Generalized Schwartz–Gunning theorem and Weyl type lemmas for the Schrödinger operator

Abstract

In this work, we study Schrödinger type equations with domain in a complex space and their solutions in the classical as well as in the sense of the space of currents. For such an equation with a (possibly) complex parameter and a (possibly) nonzero potential, conditions are given under which an analogue of the Weyl's lemma holds, meaning that each solution within the space of currents is distributionally realizable (in a suitable sense) by a semismooth, or by a semi-real-analytic function. Motivated by the Schwartz–Gunning theorem on the holomorphy of a locally integrable function, conditions guaranteeing a regular current on a general domain to be realized by a function which is (locally) a solution of the Schrödinger equation are obtained. Also, generalized Helmholtz formulas are derived and utilized in studying some special classes of zero-degree currents in regard to the property of μ-harmonicity or weak-harmonicity.

Keywords:

• $(\mu ,V)$-harmonic
• elementary solution
• solution within the space of currents of the Schrödinger equation

AMS SUBJECT CLASSIFICATIONS:

• Primary: 32C15
• 35J10
• Secondary: 32C30
• 31B35

Acknowledgments

The author wishes to acknowledge his appreciation for the hospitality of the Department of Mathematics, University of Padova. Part of the research of this paper was done while on sabbatical there in November 2016. The author is also grateful to the Department of Mathematics, Minnesota State University for granting a sabbatical in Fall 2016.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This method requires the use of the Green's mapping for the Helmholtz operator (Proposition 3.2); for example see the proof of Theorem 1.2.

2 Compare Chung [14, 30], Colton and Kress [31], Lyubich [32, p.50], Mizohata [33, Chapter 3 and 8].

3 After all, by stretching a point made by R. Penrose [34, p.230], one might imagine that ‘as though complex domains were there all the time’, hoping thereby to gain some new insights which, when restricted to a real setting, may lie hidden. For a possible example, see the formula (1.7) of [8], which remains valid in the case $V\not\equiv 0$ (see p.1261, ibid.).

4 Compare Edwards [35, p.357] for a related notion of ‘metaharmonicity’.

5 For a proof of this fact, see Lemma A.3.

6 In Chung [14] such a function (on a Euclidean domain) is taken in a weak sense.

7 See the equation (1.7); and, hereafter, the subscript V appearing as part of a notation will be dropped if $V\equiv 0$.

8 For the definition and basic properties of differential forms on a singular space see [9, §4.1].

9 The Sobolev space $W^{1,2}\left(D\right)$ can be defined by taking a suitable completion of the space ${\mathcal{C}}^1\left(\overline{D}\right)$. For more details see [8] or [15].

10 This action exists under other conditions, for instance, if $T=\left[\psi \right]$ is regular with $\psi \in {\mathcal{C}}^{\mathfrak{m}\cap \text{β}}\left(D\right)$ and $V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1\left(D\right),$ or if $T=\left[\psi \right]$ with both ψ and V belonging to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^2\left(D\right)$.

11 The right-hand limit in (9(9) $\begin{array}{rl}\prec {\tilde{\mathcal{S}}}_{p,\mu ,V}T,v\succ & =\prec T,{\mathcal{S}}_{p,\mu ,V}\left(v^0\right)\succ \\ & :=\underset{n\to \mathrm{\infty }}{lim}(T,{\chi }_{-{\mathrm{△}}_p{\psi }_n+(\mu -V){\psi }_n})\end{array}$(9) ) is independent of the choice of the sequence $\left\{{\psi }_n\right\}$.

12 The rationale behind the definition (10(10) ${\mathcal{E}}_{w,\mu ,V}\left(\zeta \right)=p_U^{\ast }{\mathfrak{q}}_{w,\mu ,V}^a\left(\zeta \right),\zeta \in \overline{U},{\zeta }^{\prime }\ne w^{\prime },$(10) ) (and Theorems 3.1 and 4.1) may be brought better to light by recalling a classical result, due to L. Schwartz, which states that ‘a necessary and sufficient condition for a linear, constant coefficient differential operator to be hypoelliptic in the entire ambient space is that the named operator possesses a fundamental solution with singular support consisting of the origin alone’, see [36].

13 A semi-${\mathcal{C}}^{\mathrm{\infty }}$ function $\mathfrak{h}$ on Y means that (locally) the function $\mathfrak{h}$ is ${\mathcal{C}}^{\mathrm{\infty }}$ off a thin analytic subset of a neighborhood of each point of Y.

14 Compare [9, 1.21] or Gunning [19, Theorem 10,p.192], where a characterization of the local equivalence at a point is given in terms the existence of a light holomorphic mapping with multiplicity equal to 1 at the point.

15 The author is grateful to Prof. M. Lanza de Cristoforis for his helpful discussions on the complex Helmholtz equation and for providing the reprints of [20, 21] and his unpublished lecture notes. Although the proof given here does not depend on the results therein, the inspiration drawn from reading them has helped to complete this proof. For a related discussion of the Euclidean case, see also Trèves [37, p.259].

16 The assertion (2) holds for a general space Y on which T is locally restrictable.

17 Mean-value representations hold in different forms for different elliptic differential equations (cf. [[38, p.286–289, in part. (39)], [39, (2.17), p.40]]) for ${\mathcal{C}}^2$-μ-harmonic functions, and [40] (resp. [36, p.237]) for pluri- (resp. bi-) harmonic functions; also [41, Theorem 9.9,p.252] for a μ-subharmonic mean-value inequality.

18 For a definition see [11, p.704].

19 Note that (under the condition (a) the formula (31(31) ${\nu }_p\left(z\right)\mathfrak{h}\left(z\right)=l_m{\int }_{\mathrm{\partial }{U}_{\left[z\right]}\left(r\right)}({\gamma }_{\mu }^z{\mathrm{\partial }}_{\nu }\mathfrak{h}-\mathfrak{h}{\mathrm{\partial }}_{\nu }{\gamma }_{\mu }^z)\mathrm{d}\sigma \mathrm{\forall }r\in (0,r_z).$(31) ) is valid with respect to every pseudoball (of small pseudoradius) in the domain of continuity of $\mathfrak{h}$.

20 This means that T corresponds to such a function.

21 It follows from Lemma A.5 that if this property holds in terms of ${\mathfrak{p}}_{\left\{1\right\}}=p_U$ for a pseudoball U, then it holds in terms of any continuous, proper map ${\mathfrak{p}}_{\left\{2\right\}}={\alpha }^{-1}$ of a domain ${\mathrm{\Omega }}^{\prime }\subseteq {\mathbb{C}}^m$ onto U arising from a homeomorphic normalization mapping $\alpha :{\mathrm{\Omega }}^{\prime }\to U,$ making use of the uniform boundedness (in $\hat{U}$) of the Jacobian determinants of the local transition maps between sheets of ${\mathfrak{p}}_{\left\{1\right\}}$ and ${\mathfrak{p}}_{\left\{2\right\}}$

22 See Lemma A.4.

23 The equivalence between the formulas (36(36) $l_m{\int }_{d{U}_{\left[a\right]}\left(r\right)}\left({\gamma }_0^a\right({\mathrm{\partial }}_{\nu }\mathfrak{h})-\mathfrak{h}{\mathrm{\partial }}_{\nu }{\gamma }_0^a)\mathrm{d}\sigma ={\nu }_p\left(a\right)\mathfrak{h}\left(a\right),\mathrm{\forall }r\in (0,r_a).$(36) ) and (37(37) $\frac{1}{r^{2m}}{\int }_{U_{\left[z\right]}\left(r\right)}\mathfrak{h}{\upsilon }_p^m={\nu }_p\left(z\right)\mathfrak{h}\left(z\right),0<r<r_z.$(37) ) is a consequence of [26, (5.7)].

24 Each of these two statements gives a correction to the implication ‘(1) ⇒ (2)’ of [26, Theorem 4.2] (cf. [27]).