On an Inclusion of the Essential Spectrum of Laplacians under Non-Compact Change of Metric

It is shown the stability of the essential self-adjointness, and an inclusion of the essential spectra of Laplacians under the change of Riemannian metric on a subset K of M. The set K may have infinite volume measured with the new metric and its completion may contain a singular set such as fractal, to which the metric is not extendable.


Introduction
Let (M, g) be a connected smooth Riemannian manifold without boundary. The Laplacian ∆ of g is called essentially self-adjoint if it has the unique self-adjoint extension ∆. In [4], Furutani showed that if ∆ with the domain C ∞ 0 (M ) is essentially self-adjoint and, if g is changed on a compact set K ⊂ M to another smooth metric g ′ on M , then the Laplacian ∆ ′ of g ′ with the domain C ∞ 0 (M ) is essentially self-adjoint; and the essential spectrum are stable under this change. In particular, the second result forms a strong contrast to the behavior of the eigenvalues, since eigenvalues change continuously with the perturbation of the metric in a certain way (see for e.g [1]).
Needless to say, there are many important Riemannian manifolds with singularity, by which, we mean that g does not extend to the Cauchy boundary (the difference between the completion of M and M ); such as algebraic varieties, cone manifolds, edge manifolds, Riemannian orbifolds. In general, the analysis on such a singular space is complicated, and one of the methods to overcome the difficulties is to modify the singularity to a simpler one by the perturbation of the Riemannian metric. The crucial steps in this process is to study the stability of the essential self-adjointness of the Laplacian, and to understand the behavior of its spectral structure under the perturbation.
Motivated by these facts, we extend Furutani's theorem to more general K so that K is not compact and its completion K includes the singular set. In this setting, the natural domain D(∆) for the Laplacian is the following: (We suppress M and the Riemann measure dµ g for the sake of simplicity.) Indeed, if the Cauchy boundary ∂ C M is almost polar; namely, Cap(∂ C M ) = 0, Key words and phrases. essential self-adjointness, incomplete manifolds, essential spectrum, perturbations.
(see Section 2 for the definition. See also for e.g. [3]) then M has negligible boundary [9], and by the Gaffney theorem [6], ∆ is essentially self-adjoint. All through the article, we assume that the Laplacians have the domain defined in (1). The following is our main result: Theorem 1. Let g and g ′ be Riemannian metrics on M such that g = g ′ outside a subset K of M . If ∆ is essentially self-adjoint in L 2 and the Cauchy boundary of K with respect to g ′ is almost polar, then ∆ ′ is essentially self-adjoint in L 2 (M ; dµ g ′ ).
Additionally, if there is a function χ on M satisfying where ∇ is the gradient of g, and the inclusion A special case of Theorem 1 is Corollary 1 (Furutani's stability result [4]). Let g and g ′ be Riemannian metrics on M such that g = g ′ outside a compact subset K of M . If ∆ is essentially self-adjoint in L 2 then ∆ ′ is essentially self-adjoint in L 2 (M ; dµ g ′ ), and (4) σ ess (∆) = σ ess (∆ ′ ).
A typical example of manifolds which satisfies the condition of Theorem 1 is given as follows: Corollary 2 (See Section 3). Let M be a complete manifold and Σ ⊂ M , an almost polar compact subset. If Σ is almost polar with respect to a metric g ′ on M \ Σ and g = g ′ outside a compact set K ⊂ M , then the same conclusion in the theorem holds true.
We may apply Theorem 1 for singular manifolds: we change g to g ′ on a bounded set K ⊃ ∂ C M so that g ′ can be extended to the almost polar Cauchy boundary with respect to g, and conclude that ∆ is essentially self-adjoint in L 2 (M ; dµ g ) and σ ess (∆) ⊂ σ ess (∆ ′ ).
A sufficient condition for ∂ C M to be almost polar is that it has Minkowski codimension greater than 2 [7] (if the metric of g extends to ∂ C M and ∂ C M is a manifold, then it is almost polar if ∂ C M has co-dimension 2).
The idea to prove the inclusion of the essential spectrum in Theorem 1 is to apply Weyl's criteria: a number λ belongs to σ ess (∆) if and only if there is a sequence φ n of "limit-eigenfunctions" of ∆ corresponding to λ (see Proposition 2 for details). Indeed, we show that if χ satisfies (2) and (3), then there is a subsequence φ n(k) such that (1 − χ)φ n(k) is a limit-eigenfunction of ∆ ′ .
Our results differ from Furutani's original results in the following two points. In order to explain those differences, let us employ an example. Let M be where K = {(x, y, z) ∈ S 2 : z ≥ 0} \ (0, 0, 1) and B(r) = {(x, y, z) ∈ R 3 : x 2 + y 2 ≤ r 2 , z = 0}. Namely, M is the S 2 with flat bottom with deleted the north point. (To be more precise, we need to smooth the intersection of K and B(1) so that M is a smooth Riemannian manifold.) Since the Cauchy boundary of M is the north point and it has null capacity, the Laplacian ∆ is essentially self-adjoint. We modify M by the stereographic projection so that (M, g ′ ) is the 2-dimensional Euclidean space R 2 . Since (M, g ′ ) is complete, the Cauchy boundary is empty, and the Laplacian ∆ ′ is essentially self-adjoint. Next, we find the function χ which satisfies condition (2) as follows: χ ∈ C ∞ 0 (M \ B(1/3)) and χ = 1 on K. (3) is satisfied. Indeed, since the north point has null capacity, the spectrum of the Laplacian on M consists only of the eigenvalues with finite multiplicity, whereas ∆ ′ has only essential spectrum. This proves that the inclusion in Theorem 1 holds. This example also shows that the assumptions in Corollary 1 is sharp in the sense that we may not drop the assumption such that K is compact to obtain (4); that is, Furutani's stability result. Indeed, if we modify the metric of R 2 to obtain M , then there is no subset N of R 2 which satisfies condition (3).
The second difference is that the essential self-adjointness of ∆ does not need to imply that of ∆ restricted to C ∞ 0 (M ); for instance, ∆ on M is essentially selfadjoint, but ∆ restricted to C ∞ 0 (M ) has infinitely many self-adjoint extensions, in particular, it is not essentially self-adjoint (see for e.g. [2]).
We organize the article in the following manner: In Section 2, we prove Theorem 1, and in Section 3, we present the examples.

Proofs
In this section we recall some definitions and prove Theorem 1 and Corollary 1. For the sake of the simplicity, we often suppress the symbols M and dµ g .
We denote by (·, ·) and (·, ·) 1 the inner product in L 2 and the Sobolev space We will use Proposition 1 (Lemma 2.1.1 [3]). If L A = φ for A ∈ O, there exists a unique element e A ∈ L A called the equilibrium potential of A such that (i) e A We prove Theorem 1. We start from: Proof of the essential self-adjointness of ∆ ′ . For arbitrary u ∈ H 1 (M ; dµ g ′ ), we have to findû n ∈ H 1 (M ; dµ g ′ ) which converges to u in H 1 (M ; dµ g ′ ). Indeed, this implies that (M, g ′ ) has negligible boundary, and hence, ∆ ′ is essentially selfadjoint by Gaffney theorem [6]. Since we may assume that u ∈ L ∞ (M ) without the loss of generality. Let where r is the distance from K. The function ψ ∈ L ∞ (M ) enjoys the property: ψ| K = 1 and ∇ψ L ∞ ≤ 1. Since for almost every x ∈ M , it follows that (1 − ψ)u ∈ H 1 . Recalling that the essential self-adjointness of ∆ implies . Because the Cauchy boundary ∂ C M of M associated to g ′ is almost polar, there is a sequence of the equilibrium potentials e n of O n ⊃ ∂ C M such that ∩ n>1 O n = ∂ C M and e n H 1 (M;dµ g ′ ) → 0 as n → ∞.
Next, we prove the inclusion of the essential spectrum and complete the proof of Theorem 1. We use the following characterization: Proposition 2 (Weyl's criterion). A number λ belongs to the essential spectrum of ∆ if and only if there is a sequence of orthonormal vectors {φ n } of L 2 such that (∆ − λ)φ n L 2 → 0 as n → ∞.
Proof of the inclusion of the essential spectrum. We assume (2) and (3) to prove the inclusion of the essential spectrum. Hereafter, we denote ∆ = ∆ and ∆ ′ = ∆ ′ because of their essential self-adjointness. Let λ ∈ σ ess (∆) and φ n ∈ D(∆) such that where · = (·, ·). Let χ be the function satisfying (2) and φ be the function defined as wherer is the distance from the support of χ. Clearly, we have Moreover, taking into account that φ n ∈ D(∆) implies Hence, Now, specifying ǫ > 0 as in the statement, by (7) and the fact that the embedding However, if f ∈ L 2 , then f φ ∈ L 2 and (f, φφ n ) = (f φ, φ n ) → 0 as n → ∞; hence, φφ n(k) → 0 weakly in L 2 as k → ∞. Because of the uniqueness of the weak-limits, it follows that φ ′ = 0, and we may assume: without the loss of generality. Since φ = 1 on supp(χ), The first and third terms in the last line tend to 0 as n → ∞ because of (8) and (6). The second term can be estimated as ∇χ L ∞ ∇(φφ n ) ≤ ∇χ L ∞ ∆φ n φφ n → 0 as n → ∞.
Finally, we assume that K is compact to prove the essential self-adjointness of the Laplacian and the stability of the essential spectrum; namely, Corollary 1.
Proof of Corollary 1. We will show that the Laplacian ∆ ′ is essentially self-adjoint and that there exist the function χ and the subset N of M which satisfy conditions (2) and (3) for each metric g and g ′ . This will imply σ ess (∆) = σ ess (∆ ′ ) by Theorem 1.
Recall that if K is compact, then its Cauchy boundary is empty so that the Laplacian ∆ ′ is essentially self-adjoint.
Since K is compact and g is smooth, there exists ǫ 0 > 0 such that for any 0 < ǫ < ǫ 0 , the metric g and its higher order (up to 2nd) derivatives are bounded on N = N (K; 2ǫ) = {x ∈ M : d(x, K) < 2ǫ}. Let wherer is the distance from K. The functionχ is 1 on N (K; ǫ/3) and has support in N (K; 2ǫ/3), and it satisfies ∇χ L ∞ ≤ 3/ǫ. However, sinceχ does not need to be in the Sobolev space H 2 of order (2, 2), we apply the Friedrichs mollifier j with radius δ > 0 forχ to find the smooth function χ = j * χ. If δ < ǫ/3, then χ satisfies: namely, condition (2). On the other hand, since K is compact, N ǫ and N are relatively compact in M with sufficiently small ǫ > 0. Hence, N has finite volume and finite diameter, and the Poincaré inequality holds on N , it follows that the embedding H 1 0 (N ; dµ g ) ⊂ L 2 (N ; dµ g ) is compact; that is, condition (3). We obtained the inclusion: σ ess (∆) ⊂ σ ess (∆ ′ ).
This argumentation holds true if we replace g by g ′ and we arrive at the conclusion.

Examples
In this section, we present examples of manifolds for which Theorem 1 can be applied.
Example 1 (see [8]). Let (M, g) be an m-dimensional complete Riemannian manifold, Σ ⊂ M be an n-dimensional compact manifold with m ≥ n + 2. Assume that M has the product structure M m−n × M n near Σ and g can be diagonalized. Choose local coordinates in a neighborhood K of Σ so that g = g 1 ⊕ g 2 in K, where g 1 is a metric on M m−n and g 2 is a metric on M n . Let g ′ be another smooth metric on M \ Σ so that If m = 2, assume that f ∈ L 2+ǫ (K; dµ g ) for some ǫ ∈ (0, ∞). If m = 3, assume that inf(f ) > 0 and f ∈ L (m(m−2)/2)+ǫ (K; dµ g ) for some ǫ ∈ (0, ∞).
Then the manifold M \Σ with metrics g and g ′ satisfy the assumption of Theorem 1. In particular, if M is compact, ∆ ′ on (M \ Σ, g ′ ) has discrete spectrum which satisfies the Weyl asymptotic formula [8].
In the next example, manifold has fractal singularity.
The compact inclusion: can be seen as follows. By definition, H 1 0 (B \ Σ) ⊂ H 1 0 (B), and the inclusion: is dense, we may assume that u ∈ L ∞ without loss of generality. Let e n be the equilibrium potential as in the proof of Theorem 1. Then u n = u(1 − e n ) ∈ H 1 0 (B \ Σ) and u n → u in H 1 0 (B; dµ g ), and hence, u ∈ H 1 0 (B \ Σ). The function χ can be found as the relative equilibrium potential of B(1) and B(2) applied the Friedrichs mollifier. Therefore, M \ Σ together with g and g ′ satisfy the condition of Theorem 1. Let us point out that • (M \ Σ, g ′ ) is C 1,1 and is not smooth, but Theorem 1 can be applied to this setting. • We can show the compactly imbedding H 1 0 (B \ Σ; dµ g ′ ) ⊂ L 2 (B; \Σ; dµ g ′ ) only for ǫ ≥ 0.
In the next example, K has infinite volume with g ′ . More generally, if M is a complete manifold and Σ ⊂ M is a compact set, then there is a smooth Riemannian metric g ′ on M \ Σ and a compact set K ⊂ M such that g = g ′ on M \ K, (M \ Σ; g ′ ) is complete and there exists a function χ and a subset N of M \ Σ satisfying conditions (2) and (3), respectively.

acknowledgments
The author would like to thank Professor Hajime Urakawa for several advises which improved the original manuscript.