TYPE EQUATIONS WITH BOUNDARY CONDITIONS

CHIA-CHI TUNG

This paper is concerned with the weak solvability of the boundary value problems for Schr¨odinger type equations. For such equations with a real (admissible) spectral parameter µ, the nonhomogeneous Dirichlet, respectively, homogeneous Dirichlet-Neumann problem on a weak Stokes domain lying in a Riemann domain is shown to admit a unique weak solution by making use of aSchr¨odinger product.

If the parameter (denoted by λ) is complex-valued, the weak solvability (in a
suitable sense) of a homogeneous Dirichlet-Neumann boundary value problem for
a generalized Schr¨odinger equation withL^{2}-density is proved in the case where
λ does not belong to R^{−}.

AMS 2010 Subject Classification: Primary 35J10, 35J25; Secondary 32C30, 35A01.

Key words: generalized Schr¨odinger product, Schr¨odinger functional, weak Green’s operator.

1. INTRODUCTION

One of the most important example of a singular differential operator (in mathematical physics) is the Schr¨odinger operator

S(ψ) :=− 4ψ+ (µ−V)ψ

in L^{2}(R^{n}), where µ is a real constant (usually called the spectral parameter)
and V a real-valued L^{2}-function (the potential). For wider applications, one
may wish to take µ and/or V to be complex-valued. In the following, let
Y denote a complex space of pure positive dimension m and DbY a rela-
tively compact open subset, and ˜D⊆Y an open neighborhood of D. Assume
throughout this paper that ( ˜D, p) is a Riemann subdomain of Y, meaning
that ˜D (as a subspace of Y) admits a holomorphic map p : ˜D → C^{m} with
discrete fibers. Denote by 4_{p} the pull-back under p of the Laplace operator
of the Euclidean metric on C^{m} to (the largest) open dense subset D^{∗} ⊆ D
where p is locally biholomorphic. Observe that if ψ∈C^{2}(D), theSchr¨odinger
operator S˜_{p,λ,V}[ψ] :=−4˜_{p}[ψ] + (λ−V) [ψ] associated with a constant λ∈C
and V ∈ L^{1}_{loc}(D), regarded as acting on C_{c}^{2}(D), is induced on D^{∗} by the
function

MATH. REPORTS15(65),4(2013), 497–510

(1.1) S_{p,λ,V}(ψ) :=− 4_{p}ψ+ (λ−V)ψ.

If D is a weak Stokes domain ([6], p. 568) lying in a Riemann domain, for the Schr¨odinger equation

(1.2) S_{p,λ,V}(Ψ) =− 4_{p}Ψ + (λ−V) Ψ = g

in D^{∗} with a real spectral parameter λ and density g ∈ L^{2}(D) (or more
generally, g ∈ H^{−1}(D)), sufficient conditions can be given for the solvability,
in a weak sense, of the equation (1.2) subject to a boundary condition (Theo-
rem 4.1). Similarly, one may consider, for a generalized Schr¨odinger equation,
a mixed homogeneous Dirichlet-Neumann problem relative to a partition

(1.3) ∂D\A= (∂D)D∪(∂D)N,

where A is a thin analytic subset of X (see Theorems 4.2 and 4.3). This
is done by making use of a Schr¨odinger product S_{λ,h} defined on a (suitable)
Sobolev space H_{λ,h}^{1} (D).

For the metaharmonic operator M_{p,λ} := S_{p,λ,0} with a real spectral pa-
rameter λ, it is possible to construct a Green’s operator (with explicitly de-
termined kernel) (at least) on suitable pseudoballs in a complex space. The
general question remains as to for what parameters λ the equation (1.2) ad-
mits a classical or a weak solution (in a suitable sense). If D is a domain in the
complex plane, Garnir [4] proved by an ingenious method that the equation

”Mp,λ(Ψ) =g” admits a distributional solution, provided λ belongs to C\R−.
This method can be extended to the case of ageneralized Schr¨odinger operator
P_{λ,h}w, defined for complex parameters λand elements w of a linear subspace
of L^{1}_{loc}(D) (see (3.6)), by making use of a generalized Poincar´e’s inequality
(Lemma 3.1) and introducing a Schr¨odinger map and itsweak Green’s opera-
torfor a real (admissible) spectral parameter µ(compare [4]). The solvability
of a homogeneous Dirichlet-Neumann problem (see (4.9)) for an (admissible)
L^{2}-density g can then be ascertained, in the case λ∈C\R−, in a weak sense,
namely, there exists an element w (in a Sobolev space) satisfying suitable
boundary conditions such that the associated Schr¨odinger functional satisfies
the equation

(1.4) hP_{λ,h}w, vi_{D} = h[g], vi_{D},

where v ranges over a (suitable) test space (Theorem 5.1). For Neumann prob-
lems for related Schr¨odinger type equations in a (possibly) irregular domain in
R^{n}, see [1, 2]. The author is indebted to the referee for suggestions which led
to improvements of this paper.

2.PRELIMINARIES

In what follows, every complex space is assumed to be reduced and has a
countable topology. For the definition and basic properties of differential forms
on a complex space, see ([5],§4.1). In particular, the exterior differentiationd,
the operators ∂, ∂¯ and d^{c}:= (1/4πi)(∂−∂) are well-defined ([5], Chap. 4).¯
Denote by kzk the Euclidean norm of z = (z1,· · · , zm) ∈ C^{m}, where zj =
x_{j} +i y_{j}. Let the space C^{m} be oriented so that the form υ^{m} := (dd^{c}kzk^{2})^{m}
is positive. If p:Y →C^{m} is a holomorphic map, set a^{0} :=p(a), p^{[a]}:=p−a^{0}
and ra:=kp^{[a]}k, for a∈Y. Clearly the form

υp :=dd^{c}r_{a}^{2} = ( i

2π)∂∂ r¯ _{a}^{2}
is nonnegative and independent of a.

For an open subset G of Y, denote by C_{k}^{ν}(G) the set of C-valued k-
forms of classC^{ν} on G, C_{k,c}^{ν} (G) the subspace of compactly supportedk-forms
(dropping the degree if k= 0), with ν =β to mean locally bounded, ν =m
measurable, and ν = λ locally Lipschitzian ([5], §4). Similarly, for C_{k}^{λ}(G).

A measurable function ψ on Y is said to be locally integrable (f ∈L^{1}_{loc}(Y))
if so is every 2m-form ψχ with χ ∈ C_{2m}^{0} (Y). Denote by dυ the Euclidean
volume element of C^{m} and define the semivolume form d˜υ := p^{∗}(dυ) on Y.

If f, ψ∈L^{1}_{loc}(D), set

(f, ψ)_{D} :=

Z

D

f ψd˜υ,

provided the integral exists. Each element f ∈L^{1}_{loc}(D) gives rise naturally to
a top-dimensional current, T = [f] :χ7→R

f∧χ, with induced functional
h[f], φi:= (−1)^{m(m−1)}^{2} π^{m}

m!

Z

T ∧φ υ^{m}_{p} = (f, φ)_{D}, ∀φ∈C_{c}^{1}(D).

3. THE SCHR ¨ODINGER PRODUCT

In ([5],§3) theDirichlet productof locally Lipschitz functions ψ, η:D→ C is defined:

[ψ, η]_{[D]} :=

Z

D

dψ∧d^{c}η¯∧υ^{m−1}_{p} ,

provided the integral exists. It is shown in ([7], (3.6)) that the Dirichlet product
is Hermitian symmetric. The Hilbert scalar product on L^{2}(D) is defined by
the pairing (f, g)7→(f,g)¯ _{D} (with induced norm k k_{L2(D)}). A generalization
of both the Hilbert and Dirichlet product is considered below.

If ψ = v ∈ C^{1}(D), denote by ∂_{ν}ψ,1 ≤ ν ≤ 2m, the ν-th first order
derivative _{∂t}^{∂ψ}

ν on D^{∗}, namely,

∂ψ

∂t_{ν} := ∂v

∂x˜_{k}, ifν = 2k−1, ∂ψ

∂t_{ν} := ∂v

∂y˜_{k}, if ν= 2k.

Define C^{1,1}(D) := {φ ∈ C^{1}(D)|φ_{xj}_{˜} and φ_{yj}_{˜} ∈ C^{λ}(D),1 ≤ j ≤ m}. If
ψ∈L^{1}_{loc}(D), denote by (the same) ∂νψ theν-th distributionalweak derivative
of ψ on D, provided it exists as an element of L^{1}_{loc}(D). Let M_{D} be the
subspace of L^{1}_{loc}(D) consisting of all elements w such that: (i) the first order
weak derivatives ∂_{ν}w =φ_{ν}, ν = 1,· · ·,2m, exist (as an element of L^{1}_{loc}(D)),
namely,

h∂_{˜}_{t}

ν[w], vi=−(w, ∂v

∂˜tν

)_{D}∗ = (φ_{ν}, v)_{D}∗, ∀v∈C_{c}^{1}(D);

(ii) each weak derivative ∂_{ν}w ∈ C^{β}(D). In the following, let h denote an
(2m+ 1)-tuple (h0,· · · , h2m), where each component hj is a locally integrable
function on D, with h_{j} ≥ 0 for 1 ≤ j ≤ 2m, and set V := −h_{0}. Then it
is easy to show that if w, v ∈ M_{D}, then each of the forms h_{j}(∂_{j}w)(∂_{j}¯v)d˜υ
is integrable on D. It follows that for any given complex constant λ, the
sesqilinear form

(3.1) S_{λ,h}(w, v) := ((λ−V)w,¯v)_{D}+

m

X

j=1

Z

D

h2j−1

∂w

∂x˜j

∂¯v

∂x˜j

+h2j

∂w

∂y˜j

∂¯v

∂˜yj

d˜υ

is well-defined on M_{D} × M_{D}. Since an element of C^{1}(D) gives rise to a
member of M_{D}, the Schr¨odinger form S_{λ,h} : C^{1}(D)×C^{1}(D) → C is well-
defined by (3.1). In the following, the reference to ”µ,h” will be abbreviated
simply as ”µ, V” whenever h=h^{{1}} := (−V,1,· · ·,1). Thus,

(3.2) S_{λ,V}(w, v) =S

λ,h{1}(w, v) = ((λ−V)w, ¯v)_{D} +S_{0,(0,h}

1,···,hm)(w, v).

Green’s first identity ([6], (5.9)) yields a relation between the operator
S˜_{p,λ,V} and the form S_{λ,h}:

(3.3) hS˜_{p,λ,V} [w], vi = S_{λ,V}(w,¯v), ∀(w, v)∈C^{1}(D)×C_{c}^{2}(D).

In the following, let µ denote a real constant. Then S_{µ,h} defines a Hermi-
tian symmetric, continuous form on C^{1}(D) (with induced seminorm denoted
accordingly by k k_{µ,h}). Let ∆ be the branch locus of p and set ∆^{0} =p(∆).

If a ∈ D_{0} := D\p^{−1}(∆^{0}), a connected neighborhood U b D can be chosen
so that p|U is a covering projection onto a ball U^{0} bC^{m}[1] with sheets Uν.
In terms of such U =U^{α}, a C^{∞}-partition of unity {(U_{ν}^{α}, ρ^{α}_{ν})}, each U_{ν}^{α} be-
ing a sheet of U^{α}, is said to be distinguished for D_{0}, if, for any real-valued

φ∈C^{1}(D), setting φ^{α}_{ν} :=ρ^{α}_{ν}φ, one has
Z

D0

2m

X

j=1

(X

α6=β

h_{j}∂φ^{α}_{µ}

∂tj

∂φ^{β}_{ν}

∂tj

) d˜υ≥0.^{1}

Definition 1. A pair (µ,h) (as above) is said to beadmissible for D if:

(i) h_{D} := min{ess inf_{D}hj|j= 1,· · ·,2m}>0; (ii) D0 admits a distinguished
partition of unity; and (iii) denoting by P

Cm[1] the Poincar´e’s constant for the Euclidean unit ball,

µ−ess sup_{D}V >− h_{D}
(PCm[1])^{2}.

A pair (µ, V) is said to beadmissible for D if so is (µ,h^{{1}}).

If the map p exhibits D as a covering space of an open all in C^{m}, then
the condition (ii) above holds trivially; in which case, every pair (µ,0) with
µ≥0 is admissible for D.

Lemma 3.1 (Generalized Poincar´e’s inequality). If (µ,h) is admissible
for D, then there exists a positive constantc_{D,µ,h} (depending on a distinguished
covering of D) such that

(3.4) kφk_{L2(D)} ≤ c_{D,µ,h}kφk_{µ,h}, ∀φ∈C^{1}(D).

Proof. It suffices to consider a real-valued φ ∈ C^{1}(D). In terms of a
distinguished covering for D_{0}, by the Poincar´e’s inequality for the Euclidean
unit ball, one has

h_{D}
(PCm[1])^{2}

Z

D

kφk^{2}d˜υ ≤ h_{D}
(PCm[1])^{2}

X

α,ν

Z

(U^{α})^{0}

k(φ^{α}_{ν})^{0}k^{2}dυ

≤h_{D}X

α,ν

Z

(U^{α})^{0}

k∇(φ^{α}_{ν})^{0}k^{2}dυ≤
Z

D0

X

α,ν 2m

X

j=1

hjk∂(φ^{α}_{ν})^{0}

∂t_{j} k^{2}d˜υ

≤ Z

D0

2m

X

j=1

h_{j}(X

α, β

∂(φ^{α}_{µ})^{0}

∂tj

∂φ(φ^{β}ν)^{0}

∂tj

) d˜υ=S_{0,(0,h}

1,···,hm)(φ, φ),
where (φ^{α}_{ν})^{0} : (U^{α})^{0} →C is induced by the restriction of φ to a sheet of U^{α}.
Therefore, one has

kφk^{2}

µ,h ≥ µ−ess sup_{D}V + h_{D}
(PCm[1])^{2}

kφk^{2}

L2(D) =:κ_{D,µ,h}kφk^{2}

L2(D).

1This condition is likely to be superfluous by making judicious choice of the partition functions.

In the following, assume (unless otherwise mentioned) that (µ,h) is ad-
missible for D. Then the Hermitian form S_{µ,h} is elliptic, hence defines a scalar
product on C^{1}(D). With respect to the induced norm k k_{µ,h} the completion
of C^{1}(D) (respectively, C_{c}^{∞}(D)) gives rise to theSobolev space H^{µ}:=H_{µ,h}^{1} (D)
(respectively, H_{µ,h,c}^{1} (D)) equipped with the (naturally extended) scalar prod-
uct S_{µ,h} given by the right-hand side expression of (3.1). It follows that, if
φ∈H^{µ}, then the generalized Poincar´e’s inequality (3.4) holds for all φ∈H^{µ}.
Clearly, this inequality implies that H^{µ}⊆L^{2}(D). Taking (µ, V) = (1,0), the
Sobolev spaces H^{1}(D) and H_{c}^{1}(D) are defined. Moreover, the Schr¨odinger
form (3.2) extends to a scalar product [ , ]_{1,0,D} =S_{1,(0,1,···}_{,1)} on H^{1}(D) with
associated energy norm denoted by k k_{1,0,D}.

In what follows, let λ denote a complex (possibly real) constant, and
(λ,h) a pair with h = {−V, h} where each hj ∈ C^{λ}(D) (unless otherwise
mentioned). It can be shown that, if w∈ M_{D}, the integral

(3.5) h∂_{k}[hj∂jw],φi¯ =−(hj∂jw, ∂kφ)¯ _{D}∗

exists for all φ∈C_{c}^{1}(D) and j, k = 1,· · · ,2m ([8], §3, Remark 1). Therefore,
each such element w gives rise to aSchr¨odinger functional P_{λ,h}w:C_{c}^{1}(D)→C
defined by

(3.6) hP_{λ,h}w, vi_{D} := S_{λ,h}(w, v), ∀v∈C_{c}^{1}(D).

Note also, that the right-hand side of (3.6) serves to extend (i) the action
of the functional P_{µ,h}w to H_{µ,h}^{1} (D) and (ii) its definition to w∈H_{µ,h}^{1} (D). If
w∈C^{1,1}(D), the functional

(3.7) hS_{λ,h}(w), φi_{D}∗ := ((λ−V)w,φ)¯ _{D}−X

j

Z

D^{∗}

∂_{j}(hj∂_{j}w) ¯φd˜υ

is well-defined for all elements φ∈C_{c}^{1}( ˜D). Consequently, the functional ”φ7→

hS_{µ,h}(w), φi_{D}∗” extends naturally to an operator (denoted by the same) on
H_{µ,h}^{1} (D). In the following, assume that D⊆Y is a weak Stokes domain with
nonempty (maximal) boundary manifold dD of D_{reg} (oriented to the exterior
of the manifold D_{reg} of simple points of D) ([5], p. 218).

Proposition 3.1. Assume that w∈C^{1,1}(D). Then for all φ∈C^{1,1}(D),
(3.8) hS_{λ,h}(w), φi_{D}∗ +

2m

X

j=1

Z

dD

φ¯n_{j}(h_{j}∂_{j}w) dσ =S_{λ,h}(w, φ),

where dσ denotes the surface element of dD, n the unit outward normal
to dD. If further λ = µ ∈ R, the above formula remains valid for all φ ∈
H^{1}

µ,h,c(D).

Proof. For all w, φ ∈ C^{1,1}(D), the formula (3.8) is a consequence of
the Green’s formula. Assume now that λ = µ ∈ R. An arbitrary element
φ∈H_{µ,h,c}^{1} (D) is approximable in the L^{2}-norm on L^{2}(D) by a sequence φn∈
C_{c}^{∞}(D). Then

S_{λ,h}(w, φ_{n})= ((µ−V)w,φ¯_{n})_{D}+

2m

X

j=1

Z

dD

φ¯_{n}n_{j}(hj∂_{j}w) dσ−
Z

D

φ¯_{n}∂_{j}(hj∂_{j}w) d˜υ

=hS_{µ,h}(w), φ_{n}i

D∗+

2m

X

j=1

Z

dD

φ¯_{n}n_{j}(h_{j}∂_{j}w) dσ

→ hS_{µ,h}(w), φi_{D}∗ +

2m

X

j=1

Z

dD

φ¯n_{j}(hj∂_{j}w)dσ, as n→ ∞,

thus, proving formula (3.8).

4. WEAK SOLVABILITY OF SCHR ¨ODINGER TYPE EQUATIONS

A pair (µ,h) (with h as in (3.1)) is said to bestrictly admissible for D, if

(4.1) kφk_{1,0,D} ≤Const.kφk_{µ,h}, ∀φ∈C_{c}^{∞}( ˜D),

(noting that C_{c}^{∞}( ˜D)⊂C_{c}^{1}(D)). To each such pair (µ,h) the associated ”S_{µ,h}”
is positive definite, hence defines a scalar product on H_{µ,h}^{1} (D).

It is shown in ([8], §3) that the k-th partial derivatives ∂_{x}_{˜}_{k}T and ∂_{y}_{˜}_{k}T
of a regular current T = [ψ] (where ψ∈L^{1}_{loc}(D)) can be defined as (2m−1)-
dimensional quasicurrents on D (which are currents if D is nonsingular). As
a consequence, given g= (g_{0}, . . . , g_{2m}) with g_{j} ∈L^{2}(D), j= 0,1,· · · ,2m, by
extending each g_{j} to ˜D by setting ˜g_{j} = 0 on ˜D\D, there is an associated
quasicurrent

(4.2) bgc := ˜g_{0}+

m

X

j=1

∂_{˜}_{x}_{k}[˜g_{j}] +∂_{˜}_{y}_{k}[˜g_{j}]

on ˜D, and an induced functional (denoted by the same) acting on C_{c}^{∞}( ˜D):

(4.3) hbgc, φi_{D} := (g_{0},φ)¯ _{D}∗ −

m

X

j=1

(g_{j}, ∂φ¯

∂x˜j

)_{D}∗ + (g_{j}, ∂φ¯

∂y˜j

)_{D}∗

.

Let H^{−1}(D) denote the linear space of all such 2m-tuples g. It follows
from (4.3) and H¨older’s inequality that, if (µ,h) is strictly admissible for D,

then

(4.4) |hbgc, φi_{D}| ≤Const. max

0≤j≤2mkg_{j}k_{L2(D)}kφk_{µ,h}, ∀φ∈C_{c}^{∞}( ˜D).

Thus, the linear operator bgc is continuous on C_{c}^{∞}(D) with respect to
the norm k k_{µ,h} (or k k_{µ,V}, if h = h^{{1}}). The subspace C_{c}^{∞}(D) being
dense in H_{µ,{h},c}^{1} (D), the operator bgc admits a unique continuous linear
extension (denoted by the same) to the latter (relative to k k_{µ,h}), hence also
to H_{µ,h}^{1} (D), by the Hahn-Banach’s theorem. The operator bgc is said to be
representableby an element η∈L^{2}(D), if the action of bgc on C_{c}^{∞}(D) agrees
with the (conjugated) action of [η]. Note that if each g_{j} has weak derivative

∂_{ν}gj belonging to L^{2}(D) for ν= 1,· · · ,2m, then bgc is representable by an
element of L^{2}(D). The subspace of ”Schr¨odinger functions” of type (µ, V),
namely,

S_{µ,V}(D) := {w∈H^{1}

µ,V(D)|P

µ,h{1}w= 0},

is given precisely by the orthogonal complement of H_{µ,V,c}^{1} (D) (with respect to
S_{µ,V}):

(4.5) H^{1}

µ,V(D) =H^{1}

µ,V,c(D)⊕ S_{µ,V}(D).

This is an easy consequence of the identity (3.3).

Definition 2. Let g ∈H^{−1}(D) and f ∈C^{1,1}(D). An element Ψ =w∈
H_{µ,V}^{1} (D) is called aweak solution of the Dirichlet problem

(4.6) S_{p,µ,V}(Ψ) =− 4_{p}Ψ + (µ−V) Ψ = bgc, Ψ = f on dD,
provided that w≡f mod (H_{µ,V,c}^{1} (D)) and

S_{µ,V}(w, ρ) = ((µ−V)w, ρ)¯_{D} +S_{0,0}(w, ρ) = hbgc, ρi_{D}, ∀ρ∈H_{µ,V,c}^{1} (D).

It is a consequence of Green’s first identity that, if g ∈ L^{2}(D), every
weak solution Ψ =ψ∈C^{1,1}(D) of the Dirichlet problem

(4.7) S_{p,µ,V} (Ψ) = g in D^{∗}, Ψ = 0 on dD,
is a classical solution of the Schr¨odinger equation (1.2) in D^{∗}.

Theorem 4.1. Assume that either (i) g ∈ L^{2}(D) and (µ, V) is ad-
missible for D or (ii) g ∈ H^{−1}(D) such that bgc is representable by an
element η ∈ L^{2}(D) and (µ, V) is strictly admissible for D. Then for each
f ∈C^{1,1}(D)∩C^{2}(D^{∗}) with 4_{p}f ∈L^{2}(D), the Dirichlet problem (4.6)admits
a unique weak solution in H_{µ,V}^{1} (D).

Proof. Let F := 4_{p}f. The homogeneous Dirichlet problem (4.7) with

“g” replaced by “ ˆg:=η+F−(µ−V)f”, where η=g whenever g∈L^{2}(D),

admits a weak solution ψ=w_{0} ∈H_{µ,V,c}^{1} (D), by Riesz’s representation theorem
(applied to the functional bˆgc). Then the function Ψ :=w_{0}+f gives a weak
solution of the Dirichlet problem (4.6). For the equation (3.3) (with µ = 0)
and the Hermitian symmetry of the Dirichlet product imply that

S_{0,0}(f, φ) = −(4_{p}f, φ)¯ _{D}, ∀φ∈C_{c}^{∞}(D),
hence, by virtue of definition (3.2), one has

S_{µ,V}(Ψ, φ) =S_{µ,V}(w_{0}, φ) +S_{µ,V}(f, φ)

=hbˆgc, φi_{D}+ ((µ−V)f, φ)¯ _{D}+S_{0,0}(f, φ) =hbgc, φi_{D},
thus, proving the above claim.

The difference ψ= Ψ1−Ψ_{2} of two solutions of (4.6) belongs to H_{µ,V,c}^{1} (D)
and the functional ”S_{µ,V}(ψ, )” annihilates H_{µ,V,c}^{1} (D). Hence, the decomposi-
tion (4.5) implies that ψ= 0.

To consider a mixed Dirichlet-Neumann boundary value problem with a
partition (1.3) of∂D\A, let N be an open neighborhood of (∂D)N contained
in ˜D and set ˆD := D∪ N. Denote by R_{0,D} the closure in H_{µ,h}^{1} (D) of the
subspace consisting of φ∈ C_{c}^{∞}( ˆD) that vanishes in the intersection with D
of some neighborhood of (∂D)_{D} (possibly depending on φ). Let A_{µ,h} be
a maximal linear subspace of H_{µ,h}^{1} (D) admitting a (Schr¨odinger) continuous
linear mapping s_{µ,h} :A_{µ,h} →L^{2}(D) satisfying the equation

(4.8) hP_{µ,h}w, vi_{D} = (s_{µ,h}(w),v)¯ _{D}, ∀v∈C_{c}^{∞}(D).

It can be shown that, for each w ∈ C^{1,1}(D)∩ A_{µ,h}, the (conjugated)
action of [s_{µ,h}(w)] on R_{0,D} agrees with the action of [S_{µ,h}(w)]. Define

N_{0,D}^{µ} :={w∈ A_{µ,h}∩R_{0,D}|S_{µ,h}(w, φ) = (s_{µ,h}(w),φ)¯ _{D}, ∀φ∈R_{0,D}}.

Definition 3.A weak solution to the Dirichlet-Neumann problem
(4.9) S_{µ,h}(w) = bgc, w∈ N_{0,D}^{µ} ,

where g∈H^{−1}(D), is an element w∈ N_{0,D}^{µ} satisfying the equation
(4.10) hP_{µ,h}w, φi_{D} = hbgc, φi_{D}, ∀φ∈R_{0,D}.

Theorem4.2. Assume that either(i) g∈L^{2}(D) and (µ,h) is admissible
for D or (ii) g ∈ H^{−1}(D) such that bgc is representable by an element of
L^{2}(D) and (µ,h) is strictly admissible for D. Then the Dirichlet-Neumann
problem (4.9)admits a unique weak solution.

Proof. Let B_{µ,h} be the set of all elements w∈H_{µ,h}^{1} (D) such that the anti-
linear form v 7→ hP_{µ,h}w, vi_{D} is continuous on H_{µ,h,c}^{1} (D) with respect to the
L^{2}-norm on L^{2}(D). B_{µ,h} is clearly a linear subspace of H_{µ,h}^{1} (D). According to
the Hahn-Banach’s theorem, the said anti-linear form is extendable to a unique
continuous anti-linear form on L^{2}(D), and this extension is given by taking the
scalar product of a unique element of L^{2}(D), to be denoted by s_{µ,h}(w), with
v. Thus the equation (4.8) holds for all v∈H_{µ,h}^{1} (D). With g given as either
in (i) or, in (ii), the functional φ 7→ hbgc, φi_{D} is well-defined and bounded
on R_{0,D} (by the inequality (4.4)). Hence, the Riesz’s representation theorem
implies that there is a unique element w_{0} ∈R_{0,D} satisfying the equation
(4.11) hbgc, φi_{D} =S_{µ,h}(w_{0}, φ), ∀φ∈R_{0,D}

with

(4.12) kw_{0}k_{µ,h} ≤ Const. max

0≤j≤2mkg_{j}k_{L2(D)}.

If bgc is representable (on C_{c}^{∞}(D)) by an element η ∈ L^{2}(D) (or,
if g ∈ L^{2}(D)), then by the formula (3.6), the equation (4.8) holds with
s_{µ,h}(w_{0}) =η(or g), for all v∈C_{c}^{∞}(D). One has, for such v, |hP_{µ,h}w_{0}, vi_{D}| ≤
kηk_{L2(D)}kvk_{L2(D)}, thereby showing that w_{0} ∈ B_{µ,h}. Moreover, by the equation
(4.11), this element w_{0} belongs to N_{0,D}^{µ} , thus giving a weak solution to the
problem (4.9).

The difference ψ = w_{1}−w_{2} of two weak solutions of (4.9) satisfies the
conditions ψ∈ N_{0,D}^{µ} ⊂R_{0,D} and hP_{µ,h}ψ, φi_{D} = 0 on R_{0,D}. Hence, it follows
from the Poincar´e’s inequality (3.4) (or (4.1)) that kψk_{L2(D)} = 0, and thus
ψ= 0.

Theorem 4.3. If (µ,h) is admissible for D, then the Schr¨odinger map
s_{µ,h} : A_{µ,h} → L^{2}(D) is invertible with inverse given by a continuous, self-
adjoint and strictly positive isomorphism G_{µ,h} : L^{2}(D) → N_{0,D}^{µ} (called the
weak Green’s operator for (4.9)); in particular, for g ∈ L^{2}(D), the element
w = G_{µ,h}g gives the unique weak solution to the Dirichlet-Neumann problem
(4.9). Moreover, for admissible (µ_{j},h), j= 1,2, the operators G_{µ}_{j}_{,h} are com-
mutative.

Proof. Given g ∈ L^{2}(D), the association g 7→ G_{µ,h}g := w, the unique
weak solution to Dirichlet-Neumann problem (4.9) is well-defined by Theo-
rem 4.2. The equations (4.11)–(4.12) imply that

(4.13) S_{µ,h}(G_{µ,h}g, φ) = (g,φ)¯ _{D}, ∀φ∈R_{0,D},
with

(4.14) kG_{µ,h}gk_{µ,h} ≤Const.kgk_{L2(D)}.

Furthermore, the induced mapping G_{µ,h} : L^{2}(D) → N_{0,D}^{µ} is a bounded
linear operator giving the left-inverse of s_{µ,h}|N_{0,D}^{µ} . Setting φ = G_{µ,h}f with
f ∈L^{2}(D) in (4.13) yields

(4.15) S_{µ,h}(G_{µ,h}g,G_{µ,h}f) = (g,G_{µ,h}f)_{D}.
Also, by (4.13) one has

S_{µ,h}(G_{µ,h}f,G_{µ,h}g) = (f,G_{µ,h}g)_{D}.
Consequently,

(4.16) (g,G_{µ,h}f)_{D} = (f,G_{µ,h}g)_{D} = (G_{µ,h}g,f)¯_{D},

thus proving that G_{µ,h} is a self-adjoint endomorphism of L^{2}(D). Formula
(4.15) shows that the integral (G_{µ,h}g,g)¯ _{D} is real-valued, and thus, by virtue
of (3.4), one has, by relations (4.15) and (4.16),

(G_{µ,h}g,g)¯ _{D} ≥(c_{D,µ,h})^{−2}kG_{µ,h}gk^{2}

L2(D).

On the other hand, the H¨older’s and the generalized Poincar´e inequalities imply that

(G_{µ,h}g,g)¯ _{D} ≤Const.kG_{µ,h}gk_{µ,h}kgk_{L2(D)}.
Therefore, by the inequality (4.14),

kG_{µ,h}gk^{2}

L2(D) ≤Const.kgk^{2}

L2(D),

so that G_{µ,h}, is a continuous endomorphism of L^{2}(D). Furthermore, G_{µ,h} is
strictly positive, since in fact, the condition G_{µ,h}g= 0 gives g=s_{{µ,h}}(G_{µ,h}g) =
0, so that, if g ∈ L^{2}(D)\{0}, then (G_{µ,h}g, ¯g)_{D} > 0. Finally, for admissible
(µ_{j},h), j= 1,2, one has

G_{µ}_{1}_{,h}g=G_{µ}_{1}_{,h}(s_{µ}

0,h(G_{µ}_{0}_{,h}g)) =G_{µ}_{1}_{,h}([s_{µ}

1,h−(µ1−µ0)]G_{µ}_{0}_{,h}g)

=G_{µ}_{1}_{,h}(s_{µ}

1,h(G_{µ}_{0}_{,h}g))−(µ1−µ0)G_{µ}_{1}_{,h}(G_{µ}_{0}_{,h}g)),
and thus, the identity

(4.17) G_{µ}_{1}_{,h}− G_{µ}_{0}_{,h}=−(µ_{1}−µ_{0})G_{µ}_{1}_{,h}G_{µ}_{0}_{,h}

holds. In particular, the commutativity of the G_{µ}_{j}_{,h} follows.

5. THE GENERALIZED GARNIR’S THEOREM

In the rest of this paper, assume that h = (−V, h_{1},· · ·, h2m) is given
(as in (3.1)) with h_{D} > 0. The purpose of this section is to consider, for a
given g∈L^{2}(D), the solvability of the operator equation (1.4) with v ranging

over R_{0,D}. If λ∈C\R−, a positive number µ_{0} is called an admissible value
for (V, D, λ), provided (µ_{0}, V) is admissible for D and λ−µ_{0} ∈ C\R−.
The method of Garnir [4] (compare [3], 5.13.8) can be extended to yield the
following:

Theorem 5.1 (Generalized Garnir’s Theorem). Let λ∈C\R− and µ0

be an admissible value for (V, D, λ). Then there exists a continuous endomor-
phism G_{λ,h} of L^{2}(D) (independent of the choice of µ0) taking values in N_{0,D}^{µ}^{0} ,
such that the element w:=G_{λ,h}g gives a weak solution, in the sense of (4.10),
of the Dirichlet-Neumann Problem (4.9) (relative to a partition (1.3)), for each
g∈L^{2}(D).

Proof. The existence of the endomorphism G_{λ,h}, for each λ∈C\R−, will
be shown in Lemma 6.1 (of the Appendix). Observe that for such λ one has

hP_{λ,h}G_{λ,h}g, vi_{D} =λ(G_{λ,h}g,v)¯ _{D} +hP_{0,h}G_{λ,h}g, vi_{D}

=λ(G_{λ,h}g,v)¯ _{D} + (s_{0,h}(G_{λ,h}g),¯v)_{D}.

for all g, v ∈L^{2}(D). As will be shown in Lemma 6.2, the terms on the right-
hand side of this equation are holomorphic in C\R− (as a function of λ). The
equation (1.4) itself is known to hold for all v ∈ R_{0,D}, in case λ = µ is a
real ”admissible” spectral parameter, namely, µ≥ ess sup_{D}V. Therefore, the
principle of analytic continuation ensures its validity for all λ∈C\R−.

6.APPENDIX: GREEN OPERATORS WITH COMPLEX SPECTRAL PARAMETERS

Lemma 6.1. Let λ∈C\R−. Then for each admissible µ0 for (V, D, λ), the mapping

(6.1) G_{λ,h} :=G_{µ}

0,h[I+ (λ−µ_{0})G_{µ}

0,h]^{−1}

is a continuous endomorphism of L^{2}(D) (independent of the choice of µ0),
taking values in N_{0,D}^{µ}^{0} .

Proof. Since G_{µ}_{0}_{,h} is a continuous, positive, self-adjoint endomorphism of
L^{2}(D), the inverse map [I+ (λ−µ0)G_{µ}_{0}_{,h}]^{−1} exists as a continuous endomor-
phism of L^{2}(D) (by [3], Lemma, p. 359). To show that the operator G_{λ,h} is
well-defined by (6.1), choose admissible µ_{j} >0, j = 0,1 for (V, D, λ). Then,
equivalent to the identity (4.17), one has

(6.2) [I+ (λ−µ_{0})G_{µ}_{0}_{,h}]G_{µ}_{1}_{,h} = [I+ (λ−µ_{1})G_{µ}_{1}_{,h}]G_{µ}_{0}_{,h},
One the other hand one has, clearly,

G_{µ}_{1}_{,h}[I+ (λ−µ0)G_{µ}_{0}_{,h}] = [I + (λ−µ0)G_{µ}_{0}_{,h}]G_{µ}_{1}_{,h}

= [I + (λ−µ1)G_{µ}_{1}_{,h}]G_{µ}_{0}_{,h}.
(6.3)

Multiplying (6.3) on the right by [I+ (λ−µ_{0})G_{µ}_{0}_{,h}]^{−1} and on the left by
[I+ (λ−µ_{1})G_{µ}_{1}_{,h}]^{−1} yields

[I+ (λ−µ1)G_{µ}_{1}_{,h}]^{−1}G_{µ}_{1}_{,h} = G_{µ}_{0}_{,h}[I+ (λ−µ0)G_{µ}_{0}_{,h}]^{−1}.
It will be shown below that the following identity holds:

(6.4) [I+ (λ−µ_{1})G_{µ}_{1},h]^{−1}G_{µ}_{1}_{,h} = [I+ (λ−µ_{0})G_{µ}_{0}_{,h}]^{−1}G_{µ}_{0}_{,h}.
It follows from this that

G_{µ}_{0}_{,h}[I+ (λ−µ0)G_{µ}_{0}_{,h}]^{−1} = [I+ (λ−µ0)G_{µ}_{0}_{,h}]^{−1}G_{µ}_{0}_{,h},

and the same identity holds with µ_{0} replaced by µ_{1}. Consequently from this
and (6.4) one concludes, as desired, that

G_{µ}_{0}_{,h}[I+ (λ−µ0)G_{µ}_{0}_{,h}]^{−1} = G_{µ}_{1}_{,h}[I+ (λ−µ1)G_{µ}_{1},h]^{−1}.

To prove the identity (6.4), observe that the operators I + (µ−µ_{1})G_{µ}_{1}
and I+ (µ−µ_{0})G_{µ}_{0} commute; hence, setting w:= [I+ (µ−µ_{1})G_{µ}_{1}]^{−1}v, one
has

[I+ (µ−µ1)Gµ1][I+ (µ−µ0)Gµ0]w= [I+ (µ−µ0)Gµ0]v.

Then,

[I+ (µ−µ0)Gµ0][I+ (µ−µ1)Gµ1]^{−1}v= [I+ (µ−µ1)Gµ1]^{−1}[I+ (µ−µ0)Gµ0]v.

Therefore,

[I+ (µ−µ_{0})G_{µ}_{0}][I+ (µ−µ_{1})G_{µ}_{1}]^{−1} = [I+ (µ−µ_{1})G_{µ}_{1}]^{−1}[I+ (µ−µ_{0})G_{µ}_{0}].

Consequently, the identity (6.4) will result from the relation
G_{µ}_{0}_{,h} = [I+ (λ−µ0)G_{µ}_{0}_{,h}][I+ (λ−µ1)G_{µ}_{1}_{,h}]^{−1}G_{µ}_{1}_{,h}

= [I+ (λ−µ_{1})G_{µ}_{1}_{,h}]^{−1}[I+ (λ−µ_{0})G_{µ}_{0}_{,h}]G_{µ}_{1}_{,h},
namely, the above relation (6.2).

Lemma 6.2. Given a pair (g, ρ)∈L^{2}(D)×L^{2}(D), the function g_{h}(λ) :=

(s_{0,h}(G_{λ,h}g),ρ)¯_{D} is holomorphic in C\R−. The same is true for the function
ˆ

g_{h}(λ) := (G_{λ,h}g,ρ)¯_{D}.

Proof. Let λ0 ∈ C\R− be given and choose µ0 for (V, D, λ0) as in
Lemma 6.1. Write λ^{0} =λ_{0}+ε. Taking ε with sufficiently small modulus, the
same µ_{0} can be used in defining G_{λ}^{0}_{,h}. With K:= [I+ (λ_{0}−µ_{0})G_{µ}_{0}_{,h}]^{−1}, the
definition (6.1) implies that

g_{h}(λ^{0})−g_{h}(λ0) = (s_{0,h}(G_{µ}_{0}_{,h}(K^{−1}+εG_{µ}_{0}_{,h})^{−1}g),ρ)¯ _{D}−(s_{0,h}(G_{µ}_{0}_{,h}Kg),ρ)¯ _{D}

= (s_{0,h}(G_{µ}_{0}_{,h}(I +εKG_{µ}_{0}_{,h})^{−1}Kg),ρ)¯_{D} −(s_{0,h}(G_{µ}_{0}_{,h}Kg),ρ)¯_{D}.

If |ε| is so small that

|ε| kG_{µ}_{0}_{,h}k

L2(D) ≤(kKk

L2(D))^{−1},
then the series P

n≥0(−1)^{n}ε^{n}(KG_{µ}_{0}_{,h})^{n}) converges (relative to the L^{2} operator
norm) and its sum is none other than (I+εKG_{µ}_{0}_{,h})^{−1}. Consequently, one has

s_{0,h}(g_{h}(λ^{0}))−s_{0,h}(g_{h}(λ0)) =X

n≥1

(−1)^{n}ε^{n}(s_{0,h}◦ G_{µ}_{0}_{,h}) (KG_{µ}_{0}_{,h})^{n}Kg),ρ)¯_{D},
the series on the right being necessarily convergent for ε with sufficiently
small modulus. The holomorphy of g_{h}(λ) in a neighborhood of λ0 is thereby
established.

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Received 1 August 2013 Minnesota State University, Department of Mathematics and Statistics,

Mankato, MN 56001 chia.tung@mnsu.edu