Abstract
We use the mirror coupling of Brownian motion to show that under a \(\beta \in (0,1)\)-dependent Kato-type assumption on the possibly nonsmooth electromagnetic potential, the corresponding magnetic Schrödinger semigroup in \({\mathbb {R}}^d\) has a global \(L^{p}\)-to-\(C^{0,\beta }\) Hölder smoothing property for all \(p\in [1,\infty ]\); in particular, his all eigenfunctions are uniformly \(\beta \)-Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a molecule in a magnetic field are uniformly \(\beta \)-Hölder continuous under weak \(L^q\)-assumptions on the magnetic potential.
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1 Introduction
Kato [7]Footnote 1has shown that each eigenfunction \(\Psi \) of a multi-particle Schrödinger operator \(H=-\Delta +W\) in \(L^2({\mathbb {R}}^{3m})\) with a potential \(W:{\mathbb {R}}^{3m}\rightarrow {\mathbb {R}}\) of the form
is uniformly \(\beta \)-Hölder continuous for all \(0<\beta < 2-3/p\), that is,
where we have written points in \({\mathbb {R}}^{3m}\) in the form \(x=(\mathbf {x}_1,\dots , \mathbf {x}_m)\), where \(\mathbf {x}_j\in {\mathbb {R}}^3\) for \(j=1,\dots ,m\). In particular, an application of this result to multi-particle Coulomb-type potentials shows that all molecular Hamilton operators (in the infinite mass limit) are uniformly \(\alpha \)-Hölder continuous for all \(0<\alpha < 1\). Kato’s proof relies on the Fourier transform and so does not apply directly to magnetic Schrödinger operators (even if one assumes a Coulomb gauge). The aim of this paper is to use probabilistic techniques to find a variant of Kato’s regularity result that applies to a magnetic Schrödinger operator H(A, V) with magnetic potential \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\) and electric potential \(V:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\). To this end, we prove the following smoothing result (cf. Theorem 2.5):
Let \(\beta \in (0,1)\) and let \(C^{0,\beta }({\mathbb {R}}^d)\) denote the space of uniformly \(\beta \)-Hölder continuous functions on \({\mathbb {R}}^d\), with its seminorm given by
and consider for \(q\in [1,\infty ]\) the Banach space \(C^{0,\beta }({\mathbb {R}}^d)\cap L^q({\mathbb {R}}^d)\) with its norm
Then for all Borel functions \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\), \(V:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) with
and all \(t>0\), \(1\le p\le q\le \infty \) one has
and the norm of this operator can be estimated explicitly.
Above, \(\mathcal {K}^{\beta }({\mathbb {R}}^d)\), \(\beta \in [0,1]\), denotes the \(\beta \)-Kato class (cf. Definition 2.1) of Borel functions \({\mathbb {R}}^d\rightarrow {\mathbb {R}}\) which has been introduced in [5], so that \(\mathcal {K}({\mathbb {R}}^d):=\mathcal {K}^{0}({\mathbb {R}}^d)\) is the classical Kato class [1] and one has \(\mathcal {K}^{\beta }({\mathbb {R}}^d)\subset \mathcal {K}^{\alpha }({\mathbb {R}}^d)\) if \(\beta \ge \alpha \). Note also that \( H (A,V)\Psi =\theta \Psi \) implies \(e^{-t H (A,V)}\Psi =e^{-t \theta }\Psi \), so that one also obtains global \(\beta \)-Hölder regularity for eigenfunctions.
The mapping property
is well known [2] and only requires a local Kato assumption on \(\left| A\right| ^2\), \(|\mathrm {div}( A)|\) and the positive part of V, and a global Kato assumption on the negative part of V.
The proof of (2) uses Brownian mirror coupling techniques (cf. Sect. 2 for the basic definitions) to deal with the magnetic potential A. Let us mention here that the use of Brownian coupling techniques in the context of Hölder estimates for semigroups that generate diffusion (which in our case would correspond to taking \(V=0\), \(A=0\)) has a long history, also on Riemannian manifolds (cf. [3, 9, 10] for some classical results).
Our main tool (cf. Theorem 2.3) for the proof of (2) is provided by the following estimate:
There exists a universal constant \(c_0<\infty \), such that for every \(q\in (1,\infty )\), every Borel function \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\) with
every \(t>0\), \(x\ne y\) in \({\mathbb {R}}^d\), and every mirror coupling \((\mathsf {X} ,\mathsf {Y} )\) of Brownian motions from (x, y) one has
where for any Brownian motion \(\mathsf {Z} \), the process \(\mathscr {S}_t\left( A|\mathsf {Z} \right) \) denotes the magnetic Euclidean action functional (cf. (7) ) which appears in the Feynman–Kac–Itô formula and where the constant \(C(A,t,q)<\infty \) can be computed explicitly.
This estimate is then combined with the Feynman–Kac–Itô formula (and perturbation theory to deal with V) to finally obtain (2).
Let us mention that locally uniform \(\beta \)-Hölder continuity results for nonmagnetic Schrödinger eigenfunctions under \(L^q\)-assumptions on V have also been obtained in [8] (cf. Theorem 11.7 therein) using straightforward Sobolev embedding techniques. In addition, in [11] (cf. Theorem B.3.5 therein) it is shown that \(\beta \)-dependent (Kato-type) assumptions in V leading to locally uniform \(\beta \)-Hölder smoothing results for nonmagnetic Schrödinger semigroups. In the latter case, the ultimate argument relies on potential theory, while Brownian motion only enters through the Feynman–Kac formula in order to show that the Schrödinger semigroup is \(L^{\infty }\)-smoothing.
Using \(L^p\)-criteria for the \(\beta \)-Kato class (cf. Remark 2.2), we show that o result directly implies the following generalization of Kato’s result for multi-particle Schrödinger operators in \({\mathbb {R}}^{3n}\) to magnetic multi-particle Schrödinger operators:
Assume there exists \(\beta \in (0,1)\), \(l\in {\mathbb {N}}\) and Borel functions \(a:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{3}\), \(v_{i},v_{ij}:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) with
where \(\mathrm {div}\) is defined in the distributional sense, and define a vector potential, resp. a magnetic potential on \({\mathbb {R}}^{3n}\) through
Then for all \(t>0\) and \(p\in [1,\infty ]\) one has
To the best of our knowledge, this is the first global Hölder-regularity result for multi-particle magnetic Schrödinger operators.
Let us finally explain how this result applies to molecules in a magnetic field: Given \(R\in {\mathbb {R}}^{3n}\), \(l\in {\mathbb {N}}\), \(Z\in [0,\infty )^l\), consider the potential
Given \(a:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{3}\) (sufficiently well behaved), set as above \(A(x):=\sum ^n_{i=1} a(\mathbf {x}_i)\). Then the operator
is the Hamilton operator corresponding to a molecule (in the infinite mass limit) with l protons and with n electrons, where the jth nucleus is located in \(\mathbf {R}_j\), and has \(Z_j\) protons, and the electrons interact with the magnetic field induced by A. Then given an arbitrary \(\beta \in (0,1)\), one has (5) for
so that the previous result gives that for all \(t>0\) and \(p\in [1,\infty ]\) one has
as long as
2 Main results
We start by recalling the definition of the mirror coupling of Brownian motions as presented in [6] and follow their exposition (pages 1-3 therein) closely before presenting our main results.
A continuous process \((\mathsf {X} ,\mathsf {Y} )\) with values in \({\mathbb {R}}^d\times {\mathbb {R}}^d\) is called a coupling of Brownian motions from \(\left( x,y\right) \in {\mathbb {R}}^d\times {\mathbb {R}}^d\), if \(\mathsf {X} \) and \(\mathsf {Y} \) are Brownian motions starting in x and y, respectively. Then, with the coupling time
the coupling \((\mathsf {X} ,\mathsf {Y} )\) is said to be maximal, if for all \(t>0\) one has
with
the transition density of Brownian motion starting in a. The reason for this notion of maximality is that for an arbitrary coupling of Brownian motions one has \(\le \) in (6).
Let x and y be two distinct points of \({\mathbb {R}}^{d}\). Then
is the hyperplane orthogonal on and bisecting the segment \(\underline{xy}\). Furthermore, define the affine map
This is the reflection at the hyperplane \(N_{x,y}\). Let \(L_{x,y}\) be the linear part of \(R_{x,y}\). Note that \(L_{x,y}\) is symmetric and idempotent.
A coupling \((\mathsf {X} ,\mathsf {Y} )\) of Brownian motions from (x, y) is called a mirror coupling, if
where
is the hitting time of \(\mathsf {X} \) with respect to \(N_{x,y}\). In other words, \(\mathsf {Y} \) is equal to the reflection of \(\mathsf {X} \) at \(N_{x,y}\) before \(\mathsf {X} \) hits \(N_{x,y}\), and is then equal to \(\mathsf {X} \). It follows that \(\tau (\mathsf {X} ,\mathsf {Y} )=\tau _{x,y}(\mathsf {X} )\), which by an explicit calculation of \({\mathbb {P}}(t\le \tau _{x,y}(\mathsf {X} ))\) implies that every mirror coupling is maximal.
Whenever well defined, we consider the following action functional on the paths of any Brownian motion \(\mathsf {Z} \), which depends on a sufficiently regular function \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\):
Above, i is the imaginary unit, \(\mathrm {div}( A)\) denotes the divergence of A understood in the distributional sense and the stochastic integral is understood in Itô’s sense.
Let \({\mathbb {P}}_a\) denote the law of Brownian motion starting in a, which is considered as a probability measure on the space of continuous paths \(\omega :[0,\infty )\rightarrow {\mathbb {R}}^d\). Generalizing the Kato class, the following hierarchy of Kato classes has been introduced in [5]:
Definition 2.1
Given \(\alpha \in [0,1]\), a Borel function \(f:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) is said to be in the \(\alpha \)-Kato class \(\mathcal {K}^{\alpha }({\mathbb {R}}^d)\), if
Note that
Remark 2.2
1) Each \(\mathcal {K}^{\alpha }({\mathbb {R}}^d)\) is a linear space and \(\mathcal {K}({\mathbb {R}}^d):=\mathcal {K}^{0}({\mathbb {R}}^d)\) is the usual Kato class.
2) One trivially has
and using
one gets
3) For all \(q\in [1,\infty )\) with \(q> d/(2-\alpha )\), one has
which follows straightforwardly (cf. Lemma 3.9 in [5]) from
4) For every natural \(D\ge d\), every linear surjective map \(\pi :{\mathbb {R}}^D\rightarrow {\mathbb {R}}^d\), and every \(f\in \mathcal {K}^{\alpha }({\mathbb {R}}^d)\), one has \(f\circ \pi \in \mathcal {K}^{\alpha }({\mathbb {R}}^D)\), cf. [5].
5) For every \(W\in \mathcal {K}({\mathbb {R}}^d)\), \(z\in {\mathbb {R}}^d\), \(t>0\), one has (cf. Lemma 3.9 in [5])
and if \(W\in \mathcal {K}^{\beta }({\mathbb {R}}^d)\), then also (cf. the proof of Theorem 3.10 in [5])
The following probabilistic estimate is our main technical result:
Theorem 2.3
There exists a universal constant \(c_0<\infty \), such that for every \(q\in (1,\infty )\), every Borel function \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\) with
every \(t>0\), \(x\ne y\) in \({\mathbb {R}}^d\), and every mirror coupling \((\mathsf {X} ,\mathsf {Y} )\) of Brownian motions from (x, y) one has
where \(1/q^*+1/q=1\) and
Remark 2.4
1) As every Kato function is locally integrable (cf. Lemma VI.5 c) in [4]), \(\mathrm {div}(A)\) exists as a distribution in the above situation.
2) Remark 2.2.5 easily shows that \(C(A,t,q)<\infty \) under the assumptions of Theorem 2.3 and that \(\int _{0}^{\cdot }\langle A\left( \mathsf {Z} _{s}\right) ,{{\mathrm{d}}}\mathsf {Z} _{s}\rangle \) is a continuous \(L^2\)-martingale for every Brownian motion \(\mathsf {Z} \) having a deterministic initial value. In particular, the process \(\mathscr {S}( A|\mathsf {Z} )\) is a continuous semimartingale.
3) The function \(t\mapsto C(A,t,q)\) is locally bounded under the assumptions of Theorem 2.3: The easiest way to see this is to refer to Khashminiski’s lemma, which implies that for every \(W\in \mathcal {K}({\mathbb {R}}^d)\) one has
and so trivially
Proof of Theorem 2.3
Let \(x\ne y\) in \({\mathbb {R}}^{d}\) and \(t>0\) be fixed. We set
Given a Brownian motion \(\mathsf {Z} \), we split
into
Clearly we a.s. have
Likewise, heuristically, for \(s<\tau \) one has \({{\mathrm{d}}}{\mathsf {Y} }_{s}=L{{\mathrm{d}}}{\mathsf {X} }_{s}\), while for \(s\ge \tau \) one has \({{\mathrm{d}}}{\mathsf {Y} }_{s}={{\mathrm{d}}}{\mathsf {X} }_{s}\), and we therefore expect that
holds a.s., where
To show that Eq. (13) holds, by replacing
with the sequence
and using the Itô isometry and dominated convergence, we can assume that A is bounded. By Theorem 6.5 in [12] we have the \(L^2\)-convergence of the dyadic approximations
where \(t_{i}:=\frac{it}{2^{n}}\) for \(i=0,\ldots ,2^{n}\). We immediately note that in case \(t<\tau \), we have \(\mathsf {Y} _{s}=R\mathsf {X} _{s}\) on \(\left[ 0,t\right] \); hence, in that case by the above limits we conclude that:
If we now assume that \(\tau \in \left( t_{k},t_{k+1}\right] \) for some \(k=0,\ldots ,2^{n}-1\), we get the following expressions for the summands in the above limits:
For \(i\le {k-1}\):
In the last step, we have used that L is self-adjoint and \(Rv-Rw=L\left( v-w\right) \).
For \(i=k\):
For \(i=k+1\):
For \(i\ge {k+2}\), the summands vanish. Compiling these equations allows us to make the following estimates,
where \(\gamma \left( z\right) :=L A\left( Rz\right) - A\left( Rz\right) \). Note that
because \(R\mathsf {X} _{\tau }=\mathsf {X} _{\tau }\). In particular, since L is self-adjoint and idempotent,
Since A is bounded by some \(\kappa >0\), and so \(|\gamma |\le 2\kappa \), using
we can thus estimate as follows,
Since \(\tau \) is an \(\mathsf {X} \)-stopping time, we conclude by the Markov property of \(\mathsf {X} \), using
and once more \(t_{j+1}-t_j=\frac{t}{2^n}\), that
Altogether, we have found that under the assumptions of the theorem one has
We are now going to estimate the \(L^1\)-norms of \(M_ {t}\) and \(I_t\). Let us start with \({\mathbb {E}}\left( \left| I_t\right| \right) \): setting
we have
where
In view of
we conclude
where from here on \(C<\infty \) denotes a universal constant whose value may change from line to line. Now let us turn to the estimate for \({\mathbb {E}}\left( \left| M_{t}\right| \right) \): Define
We note that \(h\left( r\right) \ge \left| r\right| \) and that h is in \(C^{2}\left( {\mathbb {R}}\right) \) with
In particular, we have \(|h''|\le \mathbb {1}_{\left[ -2,2\right] }\) and \(\left| h'\right| \le {1}\). We conclude by Itô’s formula
where
is an (\(L^2\))-martingale, as follows from the assumption on A and the boundedness of \(h'\). Thus,
and we have
Hence, similarly to the Lebesgue integral \(I_t\), we conclude
where
Thus, we have shown
Finally, noting that for all purely imaginary \(z,z'\) one has the elementary estimate
and \(\mathfrak {R}(\mathscr {S}_t\left( \mathsf {Z} \right) )=0\), the proof is complete. \(\square \)
If \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\) and \(V:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}\) are Borel functions with
then the symmetric sesquilinear form
in \(L^2({\mathbb {R}}^d)\) with domain of definition \(C^{\infty }_c({\mathbb {R}}^d)\) is semibounded from below and closable [2]. Thus, the closure of this form induces a self-adjoint semibounded from below operator H(A, V) in \(L^2({\mathbb {R}}^d)\). The corresponding magnetic Schrödinger semigroup is given by the Feynman–Kac–Itô formula [2]
where \(\mathsf {Z} (x)\) is an arbitrary Brownian motion in \({\mathbb {R}}^d\) starting in \(x\in {\mathbb {R}}^d\). Using the Feynman–Kac–Itô formula for \(V=0\) with Theorem 2.3 to deal with the magnetic potential A, and perturbation theory to deal with the electric potential V, we can now establish:
Theorem 2.5
Let \(\beta \in (0,1)\), let \( A:{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}^{d}\), \(V:{\mathbb {R}}^d\rightarrow {\mathbb {R}}\) be Borel functions which satisfy
and let \(t>0\), \(1\le p\le q\le \infty \). Then one has
continuously, and there exists a universal constant \(c_0<\infty \) and a constant \(C_V<\infty \) which only depends on V, such that
where
is the locally bounded function from Theorem 2.3, and
Remark 2.6
1) Using monotone convergence, one finds
which is finite for all \(t>0\) by Remark 2.2.5), so that a posteriori one also has \(D_V<\infty \) a.e.
2) As our proof shows, the constant \(C_V\) can be chosen to be any constant which satisfies that for all \(1\le p\le q\le \infty \), \(r>0\) one has
The existence of such a uniform constant has been shown in [2].
3) Using \(\Psi =e^{t {\mathbb {R}}^d}e^{-t H(A,V)}\Psi \) for eigenfunctions \(\Psi \) of H(A, V), one obtains explicit \(L^r\rightarrow C^{0,\beta }\)-estimates for eigenfunctions.
Proof of Theorem 2.5
We start by noting that the assumptions on A together with Jensen’s inequality, and that \(\mathcal {K}^{\beta }({\mathbb {R}}^d)\subset \mathcal {K}({\mathbb {R}}^d)\) shows that the pair (A, V) satisfies (15).
Set \(q:=1/(1-\beta )\in (1,\infty )\) so that \(q^*=1/\beta \) and pick a mirror coupling \((\mathsf {X} ,\mathsf {Y} )\) from \((x,y)\in ({\mathbb {R}}^d\times {\mathbb {R}}^d){\setminus }\mathrm {diag}({\mathbb {R}}^d)\) and set \(\tau :=\tau (\mathsf {X} ,\mathsf {Y} )\). Then, given \(r>0\), \(\Phi \in L^2({\mathbb {R}}^d)\cap L^{\infty }({\mathbb {R}}^d)\) we can estimate as follows,
where \(c_0<\infty \) is a universal constant. Thus, we have shown
Duhamel’s formulaFootnote 2 states that
and so
There exists [2] a constant \(C_V\) such that for all \(1\le p\le q\le \infty \), \(r>0\) one has
so that
Moreover, by what we have shown above,
Given \(f\in L^{\infty }({\mathbb {R}}^d)\), \(x\in {\mathbb {R}}^d\), and a Brownian motion \(\mathsf {Z} (x)\) in \({\mathbb {R}}^d\) starting in x we have, using \(|e^{-\mathscr {S}_{s/2}(A|\mathsf {Z} (x)) } |=1\), the estimate
so that
Combining (17), (19), (20), (21), we have shown that for all \(\Phi \in L^{\infty }({\mathbb {R}}^d)\),
and so
The above estimate together with \(\Phi =e^{-\frac{t}{2}H(A,V)}\Psi \) and (18) shows
Finally, using (18) we end up with
which completes the proof. \(\square \)
For the following result, consider the linear surjective maps
and let \(A:{\mathbb {R}}^{3n}\rightarrow {\mathbb {R}}^{3n}\), \(V:{\mathbb {R}}^{3n}\rightarrow {\mathbb {R}}\) be arbitrary functions. Remark 2.2 then shows:
Corollary 2.7
Let \(\beta \in (0,1)\), \(l\in {\mathbb {N}}\) and let \(a:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{3}\), \(v_{i},v_{ij}:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}\) be Borel functions with
and
Then for all \(t>0\) and \(p\in [1,\infty ]\) one has
continuously.
Notes
which is satisfied under a suitable \(L^q\)-assumption on the electromagnetic potential, where q depends on \(\beta \) and the dimension d.
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Fürst, O., Güneysu, B. Hölder estimates for magnetic Schrödinger semigroups in \({\mathbb {R}}^{d}\) from mirror coupling. Lett Math Phys 111, 21 (2021). https://doi.org/10.1007/s11005-021-01360-x
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DOI: https://doi.org/10.1007/s11005-021-01360-x