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

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 withL²-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 L²(Rⁿ), where µ is a real constant (usually called the spectral parameter) and V a real-valued L²-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∈C²(D), theSchr¨odinger operator S˜_p,λ,V[ψ] :=−4˜_p[ψ] + (λ−V) [ψ] associated with a constant λ∈C and V ∈ L¹_loc(D), regarded as acting on C_c²(D), is induced on D^∗ by the function

MATH. REPORTS15(65),4(2013), 497–510

(1.1) S_p,λ,V(ψ) :=− 4_pψ+ (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_pΨ + (λ−V) Ψ = g

in D^∗ with a real spectral parameter λ and density g ∈ L²(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,h¹ (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,hw, defined for complex parameters λand elements w of a linear subspace of L¹_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) L²-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,hw, 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 Rⁿ, 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^ckzk²)^m is positive. If p:Y →C^m is a holomorphic map, set a⁰ :=p(a), p^[a]:=p−a⁰ and ra:=kp^[a]k, for a∈Y. Clearly the form

υp :=dd^cr_a² = ( i

2π)∂∂ r¯ _a² 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 ∈L¹_loc(Y)) if so is every 2m-form ψχ with χ ∈ C_2m⁰ (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∈L¹_loc(D), set

(f, ψ)_D :=

Z

D

f ψd˜υ,

provided the integral exists. Each element f ∈L¹_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_c¹(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^cη¯∧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 L²(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 ∈ C¹(D), denote by ∂_νψ,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 ∈ C¹(D)|φ_xj˜ and φ_yj˜ ∈ C^λ(D),1 ≤ j ≤ m}. If ψ∈L¹_loc(D), denote by (the same) ∂νψ theν-th distributionalweak derivative of ψ on D, provided it exists as an element of L¹_loc(D). Let M_D be the subspace of L¹_loc(D) consisting of all elements w such that: (i) the first order weak derivatives ∂_νw =φ_ν, ν = 1,· · ·,2m, exist (as an element of L¹_loc(D)), namely,

h∂_˜t

ν[w], vi=−(w, ∂v

∂˜tν

)_D∗ = (φ_ν, v)_D∗, ∀v∈C_c¹(D);

(ii) each weak derivative ∂_νw ∈ 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 := −h₀. Then it is easy to show that if w, v ∈ M_D, then each of the forms h_j(∂_jw)(∂_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 C¹(D) gives rise to a member of M_D, the Schr¨odinger form S_λ,h : C¹(D)×C¹(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)∈C¹(D)×C_c²(D).

In the following, let µ denote a real constant. Then S_µ,h defines a Hermi- tian symmetric, continuous form on C¹(D) (with induced seminorm denoted accordingly by k k_µ,h). Let ∆ be the branch locus of p and set ∆⁰ =p(∆).

If a ∈ D₀ := D\p⁻¹(∆⁰), a connected neighborhood U b D can be chosen so that p|U is a covering projection onto a ball U⁰ 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 D₀, if, for any real-valued

φ∈C¹(D), setting φ^α_ν :=ρ^α_νφ, 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_Dhj|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_DV >− 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,hkφk_µ,h, ∀φ∈C¹(D).

Proof. It suffices to consider a real-valued φ ∈ C¹(D). In terms of a distinguished covering for D₀, by the Poincar´e’s inequality for the Euclidean unit ball, one has

h_D (PCm[1])²

Z

D

kφk²d˜υ ≤ h_D (PCm[1])²

X

α,ν

Z

(U^α)⁰

k(φ^α_ν)⁰k²dυ

≤h_DX

α,ν

Z

(U^α)⁰

k∇(φ^α_ν)⁰k²dυ≤ Z

D0

X

α,ν 2m

X

j=1

hjk∂(φ^α_ν)⁰

∂t_j k²d˜υ

≤ Z

D0

2m

X

j=1

h_j(X

α, β

∂(φ^α_µ)⁰

∂tj

∂φ(φ^βν)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

kφk²

µ,h ≥ µ−ess sup_DV + h_D (PCm[1])²

kφk²

L2(D) =:κ_D,µ,hkφk²

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 C¹(D). With respect to the induced norm k k_µ,h the completion of C¹(D) (respectively, C_c^∞(D)) gives rise to theSobolev space H^µ:=H_µ,h¹ (D) (respectively, H_µ,h,c¹ (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⊆L²(D). Taking (µ, V) = (1,0), the Sobolev spaces H¹(D) and H_c¹(D) are defined. Moreover, the Schr¨odinger form (3.2) extends to a scalar product [ , ]_1,0,D =S_{1,(0,1,···,1)} on H¹(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_c¹(D) and j, k = 1,· · · ,2m ([8], §3, Remark 1). Therefore, each such element w gives rise to aSchr¨odinger functional P_λ,hw:C_c¹(D)→C defined by

(3.6) hP_λ,hw, vi_D := S_λ,h(w, v), ∀v∈C_c¹(D).

Note also, that the right-hand side of (3.6) serves to extend (i) the action of the functional P_µ,hw to H_µ,h¹ (D) and (ii) its definition to w∈H_µ,h¹ (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∂_jw) ¯φd˜υ

is well-defined for all elements φ∈C_c¹( ˜D). Consequently, the functional ”φ7→

hS_µ,h(w), φi_D∗” extends naturally to an operator (denoted by the same) on H_µ,h¹ (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∂_jw) 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 ∈ H¹

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,c¹ (D) is approximable in the L²-norm on L²(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¯_nn_j(hj∂_jw) dσ− Z

D

φ¯_n∂_j(hj∂_jw) d˜υ

=hS_µ,h(w), φ_ni

D∗+

2m

X

j=1

Z

dD

φ¯_nn_j(h_j∂_jw) dσ

→ hS_µ,h(w), φi_D∗ +

2m

X

j=1

Z

dD

φ¯n_j(hj∂_jw)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_c¹(D)). To each such pair (µ,h) the associated ”S_µ,h” is positive definite, hence defines a scalar product on H_µ,h¹ (D).

It is shown in ([8], §3) that the k-th partial derivatives ∂_x˜kT and ∂_y˜kT of a regular current T = [ψ] (where ψ∈L¹_loc(D)) can be defined as (2m−1)- dimensional quasicurrents on D (which are currents if D is nonsingular). As a consequence, given g= (g₀, . . . , g_2m) with g_j ∈L²(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 := ˜g₀+

m

X

j=1

∂_˜xk[˜g_j] +∂_˜yk[˜g_j]

on ˜D, and an induced functional (denoted by the same) acting on C_c^∞( ˜D):

(4.3) hbgc, φi_D := (g₀,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_jk_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},c¹ (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,h¹ (D), by the Hahn-Banach’s theorem. The operator bgc is said to be representableby an element η∈L²(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

∂_νgj belonging to L²(D) for ν= 1,· · · ,2m, then bgc is representable by an element of L²(D). The subspace of ”Schr¨odinger functions” of type (µ, V), namely,

S_µ,V(D) := {w∈H¹

µ,V(D)|P

µ,h{1}w= 0},

is given precisely by the orthogonal complement of H_µ,V,c¹ (D) (with respect to S_µ,V):

(4.5) H¹

µ,V(D) =H¹

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,V¹ (D) is called aweak solution of the Dirichlet problem

(4.6) S_p,µ,V(Ψ) =− 4_pΨ + (µ−V) Ψ = bgc, Ψ = f on dD, provided that w≡f mod (H_µ,V,c¹ (D)) and

S_µ,V(w, ρ) = ((µ−V)w, ρ)¯_D +S_0,0(w, ρ) = hbgc, ρi_D, ∀ρ∈H_µ,V,c¹ (D).

It is a consequence of Green’s first identity that, if g ∈ L²(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 ∈ L²(D) and (µ, V) is ad- missible for D or (ii) g ∈ H⁻¹(D) such that bgc is representable by an element η ∈ L²(D) and (µ, V) is strictly admissible for D. Then for each f ∈C^1,1(D)∩C²(D^∗) with 4_pf ∈L²(D), the Dirichlet problem (4.6)admits a unique weak solution in H_µ,V¹ (D).

Proof. Let F := 4_pf. The homogeneous Dirichlet problem (4.7) with

“g” replaced by “ ˆg:=η+F−(µ−V)f”, where η=g whenever g∈L²(D),

admits a weak solution ψ=w₀ ∈H_µ,V,c¹ (D), by Riesz’s representation theorem (applied to the functional bˆgc). Then the function Ψ :=w₀+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_pf, φ)¯ _D, ∀φ∈C_c^∞(D), hence, by virtue of definition (3.2), one has

S_µ,V(Ψ, φ) =S_µ,V(w₀, 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−Ψ₂ of two solutions of (4.6) belongs to H_µ,V,c¹ (D) and the functional ”S_µ,V(ψ, )” annihilates H_µ,V,c¹ (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,h¹ (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,h¹ (D) admitting a (Schr¨odinger) continuous linear mapping s_µ,h :A_µ,h →L²(D) satisfying the equation

(4.8) hP_µ,hw, 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,hw, φi_D = hbgc, φi_D, ∀φ∈R_0,D.

Theorem4.2. Assume that either(i) g∈L²(D) and (µ,h) is admissible for D or (ii) g ∈ H⁻¹(D) such that bgc is representable by an element of L²(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,h¹ (D) such that the anti- linear form v 7→ hP_µ,hw, vi_D is continuous on H_µ,h,c¹ (D) with respect to the L²-norm on L²(D). B_µ,h is clearly a linear subspace of H_µ,h¹ (D). According to the Hahn-Banach’s theorem, the said anti-linear form is extendable to a unique continuous anti-linear form on L²(D), and this extension is given by taking the scalar product of a unique element of L²(D), to be denoted by s_µ,h(w), with v. Thus the equation (4.8) holds for all v∈H_µ,h¹ (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 w₀ ∈R_0,D satisfying the equation (4.11) hbgc, φi_D =S_µ,h(w₀, φ), ∀φ∈R_0,D

with

(4.12) kw₀k_µ,h ≤ Const. max

0≤j≤2mkg_jk_L2(D).

If bgc is representable (on C_c^∞(D)) by an element η ∈ L²(D) (or, if g ∈ L²(D)), then by the formula (3.6), the equation (4.8) holds with s_µ,h(w₀) =η(or g), for all v∈C_c^∞(D). One has, for such v, |hP_µ,hw₀, vi_D| ≤ kηk_L2(D)kvk_L2(D), thereby showing that w₀ ∈ B_µ,h. Moreover, by the equation (4.11), this element w₀ belongs to N_0,D^µ , thus giving a weak solution to the problem (4.9).

The difference ψ = w₁−w₂ of two weak solutions of (4.9) satisfies the conditions ψ∈ N_0,D^µ ⊂R_0,D and hP_µ,hψ, 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 → L²(D) is invertible with inverse given by a continuous, self- adjoint and strictly positive isomorphism G_µ,h : L²(D) → N_0,D^µ (called the weak Green’s operator for (4.9)); in particular, for g ∈ L²(D), the element w = G_µ,hg gives the unique weak solution to the Dirichlet-Neumann problem (4.9). Moreover, for admissible (µ_j,h), j= 1,2, the operators G_µj,h are com- mutative.

Proof. Given g ∈ L²(D), the association g 7→ G_µ,hg := 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,hg, φ) = (g,φ)¯ _D, ∀φ∈R_0,D, with

(4.14) kG_µ,hgk_µ,h ≤Const.kgk_L2(D).

Furthermore, the induced mapping G_µ,h : L²(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,hf with f ∈L²(D) in (4.13) yields

(4.15) S_µ,h(G_µ,hg,G_µ,hf) = (g,G_µ,hf)_D. Also, by (4.13) one has

S_µ,h(G_µ,hf,G_µ,hg) = (f,G_µ,hg)_D. Consequently,

(4.16) (g,G_µ,hf)_D = (f,G_µ,hg)_D = (G_µ,hg,f)¯_D,

thus proving that G_µ,h is a self-adjoint endomorphism of L²(D). Formula (4.15) shows that the integral (G_µ,hg,g)¯ _D is real-valued, and thus, by virtue of (3.4), one has, by relations (4.15) and (4.16),

(G_µ,hg,g)¯ _D ≥(c_D,µ,h)⁻²kG_µ,hgk²

L2(D).

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

(G_µ,hg,g)¯ _D ≤Const.kG_µ,hgk_µ,hkgk_L2(D). Therefore, by the inequality (4.14),

kG_µ,hgk²

L2(D) ≤Const.kgk²

L2(D),

so that G_µ,h, is a continuous endomorphism of L²(D). Furthermore, G_µ,h is strictly positive, since in fact, the condition G_µ,hg= 0 gives g=s_{µ,h}(G_µ,hg) = 0, so that, if g ∈ L²(D)\{0}, then (G_µ,hg, ¯g)_D > 0. Finally, for admissible (µ_j,h), j= 1,2, one has

G_µ1,hg=G_µ1,h(s_µ

0,h(G_µ0,hg)) =G_µ1,h([s_µ

1,h−(µ1−µ0)]G_µ0,hg)

=G_µ1,h(s_µ

1,h(G_µ0,hg))−(µ1−µ0)G_µ1,h(G_µ0,hg)), and thus, the identity

(4.17) G_µ1,h− G_µ0,h=−(µ₁−µ₀)G_µ1,hG_µ0,h

holds. In particular, the commutativity of the G_µj,h follows.

5. THE GENERALIZED GARNIR’S THEOREM

In the rest of this paper, assume that h = (−V, h₁,· · ·, h2m) is given (as in (3.1)) with h_D > 0. The purpose of this section is to consider, for a given g∈L²(D), the solvability of the operator equation (1.4) with v ranging

over R_0,D. If λ∈C\R−, a positive number µ₀ is called an admissible value for (V, D, λ), provided (µ₀, V) is admissible for D and λ−µ₀ ∈ 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 L²(D) (independent of the choice of µ0) taking values in N_0,D^µ0 , such that the element w:=G_λ,hg 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∈L²(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,hG_λ,hg, vi_D =λ(G_λ,hg,v)¯ _D +hP_0,hG_λ,hg, vi_D

=λ(G_λ,hg,v)¯ _D + (s_0,h(G_λ,hg),¯v)_D.

for all g, v ∈L²(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_DV. 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−µ₀)G_µ

0,h]⁻¹

is a continuous endomorphism of L²(D) (independent of the choice of µ0), taking values in N_0,D^µ0 .

Proof. Since G_µ0,h is a continuous, positive, self-adjoint endomorphism of L²(D), the inverse map [I+ (λ−µ0)G_µ0,h]⁻¹ exists as a continuous endomor- phism of L²(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−µ₀)G_µ0,h]G_µ1,h = [I+ (λ−µ₁)G_µ1,h]G_µ0,h, One the other hand one has, clearly,

G_µ1,h[I+ (λ−µ0)G_µ0,h] = [I + (λ−µ0)G_µ0,h]G_µ1,h

= [I + (λ−µ1)G_µ1,h]G_µ0,h. (6.3)

Multiplying (6.3) on the right by [I+ (λ−µ₀)G_µ0,h]⁻¹ and on the left by [I+ (λ−µ₁)G_µ1,h]⁻¹ yields

[I+ (λ−µ1)G_µ1,h]⁻¹G_µ1,h = G_µ0,h[I+ (λ−µ0)G_µ0,h]⁻¹. It will be shown below that the following identity holds:

(6.4) [I+ (λ−µ₁)G_µ1,h]⁻¹G_µ1,h = [I+ (λ−µ₀)G_µ0,h]⁻¹G_µ0,h. It follows from this that

G_µ0,h[I+ (λ−µ0)G_µ0,h]⁻¹ = [I+ (λ−µ0)G_µ0,h]⁻¹G_µ0,h,

and the same identity holds with µ₀ replaced by µ₁. Consequently from this and (6.4) one concludes, as desired, that

G_µ0,h[I+ (λ−µ0)G_µ0,h]⁻¹ = G_µ1,h[I+ (λ−µ1)G_µ1,h]⁻¹.

To prove the identity (6.4), observe that the operators I + (µ−µ₁)G_µ1 and I+ (µ−µ₀)G_µ0 commute; hence, setting w:= [I+ (µ−µ₁)G_µ1]⁻¹v, 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]⁻¹v= [I+ (µ−µ1)Gµ1]⁻¹[I+ (µ−µ0)Gµ0]v.

Therefore,

[I+ (µ−µ₀)G_µ0][I+ (µ−µ₁)G_µ1]⁻¹ = [I+ (µ−µ₁)G_µ1]⁻¹[I+ (µ−µ₀)G_µ0].

Consequently, the identity (6.4) will result from the relation G_µ0,h = [I+ (λ−µ0)G_µ0,h][I+ (λ−µ1)G_µ1,h]⁻¹G_µ1,h

= [I+ (λ−µ₁)G_µ1,h]⁻¹[I+ (λ−µ₀)G_µ0,h]G_µ1,h, namely, the above relation (6.2).

Lemma 6.2. Given a pair (g, ρ)∈L²(D)×L²(D), the function g_h(λ) :=

(s_0,h(G_λ,hg),ρ)¯_D is holomorphic in C\R−. The same is true for the function ˆ

g_h(λ) := (G_λ,hg,ρ)¯_D.

Proof. Let λ0 ∈ C\R− be given and choose µ0 for (V, D, λ0) as in Lemma 6.1. Write λ⁰ =λ₀+ε. Taking ε with sufficiently small modulus, the same µ₀ can be used in defining G_λ⁰_,h. With K:= [I+ (λ₀−µ₀)G_µ0,h]⁻¹, the definition (6.1) implies that

g_h(λ⁰)−g_h(λ0) = (s_0,h(G_µ0,h(K⁻¹+εG_µ0,h)⁻¹g),ρ)¯ _D−(s_0,h(G_µ0,hKg),ρ)¯ _D

= (s_0,h(G_µ0,h(I +εKG_µ0,h)⁻¹Kg),ρ)¯_D −(s_0,h(G_µ0,hKg),ρ)¯_D.

If |ε| is so small that

|ε| kG_µ0,hk

L2(D) ≤(kKk

L2(D))⁻¹, then the series P

n≥0(−1)ⁿεⁿ(KG_µ0,h)ⁿ) converges (relative to the L² operator norm) and its sum is none other than (I+εKG_µ0,h)⁻¹. Consequently, one has

s_0,h(g_h(λ⁰))−s_0,h(g_h(λ0)) =X

n≥1

(−1)ⁿεⁿ(s_0,h◦ G_µ0,h) (KG_µ0,h)ⁿKg),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