The Eigenvalue Counting Function for Krein-von Neumann ... · Outline 1 Topics Discussed 2 On Weyl Asymptotics 3 A Bit of Operator Theory 4 Mark Krein’s 1947 Result 5 Perturbed

The Eigenvalue Counting Function for Krein-vonNeumann Extensions of Elliptic Operators

Fritz Gesztesy (Baylor University, Waco, TX)

Based on joint work with

Mark Ashbaugh (University of Missouri, Columbia, MO)

Ari Laptev (Mittag-Le✏er Instritute, Sweden)

Marius Mitrea (University of Missouri, Columbia, MO)

Selim Sukhtaiev (Rice University, Houston, TX)

Seminar

Department of Mathematics, UCCS

September 29, 2017

Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 1 / 33

Outline

1 Topics Discussed

2 On Weyl Asymptotics

3 A Bit of Operator Theory

4 Mark Krein’s 1947 Result

5 Perturbed Krein Laplacians HK ,⌦ = ��K ,⌦ + V on Bounded Domains

6 Spectral Properties of HK ,⌦

7 The Perturbed Krein Laplacian Connected to a Buckling Problem

8 Weyl Asymptotics for HK ,⌦

9 Bounds on Eigenvalue Counting Functions


Topics Discussed

Topics involved:

Weyl asymptotics: a bit of history (used as a warm-up).

Self-adjoint extensions of semibounded operators in a Hilbert space,particularly, Friedrichs and Krein–von Neumann extensions.

Examples.

Spectral connections between the Perturbed Krein Laplacian in boundedLipschitz domains in Rn, n 2 N, and the buckling of a clamped plate(! elasticity applications).

Weyl asymptotics of perturbed Krein Laplacians in bounded Lipschitzdomains in Rn.

Eigenvalue Counting Function Bounds for perturbed Krein Laplaciansin arbitrary finite volume domains in Rn.


Topics Discussed

Based on various collaborations since 2010:

[AGMST10] M. S. Ashbaugh, F.G., M. Mitrea, R. Shterenberg, and G.Teschl, The Krein—von Neumann extension and its connection to an abstractbuckling problem, Math. Nachr. 283, 165–179 (2010).

[AGMT10] M. S. Ashbaugh, F.G., M. Mitrea, and G. Teschl, Spectraltheory for perturbed Krein Laplacians in nonsmooth domains, Adv. Math. 223,1372–1467 (2010).

[GLMS15] F.G., A. Laptev, M. Mitrea, and S. Sukhtaiev, A bound for theeigenvalue counting function for higher-order Krein Laplacians on open sets, inMathematical Results in Quantum Mechanics, Proceedings of the QMath12Conference, P. Exner, W. Konig, and H. Neidhardt (eds.), World Scientific,Singapore, 2015, pp. 3–29.

[BGMM16] J. Behrndt, F.G., T. Micheler, and M. Mitrea, The Krein-vonNeumann realization of perturbed Laplacians on bounded Lipschitz domains,IWOTA 2014 proceedings, Birkhauser, OTAA 255, 49–66 (2016).

[AGLMS16] M. Ashbaugh, F.G., A. Laptev, M. Mitrea, and S. Sukhtaiev,A bound for the eigenvalue counting function for Krein–von Neumann andFriedrichs extensions, Adv. Math. 304, 1108–1155 (2017).


On Weyl Asymptotics

Weyl asymptotics for the Laplacian ��:

Let ⌦ ✓ Rn, n 2 N, be a bounded, smooth domain, and ��BC ,⌦ be theLaplacian on ⌦ in the Hilbert space L2(⌦) with appropriate classical boundaryconditions (BC), e.g., Dirichlet, Neumann, Robin-type, on the boundary @⌦.

Let �BC ,⌦,j , j 2 N, be the eigenvalues of ��BC ,⌦ enumerated according to theirmultiplicity.

Introduce the eigenvalue counting function for ��BC ,⌦ by

N(�;��BC ,⌦) = #{j 2 N | 0 <�BC ,⌦,j �}, � 2 R.

In 1911, Weyl proved his celebrated leading order asymptotics

N(�;��BC ,⌦) = (2⇡)�nvn|⌦|�n/2 + o��n/2

�as �!1,

where, vn = ⇡n/2/�((n/2) + 1) is the volume of the unit ball in Rn.

Note. The leading order is shape and BC-independent for a large class of BCsand only depends on the volume |⌦| of ⌦.Weyl also initiated a study of the remainder term in his asymptotics.


On Weyl Asymptotics

Weyl asymptotics for the Laplacian �� (contd.):

A bit of history: it all started in Theoretical Physics

The problem to determine the eigenvalue asymptotics was apparentlyindependently posed by

Arnold Sommerfeld (September 1910)

and

Hendrik Antoon Lorentz (October 1910), Nobel Prize in Physics in 1902.

Hermann Weyl (1885–1955) solved it in less than 4 months by February 1911(in the Dirichlet case for n = 2) and wrote 6 papers on this subject from 1911–15extending his result to other boundary conditions and to n = 3.

Rumor has it that David Hilbert (1862–1943) did not think he would live to seethe problem solved ......


On Weyl Asymptotics

Weyl asymptotics for the Laplacian �� (contd.):

The problem originated in Physics in connection with (forced) vibrations andblack body radiation (standing electromagnetic waves in a cavity with reflectingsurface walls):

Gustav Kirchho↵ (1859), Josef Stefan (1879), Ludwig Boltzmann (1884),Wilhelm Wien (1893, 1896), Lord Rayleigh (1900, 1905), Sir James Jeans(1905), Albert Einstein (1905), especially,

Max Planck (October 1900), who was led to the Discovery of QuantumTheory (E = h⌫, h - Planck’s constant, energy quantization, one of the fathersof quantum physics, Nobel Prize in Physics in 1918).

This reads like a list of Who’s Who in Theoretical Physics just before QuantumMechanics entered the scene.

The subject is still so popular as there are numerous applications/ramificationsin areas such as: Acoustics (Music: Lord Raleigh, high overtones for violins),Radiation, (Inverse) Spectral Geometry (Mark Kac’s “Can one hear the shape ofa drum?” in 1966, perhaps, one of the most cited spectral theory papers ?),Analytic Number Theory (automorphic forms), etc.


A Bit of Operator Theory

Notation for the abstract part:

Let H be a separable complex Hilbert space with scalar product (·, ·)H andidentity operator IH in H.

+ denotes the direct sum (not necessarily orthogonal) in H.

S denotes a symmetric, closed, densely defined operator in H bounded frombelow (typically, S � 0 or S � "IH for some " > 0).

eS a self-adjoint extension of S .

SF denotes the Friedrichs extension of S .

SK denotes the Krein-von Neumann extension of S .



Basic notions: symmetric and nonnegative operators

A linear operator S : dom(S) ✓ H! H is called non-negative, S � 0, if

(u, Su)H � 0, u 2 dom(S).

S is called strictly positive, if for some " > 0, (u, Su)H � "kuk2H, u 2 dom(S).One then writes S � "IH. Similarly, one defines S � T (but there are technicaldetails concerning (quadratic form) domains........)

Notation: A✓B denotes dom(A)✓ dom(B) and Af=Bf for all f 2 dom(A).I.e., B is an extension of A (equivalently, A is a restriction of B).

S symmetric in H: S ✓ S⇤.

T self-adjoint in H: T = T ⇤.



Basic principle of self-adjoint extensions:

Let Sj , j = 1, 2, be symmetric, S1 ✓ S2 =) S⇤2 ✓ S⇤

1

Thus,S1 ✓ S2 ✓ S⇤

2 ✓ S⇤1

Existence of self-adjoint extensions was completely settled in

J. von Neumann, Allgemeine Eigenwerttheorie HermitescherFunktionaloperatoren, Math. Ann. 102, 49–131 (1929–30)

in terms of equal deficiency indices.

This marked the birth of Mathematical Quantum Theory.

In particular, he introduced the notion of closed operators and hence enabled thenotion of spectral theory for unbounded linear operators in Hilbert spaces.



John von Neumann:

John von Neumann (December 28, 1903 – February 8, 1957):

Regarded as the foremost math-ematician of his time and prob-ably “the last representativeof the great mathematicians”like Euler, Gauss, Poincare, orHilbert, he was a pioneer of theapplication of operator theoryto quantum mechanics, in thedevelopment of functional anal-ysis, a key figure in the devel-opment of game theory and theconcepts of cellular automata,and the digital computer.


Mark Krein’s 1947 Result

Mark Krein’s 1947 result:

Theorem (M. Krein, Mat. Sb. 20 (1947)).

S � 0 densely defined, closed in H. Then, among all non-negative self-adjointextensions of S , there exist two distinguished (extremal ) ones, SK and SF ,which are the smallest and largest (in the sense of order between the quadraticforms associated with self-adjoint operators) such extensions. Any non-negativeself-adjoint extension eS � 0 of S necessarily satisfies

0 SK eS SF , i.e., 8a > 0, (SF + aIH)�1 (eS + aIH)�1 (SK + aIH)�1.

In particular, this determines SK and SF uniquely.In addition, if S � "IH for some " > 0, one has

dom(SF ) = dom(S)u (SF )�1 ker(S⇤),

dom(SK ) = dom(S)u ker(S⇤),

dom(S⇤) = dom(S)u (SF )�1 ker(S⇤)u ker(S⇤),

ker(SK ) = ker�(SK )

1/2�= ker(S⇤) = ran(S)? = def(S),

this null space might well be infinite-dimensional (it is for PDEs ) !!!Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 12 / 33


Mark Krein:

Mark Grigorievich Krein (April 3, 1907 – October 17, 1989):

Mathematician Extraordinaire:One of the giants of 20th cen-tury mathematics, Wolf Prize inMathematics in 1982; developedhis own approach to the theoryof self-adjoint extensions around1947.



1d Example: ⌦ = (a, b), �1 < a < b <1, V = 0 :

��min,(a,b) = � d2

dx2

��C10 ((a,b))

, dom(��min,(a,b)) = W 2,20 ((a, b)),

��F ,(a,b)u = ��D,(a,b)u = �u00,u 2 dom(��F ,(a,b)) =

�v 2 L2((a, b))

�� v , v 0 2 AC ([a, b]);

Dirichlet b.c.s v(a) = v(b) = 0; v 00 2 L2((a, b)) ,

��N,(a,b)u = �u00,u 2 dom(��N,(a,b)) =

�v 2 L2((a, b))

�� v , v 0 2 AC ([a, b]);

Neumann b.c.s v 0(a) = v 0(b) = 0; v 00 2 L2((a, b)) ,

��K ,(a,b)u = �u00,u 2 dom(��K ,(a,b)) =

�v 2 L2((a, b))

�� v , v 0 2 AC ([a, b]);

Krein b.c.s v 0(a) = v 0(b) = [v(b) � v(a)]/(b � a); v 00 2 L2((a, b)) .

This settles the one-dimensional case n = 1. (Hence, n � 2 from now on.)Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 14 / 33

Perturbed Krein Laplacians HK,⌦ = ��K,⌦ + V on Bounded Domains

Perturbed Krein Laplacians on bounded Lipschitz(resp., minimally smooth) domains ⌦ ⇢ Rn, n � 2:

We assume for the remainder of the Weyl asymptotics part:

Hypothesis.

Let n 2 N, n � 2.

(i) ⌦ ⇢ Rn is a nonempty, open, bounded, Lipschitz domain.

(ii) V 2 L1(⌦) is nonnegative, V � 0.

The case n = 1, ⌦ = (a, b), �1 < a < b <1, was settled in the previousexample.

This permits one to make the step from the Laplacian, ��, to a

perturbed Laplacian, ��+ V , i.e., a Schrodinger-type operator.



Bounded Lipschitz domains:

Definition.

Let ; 6= ⌦ ⇢ Rn open and R > 0 fixed. ⌦ is called a bounded Lipschitz domain,if there exists r 2 (0,R) such that for every x0 2 @⌦ one can find a rigidtransformation T : Rn ! Rn and a Lipschitz function ' : Rn�1 ! R with

T�⌦ \ B(x0, r)

�= T

�B(x0, r)

� \ �(x 0, xn) 2 Rn�1 ⇥ R�� xn > '(x 0)

.

Examples of bounded Lipschitz domains include:

(i) All bounded (geometrically) convex domains.

(ii) All open sets which are the image of a domain as in (i) above under aC 1,1-di↵eomorphism.

(iii) All bounded domains of class C 1,r for some r > 1/2.

Considering only smooth ⌦ could not even handle a rectangle in R2 !!!

So studying Lipschitz domains is not just a technicality.Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 16 / 33


The domain of the perturbed Krein Laplacian HK ,⌦:

The minimal and maximal perturbed Laplacian in L2(⌦) are defined by

Hmin,⌦u := (��+ V )u, u 2 dom(Hmin,⌦) := W 2,20 (⌦),

Hmax,⌦u := (��+ V )u,

u 2 dom(Hmax,⌦) =�v 2 L2(⌦)

��v 2 L2(⌦) .

Then

Hmin,⌦ = (��+ V )|C10 (⌦) = Hmax,⌦

⇤ and Hmax,⌦ = Hmin,⌦⇤.

The Krein–von Neumann extension HK ,⌦ of Hmin,⌦, i.e., the perturbed KreinLaplacian in L2(⌦) is then given by

HK ,⌦u := (��+ V )u,

u 2 dom(HK ,⌦) = dom(Hmin,⌦) + ker(Hmax,⌦)

= W 2,20 (⌦) +

�v 2 L2(⌦)

�� (��+ V )v = 0 in ⌦ .

(One recalls, + denotes the direct sum (not necessarily orthogonal) in L2(⌦).)



The domain of HK ,⌦ (contd.): nonlocal b.c.’s

NONLOCAL and V -DEPENDENT boundary condition for HK,⌦:

dom(HK ,⌦) :=�v 2 dom(Hmax,⌦)

�� Nv +MWTD,N,⌦,V (0)(�Dv) = 0

,

i.e., these b.c.’s are not of local Robin-type.

• �D and �N are “appropriate” Dirichlet and Neumann trace operators in L2(@⌦),i.e., suitable generalizations of

�Du = u|@⌦ and �Nu = ⌫ ·ru|@⌦,with ⌫ the outward pointing unit vector on @⌦ (which exists a.e. on @⌦).

• MWTD,N,⌦,V (z) is an “appropriate” z-dependent Dirichlet-to-Neumann operator

(i.e., an operator-valued Weyl–Titchmarsh operator) in L2(@⌦),

MWTD,N,⌦,V (z) “ = ” �N

⇥�N(HD,⌦ � zIL2(⌦))

�1⇤⇤, z 2 ⇢(HD,⌦).

Formally, if ⌦ is smooth, and in the special case V = 0, this nonlocal b.c. is ofthe form

@v

@⌫(x)� @(Hv)

@⌫(x) = 0, x 2 @⌦,

where (Hv) denotes the harmonic extension of v |@⌦ into ⌦.Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 18 / 33

Spectral Properties of HK,⌦

General spectral properties of HK ,⌦:

Theorem ([AGMT10], [BGMM16]).

The perturbed Krein Laplacian HK ,⌦ is a self-adjoint operator on L2(⌦) whichsatisfies

HK ,⌦ � 0 and Hmin,⌦ ✓ HK ,⌦ ✓ Hmax,⌦,

HK ,⌦ has a purely discrete spectrum in (0,1),

�ess(HK ,⌦) = {0} if n � 2.

For any non-negative self-adjoint extension eS of Hmin,⌦ = (��+ V )|C10 (⌦) on

dom(Hmin,⌦) = W 2,20 (⌦) one has

HK ,⌦ eS HD,⌦.


The Perturbed Krein Laplacian Connected to a Buckling Problem

Connection: Krein Laplacian ! Buckling Problem:

Given � 2 C, consider the eigenvalue problem for the generalized buckling of aclamped plate in ⌦ ⇢ Rn,

(��+ V )2u = � (��+ V )u in ⌦, u 2 dom(��max,⌦), �Du = 0, �Nu = 0,

where (��+ V )2u := (��+ V )(��u + Vu) in the sense of distributions in ⌦.

Equivalently,

(��+ V )2u = � (��+ V )u in ⌦, u 2W 2,20 (⌦). (⇤)

This is a generalized eigenvalue problem of the type Au = �Bu in the sense thatB 6= IL2(⌦) in general. Equivalently, it’s a linear operator pencil problem .......

One can show that necessarily, � > 0.


� > 0 is an eigenvalue (necessarily discrete) of HK ,⌦ if and only if � > 0 is aneigenvalue of the (generalized) buckling problem (⇤). Moreover, themultiplicities coincide.


The Perturbed Krein Laplacian Connected to a Buckling Problem

Notes:

(i) The equivalence:

Krein Laplacian eigenvalue problem ! buckling problem

shows that the nonlocal and V -dependent boundary condition associated withthe second-order perturbed Krein Laplacian eigenvalue problem can be turnedinto a fourth-order buckling problem with (local) Dirichlet and Neumannboundary conditions.

(ii) The (generalized) buckling problem comes from a linear operator pencil

problem Au = �Bu, where A = (��+ V )2, B = (��+ V ).

(iii) G. Grubb, J. Oper. Th. 10 (1983) developed the intimate connectionbetween SK and the (generalized) buckling problem for even-order ellipticpartial di↵erential operators on smooth bounded domains ⌦ ⇢ Rn.

We found the extension to abstract Hilbert space operators and hence, anabstract buckling problem, in 2010, so this is now a general phenomenon.

Moreover, Grubb’s results for smooth bounded domains were extended tobounded Lipschitz domains in [BGMM16], see the next result:


Weyl Asymptotics for HK,⌦

Weyl asymptotics for positive e.v.’s of HK ,⌦:


Assume that ⌦ is a bounded Lipschitz domain in Rn, n � 2, and letN(�;HK ,⌦) = #{j 2 N | 0 <�K ,⌦,j �}, � 2 R, with �K ,⌦,j enumeratedaccording to multiplicity. Then

N(�;HK ,⌦) = (2⇡)�nvn|⌦|�n/2 + O��(n�(1/2))/2

�as �!1,

where, vn = ⇡n/2/�((n/2) + 1) is the volume of the unit ball in Rn.

This is based on the one-to-one connection with the (generalized) bucklingproblem combined with results of V. A. Kozlov and new results on Dirichlet andNeumann traces for Lipschitz domains in [BGMM16].

Note. The remainder estimate O��(n�(1/2))/2

�is possible because of the

Lipschitz hypothesis on ⌦. For arbitrary bounded, open ⌦ one obtains onlyo��n/2

�for the remainder estimate.


Weyl Asymptotics for HK,⌦

Epilogue: All this was inspired by

A. Alonso and B. Simon, The Birman–Krein–Vishik theory of selfadjointextensions of semibounded operators, J. Oper. Th. 4, 251–270 (1980).

“It seems to us that the Krein extension of ��, i.e., �� with the boundarycondition @⌫u = @⌫H(u) on @⌦, is a natural object and therefore worthy offurther study. For example: Are the asymptotics of its nonzero eigenvalues givenby Weyl’s formula?”

First solved 3 years later by G. Grubb, Spectral asymptotics for the “soft”selfadjoint extension of a symmetric elliptic di↵erential operator, J. Oper. Th. 10,9–20 (1983), for the case of even-order elliptic partial di↵erential operators onsmooth, bounded domains ⌦ ⇢ Rn.

V. A. Kozlov, Estimates of the remainder in formulas for the asymptotic behaviorof the spectrum for lin. operator bundles, Funct. Anal. Appl. 17, 147-149 (1983).

Our Weyl asymptotics results for HK ,⌦ in 2010 ([AGMT10], Adv. Math.) were thefirst for certain nonsmooth bounded domains ⌦ (close to Lipschitz); thecorresponding bounded Lipschitz domain results are from 2015 (see [BGMM16]).


Bounds on Eigenvalue Counting Functions

Back to Eigenvalue Counting Functions:

Hypothesis.

Let n 2 N, n � 2, ; 6= ⌦ ⇢ Rn open and of finite (Euclidean) volume.

Note. (i) No assumptions are made on @⌦.

(ii) Since W 2m,20 (⌦) embeds compactly into L2(⌦), ��F ,⌦ has purely discrete

spectrum and hence so does ��K ,⌦ in (0,1) (i.e., away from 0).

For simplicity, we start with the Laplacian (i.e., V = 0): Recall the eigenvaluecounting function for ��K ,⌦,

N(�;��K ,⌦) = #{j 2 N | 0 <�K ,⌦,j �}, � 2 R,

�K ,⌦,j , j 2 N, the eigenvalues of ��K ,⌦ enumerated according to multiplicity.

Theorem (Krein Laplacian, [GLMS15]).

N(�;��K ,⌦) (2⇡)�nvn|⌦|{1 + [2/(2 + n)]}n/2�n/2, � > 0,

where vn := ⇡n/2/�((n + 2)/2) is the (Euclidean) volume of the unit ball in Rn.

This seemed to be a new result, no matter what assumptions on @⌦.Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 24 / 33


Eigenvalue Counting Functions (contd.):

Sketch of proof.

Introduce the minimal operator ��min,⌦ = ��, dom(��min,⌦) = W 2,20 (⌦),

then ��min,⌦ � "I⌦, and introduce the symmetric forms in L2(⌦),a⌦(f , g) = (��min,⌦f ,��min,⌦g)L2(⌦), f , g 2 dom(a⌦) = dom(��min,⌦),

b⌦(f , g) = (f ,��min,⌦g)L2(⌦), f , g 2 dom(b⌦) = dom(��min,⌦).

On can show (and this is key, but requires some e↵orts!) that

N(�;��K ,⌦) max�dim

�f 2 dom(��min,⌦)

�� a⌦(f , f )� � b⌦(f , f ) < 0 �

.

To further analyze this we now fix � 2 (0,1) and introduce the auxiliary operator

L⌦,� := (��min,⌦)⇤(��min,⌦)� �(��min,⌦).

Then L⌦,� is self-adjoint, bounded from below, with purely discrete spectrum asits form domain, W 2,2

0 (⌦), embeds compactly into L2(⌦). We will study theauxiliary eigenvalue problem,

L⌦,�'j = µj'j , 'j 2 dom(L⌦,�),

where {'j}j2N represents an orthonormal basis of eigenfunctions in L2(⌦) and werepeat the eigenvalues µj of L⌦,� according to their multiplicity.




Sketch of proof (contd.).

Since 'j 2W 2,20 (⌦), we denote by

e'j(x) :=

('j(x), x 2 ⌦,

0, x 2 Rn\⌦, their zero-extension of 'j to all of Rn.

Next, given µ > 0, one estimates

µ�1P

j2N, µj<µ(µ� µj) � µ�1P

j2N, µj<0, µj<µ(µ� µj)

� µ�1P

j2N, µj<0, µj<µ µ = n�(L⌦,�),

where n�(L⌦,�) denotes the number of strictly negative eigenvalues of L⌦,�. Thisimplies

N(�;��K ,⌦) max�dim

�f 2 dom(��min,⌦)

�� a⌦(f , f )� � b⌦(f , f ) < 0 �

= n�(L⌦,�) µ�1P

j2N, µj<µ(µ� µj) = µ�1P

j2N[µ� µj ]+, µ > 0. (⇤)Here, x+ := max (0, x), x 2 R.

Next, we focus on estimating the right-hand side of (⇤).Fritz Gesztesy (Baylor) Krein–von Neumann extensions September 29, 2017 26 / 33




N(�;��K ,⌦) µ�1X

j2N(µ� µj)+ = µ�1

X

j2N

⇥('j , (µ� µj)'j)L2(⌦)

⇤+

= µ�1X

j2N

hµk'jk2L2(⌦) �

��(��)'j

��2L2(⌦)

+ ��'j , (��)'j

�L2(⌦)

i

+

= µ�1X

j2N

hµke'jk2L2(Rn) �

��(��)e'j

��2L2(Rn)

+ ��e'j , (��)e'j

�L2(Rn)

i

+

= µ�1X

j2N

Z

Rn

⇥µ� �|⇠|4 � �|⇠|2�⇤��be'j(⇠)

��2 dn⇠

�

+

c = Fourier Transf.

µ�1X

j2N

Z

Rn

⇥µ� �|⇠|4 � �|⇠|2�⇤

+

��be'j(⇠)��2 dn⇠

= µ�1

Z

Rn

⇥µ� |⇠|4 + �|⇠|2⇤

+

X

j2N

��be'j(⇠)��2 dn⇠.





Here we used unitarity of the Fourier transform on L2(Rn), the fact that⇥µ� |⇠|4 + �|⇠|2⇤

+has compact support, and the monotone convergence theorem

in the final step. Next,P

j2N��be'j(⇠)

��2 = (2⇡)�nP

j2N��e i⇠·, e'j

�L2(Rn)

��2 = (2⇡)�nP

j2N��e i⇠·,'j

�L2(⌦)

��2

= (2⇡)�n��e i⇠·

��2L2(⌦)

= (2⇡)�n|⌦|,since {'j}j2N is an orthonormal basis in L2(⌦).

Introducing ↵ = ��2µ, changing variables, ⇠ = �1/2⌘, and taking the minimumwith respect to ↵ > 0, finally yields the bound,

N(�;��K ,⌦) (2⇡)�n|⌦|min↵>0

✓↵�1

Z

Rn

⇥↵� |⇠|4 + |⇠|2⇤

+dn⇠

◆�n/2

= (2⇡)�nvn|⌦|{1 + [2/(2 + n)]}n/2�n/2, � > 0.

Explicit minimization over ↵ is a nice but nontrivial calculus exercise! (I flunkedit, but Mark Ashbaugh was up to it!) ⇤



Eigenvalue Counting Functions (contd.): Extensions

(i) Actually, in [GLMS15] we considered higher-order Krein Laplacians inL2(⌦), denoted by ��K ,⌦,m, associated to the minimal operator

��min,⌦,m = (��)m, dom((��)m) = W 2m,20 (⌦), m 2 N,

in [GLMS15] with the corresponding estimate,

N(�;��K ,⌦,m) (2⇡)�nvn|⌦|{1 + [2m/(2m+ n)]}n/(2m)�n/(2m), � > 0, m 2 N.

(ii) More importantly, curently in [AGLMS16] we consider higher-order Kreinextensions, denoted by AK ,⌦,2m(a, b, q), of the closed, strictly positive,higher-order di↵erential operator in L2(⌦) (the minimal operator)

Amin,⌦,2m(a, b, q) = ⌧2m(a, b, q), dom(Amin,⌦,2m(a, b, q)) = W 2m,20 (⌦),

corresponding to the higher-order di↵erential expression

⌧2m(a, b, q) =⇣Pn

j,k=1(�i@j � bj(x))aj,k(x)(�i@k � bk(x)) + q(x)⌘m

, m 2 N.

Moreover, we need an “extension” of Amin,⌦,2m(a, b, q) to all of Rn, denoted by

the operator eA2m(a, b, q) in L2(Rn),

eA2m(a, b, q) = ⌧2m(a, b, q), dom� eA2m(a, b, q)

�= W 2m,2(Rn).




Hypothesis.

(i) Let ; 6= ⌦ ⇢ Rn be open and bounded, n 2 N.(ii) Let m 2 N. Assume that

b = (b1, b2, . . . , bn) 2⇥W (2m�1),1(Rn)

⇤n, bj real-valued, 1 j n,

0 q 2W (2m�2),1(Rn), a := {aj,k}1j,kn 2 C (2m�1)�Rn,Rn2

�,

and assume that there exists "a > 0, such that (uniform ellipticity ....)Pn

j,k=1 aj,k(x)yjyk � "a|y |2, x 2 Rn, y = (y1, . . . , yn) 2 Rn.

In addition, assume that the n ⇥ n matrix-valued function a equals the identity Inoutside a ball Bn(0;R) containing ⌦, that is, there exists R > 0 such that

a(x) = In, |x | � R , ⌦ ⇢ Bn(0;R).

Given these assumptions, 0 eA2m(a, b, q) is self-adjoint in L2(Rn).

We need one more spectral theory assumption on eA2m(a, b, q) so that the Fouriertransform as the eigenfunction transform of �� on dom(��) = W 2,2(Rn) canbe replaced by the eigenfunction transform, i.e., a distorted Fourier transfom,associated with eA2m(a, b, q) on dom

� eA2m(a, b, q)�= W 2m,2(Rn):




Hypothesis.

In addition to the previous hypotheses on a, b, q and ⌦, assume the following:(i) Suppose there exists � : Rn ⇥ Rn ! C, such that the operator

(Ff )(⇠) := (2⇡)�n/2RRn f (x)�(x , ⇠) dnx , ⇠ 2 Rn,

originally defined on functions f 2 L2(Rn) with compact support, can be extendedto the unitary operator in L2(Rn), such that

f 2W 2,2(Rn; dnx) if and only if |⇠|2(Ff )(⇠) 2 L2(Rn; dn⇠),

and eA2(a, b, q) = F�1M|⇠|2F,where M|⇠|2 represents the maximally defined operator of multiplication by |⇠|2 inL2(Rn; dn⇠).

(ii) In addition, suppose that sup⇠2Rn k�( · , ⇠)kL2(⌦) <1.

Note. Condition (ii) is tricky, but appropriate limiting absorption theorems canbe shown to imply it in many cases of interest. Condition (i) implies the absenceof eigenvalues and any singular continuous spectrum of eA2(a, b, q). It issimplest implemented by choosing (a� In), b, q all of compact support.




Theorem ([AGLMS16]).

Under these hypotheses, and with AK ,⌦,2m(a, b, q) and AF ,⌦,2m(a, b, q) the Kreinand Friedrichs extensions of A⌦,2m(a, b, q), the following estimates holds:

N(�;AK ,⌦,2m(a, b, q)) vn(2⇡)n

✓1 +

2m

2m + n

◆n/(2m)

sup⇠2Rn

k�( · , ⇠)k2L2(⌦)�n/(2m),

N(�,AF ,⌦,2m(a, b, q)) vn(2⇡)n

✓1 +

2m

n

◆n/(2m)

sup⇠2Rn

k�( · , ⇠)k2L2(⌦)�n/(2m),

� > 0, m 2 N.

Here vn := ⇡n/2/�((n + 2)/2) denotes the (Euclidean) volume of the unit ball inRn and �( · , · ) represent the suitably normalized generalized eigenfunctions ofeA2(a, b, q) satisfying

eA2(a, b, q)�( · , ⇠) = |⇠|2�( · , ⇠)in the distributional sense. In particular, k�( · , ⇠)k2L2(⌦) = |⌦| if a = In, b = q = 0.




Much of this is new in the Krein extension context, no matter what assumptionson the boundary @⌦, but there are some cases which overlap with earlier work byM. Sh. Birman – M. Solomyak and G. Rozenblum.

The case of the Friedrichs extension in the special case of the Laplacian (i.e., ifa = In, b = q = 0), actually, the more general case of the fractional Lalacian, isdue to A. Laptev ’97.

Thank you!


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