frac\right) ^e^_|\lambda _0|}, \quad |x|\le a, \end$$

where the last step follows by the fact that \(\frac \le 1\). Hence,

$$\begin \int _^d}|x|^p\varphi _0^2(x)\hbox x&\ge \int __a}|x|^p\varphi _0^2(x)\hbox x\\&\ge \varphi ^2_0(\textbf)(C^_)^2 _a^2 e^_|\lambda _0|}\int __a}|x|^\left( \frac\right) ^\hbox x. \end$$

Setting \((C^_)^p=(C^_)^2\int __a}|x|^\left( \frac\right) ^\hbox x\), the result follows.

Proposition 5.5

Let \(0<p<p_*(m,\alpha )\) and \(v>\lambda _a+\delta \) for some \(\delta >0\). Then, there exists a constant \(C_>0\) such that

$$\begin \Lambda _p(\varphi _0) \le C_ \, ^ \varphi ^_0(\textbf). \end$$

Proof

As in Theorem 5.1, observe that for \(|x|\ge a\) we have by Proposition 4.2

$$\begin \varphi _0(x)\le \varphi _0(0)\mathbb ^x[e^]. \end$$

(5.13)

Moreover, by Remark 5.3,

$$\begin \varphi _0(0)\le C^_\mathcal J_a \varphi _0(\textbf). \end$$

(5.14)

On the other hand, from \(v-|\lambda _0|<\lambda _a\) we get \(|\lambda _0|>v-\lambda _a>\delta \) and then

$$\begin \mathbb ^x[e^]\le \mathbb ^x[e^]\le C^_j_(|x|), \quad |x|\ge a, \end$$

(5.15)

where we used also Theorem