Eigenfunctions on an infinite Schrödinger network

Lemma 3.1

In a Schrödinger network { X , t , q } , assume that q ≥ 0 and q ≢ 0 . If there is a non-negative bounded function v such that v ≢ 0 and Δ v ( a ) ≥ q ( a ) v ( a ) , then there exists a bounded function h > 0 such that Δ h ( a ) = q ( a ) h ( a ) .

Proof

Assume that v is bounded by k . Since q ≥ 0 , the constant function 1 is a Δ q -superharmonic. Hence, v is Δ q -subharmonic function majorized by the Δ q -superharmonic function k . Therefore, there exists a Δ q -harmonic function h , 0 ≤ v ≤ h ≤ k [5, Theorem 4.1.1]. If h is zero at any vertex, it must be zero everywhere, which contradicts the assumption that v is not identically zero. Therefore, h is positive on the entire network.□

Definition 3.2

A λ -eigenfunction of the Laplacian operator Δ ( Δ q ) defined in the infinite network { X , t } ( { X , t , q } ) is a nonzero function f on the graph X such that Δ f = λ f ( Δ q f = λ f ) . The constant λ is called the eigenvalue of the eigenfunction f .

Proposition 3.3

If v is a non-negative bounded function such that v ≢ 0 and Δ v ≥ α v , for α > 0 , then any positive number β ≤ α is an eigenvalue of the Laplacian Δ with a corresponding positive, bounded eigenfunction v.

Proof

We can use Lemma 3.1, which guarantees the existence of a positive, bounded function h satisfying Δ h ( a ) = β h ( a ) on X .□

Theorem 3.4

A function σ is considered an eigenfunction associated with λ = Δ η ( a ) η ( a ) for some η > 0 if and only if σ can be expressed as the product of η and a harmonic function h on the network { X , t ∗ } , where t ∗ ( a , b ) = η ( b ) t ( a , b ) .

Proof

If Δ σ ( x ) = λ σ ( a ) , then Δ σ ( a ) = Δ η ( a ) η ( a ) σ ( a ) , which implies η ( a ) Δ σ ( a ) − σ ( a ) Δ η ( a ) = 0 . Hence,

Since η > 0 , Δ ∗ σ ( a ) η ( a ) = 0 . Thus, the function σ can be written as the product of two functions: h and η . The function h = σ ( a ) η ( a ) is a Δ ∗ - harmonic function. Therefore, the dimension of the eigenspace associated with a particular eigenvalue λ with respect to Δ is equal to the dimension of the space of harmonic functions on the network { X , t ∗ } .

Conversely, if σ is the product of h and η , where h is a Δ ∗ -harmonic function, then

Remark 1

With minor adjustments to the proof, we can show that the inequality Δ σ ( a ) ≤ λ σ ( a ) holds if and only if σ ( a ) can be expressed as the product of two functions: s and η , where Δ ∗ s ≤ 0 on { X , t ∗ } .

Proposition 4.1

If α is an eigenvalue with a corresponding non-negative eigenfunction φ that is non-identically zero, then φ > 0 and α > − [ t ( a ) + q ( a ) ] for all all points on the network { X , t , q } .

Proof

To prove this, we can use the fact that q ( a ) ≥ Δ ξ ( a ) ξ ( a ) which implies, [ t ( a ) + q ( a ) ] > 0 . Now, ∑ b t ( a , b ) φ ( b ) = α φ ( a ) + [ t ( a ) + q ( a ) ] φ ( a ) . Hence, if the eigenfunction is zero at any point c , then φ ( b ) = 0 for all b ∼ c by the connectedness of X , it must be zero everywhere, which contradicts the assumption φ ≢ 0 . Finally, since

for all a .□

Theorem 4.2

A function u is an α -eigenfunction of the Schrödinger operator Δ q if and only if α is an eigenvalue of the modified Laplacian Δ ∗ ∗ of the network { X , t ∗ ∗ } , where t ∗ ∗ ( a , b ) = η ( b ) η ( a ) t ( a , b ) . The relationship between the eigenfunctions of these two operators is given by the equation u ( a ) = v ( a ) η ( a ) , where v is an eigenfunction of Δ ∗ ∗ associated with α .

Proof

Let Δ q u ( a ) = α u ( a ) . Then,

If we divide both sides by [ η ( a ) ] 2 , we obtain

Hence, v ( a ) = u ( a ) η ( a ) is an eigenfunction of Δ ∗ ∗ associated with α .

On the other hand, if u ( a ) = v ( a ) η ( a ) , where Δ ∗ ∗ v = α v , we want to prove that Δ q u ( a ) = α u ( a ) :

Remark 2

1. By making minor adjustments to the proof, we can also show that the inequality Δ q u ( a ) ≤ α u ( a ) holds on { X , t , q } if and only if u = v η , where Δ ∗ ∗ v ≤ α v . If v satisfies the inequality Δ ∗ ∗ v ≤ α v , then the product of v and η is an α -superharmonic function on the original Schrödinger network. α -superharmonic functions are used to describe the lowest possible eigenvalue, as seen in the Agmon-Allegretto theorem. For more information, you can refer to a related article, by Lennx and Stollmann [10].

2. If α is a constant such that α ≥ Δ ∗ ∗ v v for a function v > 0 on { X , t ∗ ∗ } , then there exist α -superharmonic functions on the original Schrödinger network { X , t , q } . One example of such a function is u = v η , which is positive on { X , t , q } .

3. There exists a positive eigenfunction of the Schrödinger operator Δ q on { X , t , q } . As discussed earlier, there is a positive function h such that α = Δ ∗ ∗ v v on { X , t ∗ ∗ } . Using Theorem 4.2, we proved earlier, u ( x ) = h ( x ) η ( x ) is a positive eigenfunction of Δ q on { X , t , q } .