The Eigenvalue Counting Function for Krein-von Neumann Extensions of Elliptic Operators

Transcription

1 The Eigenvalue Counting Function for Krein-von Neumann 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, / 33

2 Outline 1 Topics Discussed 2 On Weyl Asymptotics 3 A Bit of Operator Theory 4 Mark Krein s 1947 Result 5 Perturbed Krein Laplacians H K, = K, + V on Bounded Domains 6 Spectral Properties of H K, 7 The Perturbed Krein Laplacian Connected to a Buckling Problem 8 Weyl Asymptotics for H K, 9 Bounds on Eigenvalue Counting Functions Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

3 Topics involved: Topics Discussed 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 bounded Lipschitz domains in R n, n 2 N, andthebuckling of a clamped plate (! elasticity applications). Weyl asymptotics of perturbed Krein Laplacians in bounded Lipschitz domains in R n. Eigenvalue Counting Function Bounds for perturbed Krein Laplacians in arbitrary finite volume domains in R n. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

4 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 abstract buckling problem, Math. Nachr. 283, (2010). [AGMT10] M. S. Ashbaugh, F.G., M. Mitrea, and G. Teschl, Spectral theory for perturbed Krein Laplacians in nonsmooth domains, Adv. Math. 223, (2010). [GLMS15] F.G., A. Laptev, M. Mitrea, and S. Sukhtaiev, A bound for the eigenvalue counting function for higher-order Krein Laplacians on open sets, in Mathematical Results in Quantum Mechanics, Proceedings of the QMath12 Conference, P. Exner, W. König, and H. Neidhardt (eds.), World Scientific, Singapore, 2015, pp [BGMM16] J. Behrndt, F.G., T. Micheler, and M. Mitrea, The Krein-von Neumann realization of perturbed Laplacians on bounded Lipschitz domains, IWOTA 2014 proceedings, Birkhäuser, OTAA 255, (2016). [AGLMS16] M. Ashbaugh, F.G., A. Laptev, M. Mitrea, and S. Sukhtaiev, A bound for the eigenvalue counting function for Krein von Neumann and Friedrichs extensions, Adv. Math. 304, (2017). Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

5 On Weyl Asymptotics Weyl asymptotics for the Laplacian : Let R n, n 2 N, beabounded, smooth domain, and BC, be the Laplacian on in the Hilbert space L 2 ( ) with appropriate classical boundary conditions (BC), e.g., Dirichlet, Neumann, Robin-type, on the Let BC,,j, j 2 N, be the eigenvalues of BC, enumerated according to their multiplicity. Introduce the eigenvalue counting function for BC, by N( ; BC, ) =#{j 2 N 0 < BC,,j apple }, 2 R. In 1911, Weyl proved his celebrated leading order asymptotics N( ; BC, ) =(2 ) n v n n/2 + o n/2 as!1, where, v n = n/2 / ((n/2) + 1) is the volume of the unit ball in R n. Note. The leading order is shape and BC-independent for a large class of BCs and only depends on the volume of. Weyl also initiated a study of the remainder term in his asymptotics. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

6 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 apparently independently posed by Arnold Sommerfeld (September 1910) and Hendrik Antoon Lorentz (October 1910), Nobel Prize in Physics in Hermann Weyl ( ) 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 extending his result to other boundary conditions and to n = 3. Rumor has it that David Hilbert ( ) did not think he would live to see the problem solved... Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

7 On Weyl Asymptotics Weyl asymptotics for the Laplacian (contd.): The problem originated in Physics in connection with (forced) vibrations and black body radiation (standing electromagnetic waves in a cavity with reflecting surface 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), whowasledtothediscovery of Quantum Theory (E = h, h - Planck s constant, energy quantization, one of the fathers of quantum physics, Nobel Prize in Physics in 1918). This reads like a list of Who s Who in Theoretical Physics just before Quantum Mechanics entered the scene. The subject is still so popular as there are numerous applications/ramifications in areas such as: Acoustics (Music: Lord Raleigh, high overtones for violins), Radiation, (Inverse) Spectral Geometry (Mark Kac s Can one hear the shape of a drum? in 1966, perhaps, one of the most cited spectral theory papers?), Analytic Number Theory (automorphic forms), etc. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

8 A Bit of Operator Theory Notation for the abstract part: Let H be a separable complex Hilbert space with scalar product (, ) H and identity operator I H in H. + denotes the direct sum (not necessarily orthogonal) in H. S denotes a symmetric, closed, densely defined operator in H bounded from below (typically, S 0 or S "I H for some ">0). es a self-adjoint extension of S. S F denotes the Friedrichs extension of S. S K denotes the Krein-von Neumann extension of S. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

9 A Bit of Operator Theory Basic notions: symmetric and nonnegative operators A linear operator S : dom(s) H!His called non-negative, S (u, Su) H 0, u 2 dom(s). S is called strictly positive, if for some ">0, (u, Su) H "kuk 2 H, u 2 dom(s). One then writes S "I H. Similarly, one defines S T (but there are technical details concerning (quadratic form) domains...) 0, if 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. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

10 A Bit of Operator Theory Basic principle of self-adjoint extensions: Let S j, j =1, 2, be symmetric, S 1 S 2 =) S 2 S 1 Thus, S 1 S 2 S 2 S 1 Existence of self-adjoint extensions was completely settled in J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102, ( ) 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 the notion of spectral theory for unbounded linear operators in Hilbert spaces. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

11 A Bit of Operator Theory John von Neumann: John von Neumann (December 28, 1903 February 8, 1957): Regarded as the foremost mathematician of his time and probably the last representative of the great mathematicians like Euler, Gauss, Poincaré, or Hilbert, he was a pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis, a key figure in the development of game theory and the concepts of cellular automata, and the digital computer. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

12 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-adjoint extensions of S, thereexisttwo distinguished (extremal ) ones, S K and S F, which are the smallest and largest (in the sense of order between the quadratic forms associated with self-adjoint operators) such extensions. Any non-negative self-adjoint extension S e 0 of S necessarily satisfies 0 apple S K apple e S apple S F, i.e., 8a > 0, (S F + ai H ) 1 apple ( e S + ai H ) 1 apple (S K + ai H ) 1. In particular, this determines S K and S F uniquely. In addition, if S "I H for some ">0, one has dom(s F ) = dom(s) u (S F ) 1 ker(s ), dom(s K ) = dom(s) u ker(s ), dom(s ) = dom(s) u (S F ) 1 ker(s ) u ker(s ), ker(s K )=ker (S K ) 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, / 33

13 Mark Krein: Mark Krein s 1947 Result Mark Grigorievich Krein (April 3, 1907 October 17, 1989): Mathematician Extraordinaire: One of the giants of 20th century mathematics, Wolf Prize in Mathematics in 1982; developed his own approach to the theory of self-adjoint extensions around Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

14 Mark Krein s 1947 Result 1d Example: =(a, b), 1 < a < b < 1, V =0: min,(a,b) = d 2 dx 2 C0 1 F,(a,b)u = D,(a,b)u = u 00,, dom( min,(a,b)) =W 2,2 0 ((a, b)), ((a,b)) u 2 dom( F,(a,b)) = v 2 L 2 ((a, b)) v, v 0 2 AC ([a, b]); Dirichlet b.c.s v(a) =v(b) =0; v 00 2 L 2 ((a, b)), N,(a,b)u = u 00, u 2 dom( N,(a,b)) = v 2 L 2 ((a, b)) v, v 0 2 AC ([a, b]); Neumann b.c.s v 0 (a) =v 0 (b) =0; v 00 2 L 2 ((a, b)), K,(a,b)u = u 00, u 2 dom( K,(a,b)) = v 2 L 2 ((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 L 2 ((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, / 33

15 Perturbed Krein Laplacians H K, = K, + V on Bounded Domains Perturbed Krein Laplacians on bounded Lipschitz (resp., minimally smooth) domains R n, n 2: We assume for the remainder of the Weyl asymptotics part: Hypothesis. Let n 2 N, n 2. (i) R n is a nonempty, open, bounded, Lipschitz domain. (ii) V 2 L 1 ( ) is nonnegative, V 0. The case n = 1, =(a, b), example. 1 < a < b < 1, was settled in the previous This permits one to make the step from the Laplacian,,toa perturbed Laplacian, + V,i.e.,aSchrödinger-type operator. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

16 Perturbed Krein Laplacians H K, = K, + V on Bounded Domains Bounded Lipschitz domains: Definition. Let ;6= R n open and R > 0fixed. is called a bounded Lipschitz domain, if there exists r 2 (0, R) suchthatforeveryx 0 one can find a rigid transformation T : R n! R n and a Lipschitz function ' : R n 1! R with T \ B(x 0, r) = T B(x 0, r) \ (x 0, x n ) 2 R n 1 R x n >'(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 a C 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 R 2!!! So studying Lipschitz domains is not just a technicality. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

17 Perturbed Krein Laplacians H K, = K, + V on Bounded Domains The domain of the perturbed Krein Laplacian H K, : The minimal and maximal perturbed Laplacian in L 2 ( ) aredefinedby Then H min, u := ( +V )u, u 2 dom(h min, ):=W 2,2 0 ( ), H max, u := ( +V )u, u 2 dom(h max, )= v 2 L 2 ( ) v 2 L 2 ( ). H min, = ( +V ) C 1 0 ( ) = H max, and H max, = H min,. The Krein von Neumann extension H K, of H min,,i.e.,theperturbed Krein Laplacian in L 2 ( ) is then given by H K, u := ( +V )u, u 2 dom(h K, ) = dom(h min, ) +ker(h max, ) = W 2,2 0 ( ) + v 2 L 2 ( ) ( +V )v =0in. (One recalls, + denotes the direct sum (not necessarily orthogonal) in L 2 ( ).) Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

18 Perturbed Krein Laplacians H K, = K, + V on Bounded Domains The domain of H K, (contd.): nonlocal b.c. s NONLOCAL and V -DEPENDENT boundary condition for H K, : dom(h K, ):= v 2 dom(h max, ) Nv + M WT D,N,,V (0)( D v)=0, i.e., these b.c. s are not of local Robin-type. D and N are appropriate Dirichlet and Neumann trace operators in L 2 (@ ), i.e., suitable generalizations of Du = and Nu = with the outward pointing unit vector (which exists a.e. ). MD,N,,V WT (z) is an appropriate z-dependent Dirichlet-to-Neumann operator (i.e., an operator-valued Weyl Titchmarsh operator) inl 2 (@ ), MD,N,,V WT (z) = N N(H D, zi L2 ( )) 1, z 2 (HD, ). Formally, if is smooth, andinthespecialcasev = 0, thisnonlocal b.c. is of (x) =0, where (Hv) denotes the harmonic extension of into. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

19 Spectral Properties of H K, General spectral properties of H K, : Theorem ([AGMT10], [BGMM16]). The perturbed Krein Laplacian H K, is a self-adjoint operator on L 2 ( ) which satisfies H K, 0andH min, H K, H max,, H K, has a purely discrete spectrum in (0, 1), ess(h K, )={0} if n 2. For any non-negative self-adjoint extension e S of H min, = ( +V ) C 1 0 ( ) on dom(h min, )=W 2,2 0 ( ) one has H K, apple e S apple H D,. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

20 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 a clamped plate in R n, ( +V ) 2 u = ( +V )u in, u 2 dom( max, ), Du =0, Nu =0, where ( +V ) 2 u := ( +V )( u + Vu) in the sense of distributions in. Equivalently, ( +V ) 2 u = ( +V )u in, u 2 W 2,2 0 ( ). ( ) This is a generalized eigenvalue problem of the type Au = Bu in the sense that B 6= I L 2 ( ) in general. Equivalently, it s a linear operator pencil problem... One can show that necessarily, >0. Theorem ([AGMT10], [BGMM16]). >0 is an eigenvalue (necessarily discrete) of H K, if and only if >0isan eigenvalue of the (generalized) buckling problem ( ). Moreover, the multiplicities coincide. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

21 The Perturbed Krein Laplacian Connected to a Buckling Problem Notes: (i) Theequivalence: Krein Laplacian eigenvalue problem! buckling problem shows that the nonlocal and V -dependent boundary condition associated with the second-order perturbed Krein Laplacian eigenvalue problem can be turned into a fourth-order buckling problem with (local) Dirichlet and Neumann boundary 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 connection between S K and the (generalized) buckling problem for even-order elliptic partial di erential operators on smooth bounded domains R n. We found the extension to abstract Hilbert space operators and hence, an abstract buckling problem, in 2010, so this is now a general phenomenon. Moreover, Grubb s results for smooth bounded domains were extended to bounded Lipschitz domains in [BGMM16], see the next result: Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

22 Weyl Asymptotics for H K, Weyl asymptotics for positive e.v. s of H K, : Theorem ([AGMT10], [BGMM16]). Assume that is a bounded Lipschitz domain in R n, n 2, and let N( ; H K, ) =#{j 2 N 0 < K,,j apple }, 2 R, with K,,j enumerated according to multiplicity. Then N( ; H K, ) =(2 ) n v n n/2 + O (n (1/2))/2 as!1, where, v n = n/2 / ((n/2) + 1) is the volume of the unit ball in R n. This is based on the one-to-one connection with the (generalized) buckling problem combined with results of V. A. Kozlov and new results on Dirichlet and Neumann 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 only o n/2 for the remainder estimate. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

23 Weyl Asymptotics for H K, Epilogue: All this was inspired by A. Alonso and B. Simon, The Birman Krein Vishik theory of selfadjoint extensions of semibounded operators, J. Oper. Th. 4, (1980). It seems to us that the Krein extension of,i.e., with the boundary u H(u) is a natural object and therefore worthy of further study. For example: Are the asymptotics of its nonzero eigenvalues given by 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 on smooth, bounded domains R n. V. A. Kozlov, Estimates of the remainder in formulas for the asymptotic behavior of the spectrum for lin. operator bundles, Funct. Anal. Appl. 17, (1983). Our Weyl asymptotics results for H K, in 2010 ([AGMT10], Adv. Math.) were the first for certain nonsmooth bounded domains (close to Lipschitz); the corresponding bounded Lipschitz domain results are from 2015 (see [BGMM16]). Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

24 Bounds on Eigenvalue Counting Functions Back to Eigenvalue Counting Functions: Hypothesis. Let n 2 N, n 2, ;6= R n open and of finite (Euclidean) volume. Note. (i) No assumptions are made (ii) SinceW 2m,2 0 ( ) embeds compactly into L 2 ( ), 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 eigenvalue counting function for K,, N( ; K, ) =#{j 2 N 0 < K,,j apple }, 2 R, K,,j, j 2 N, the eigenvalues of Theorem (Krein Laplacian, [GLMS15]). K, enumerated according to multiplicity. N( ; K, ) apple (2 ) n v n {1+[2/(2 + n)]} n/2 n/2, >0, where v n := n/2 / ((n + 2)/2) is the (Euclidean) volume of the unit ball in R n. This seemed to be a new result, no matter what assumptions Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

25 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Sketch of proof. Introduce the minimal operator min, =, dom( min, ) =W 2,2 0 ( ), then min, "I,andintroducethesymmetricformsinL 2 ( ), 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, ) apple 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 as its form domain, W 2,2 0 ( ), embeds compactly into L 2 ( ). We will study the auxiliary eigenvalue problem, L, ' j = µ j ' j, ' j 2 dom(l, ), where {' j } j2n represents an orthonormal basis of eigenfunctions in L 2 ( ) andwe repeat the eigenvalues µ j of L, according to their multiplicity. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

26 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Sketch of proof (contd.). Since ' j 2 W 2,2 0 ( ), we denote by ( ' j (x), x 2, e' j (x) := 0, x 2 R n \, their zero-extension of ' j to all of R n. Next, given µ>0, one estimates µ 1 P j2n, µ j <µ (µ µ j) µ 1 P j2n, µ j <0, µ j <µ (µ µ j) µ 1 P j2n, µ j <0, µ j <µ µ = n (L, ), where n (L, ) denotes the number of strictly negative eigenvalues of L,.This implies N( ; K, ) apple max dim f 2 dom( min, ) a (f, f ) b (f, f ) < 0 = n (L, ) apple µ 1 P j2n, µ j <µ (µ µ j)=µ 1 P j2n [µ µ j] +, 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, / 33

27 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Sketch of proof (contd.). N( ; K, ) apple µ X 1 (µ µ j ) + = µ X 1 ('j, (µ µ j )' j ) L2 ( ) + j2n j2n = µ X h i 1 µk' j k 2 2 L 2 ( ) ( )' j L 2 ( ) + ' j, ( )' j L 2 ( ) + j2n = µ X h i 1 µk e' j k 2 2 L 2 (R n ) ( ) e' j L 2 (R n ) + e' j, ( ) e' j L 2 (R n ) + j2n = µ X apple Z 1 µ 4 2 b e'j ( ) 2 d n c = Fourier Transf. j2n R n + apple µ X Z 1 µ 4 2 b e' + j ( ) 2 d n j2n R n = µ ZR 1 µ X b e'j ( ) 2 d n. + n j2n Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

28 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Sketch of proof (contd.). Here we used unitarity of the Fourier transform on L 2 (R n ), the fact that µ has compact support, and the monotone convergence theorem + in the final step. Next, P b j2n e' j ( ) 2 =(2 ) n P j2n ei, 2 e' j L 2 (R n ) =(2 ) n P j2n ei, 2 ' j L 2 ( ) =(2 ) n e i 2 L 2 ( ) =(2 ) n, since {' j } j2n is an orthonormal basis in L 2 ( ). Introducing = 2 µ, changing variables, = 1/2,andtakingtheminimum with respect to >0, finally yields the bound, Z N( ; K, ) apple (2 ) n min >0 + d n n/2 R n =(2 ) n v n {1+[2/(2 + n)]} n/2 n/2, > 0. Explicit minimization over is a nice but nontrivial calculus exercise! (I flunked it, but Mark Ashbaugh was up to it!) Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

29 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Extensions (i) Actually, in [GLMS15] we considered higher-order Krein Laplacians in L 2 ( ), denoted by K,,m, associated to the minimal operator min,,m =( ) m, dom(( ) m )=W 2m,2 0 ( ), m 2 N, in [GLMS15] with the corresponding estimate, N( ; K,,m) apple (2 ) n v n {1+[2m/(2m + n)]} n/(2m) n/(2m), > 0, m 2 N. (ii) More importantly, curently in [AGLMS16] we consider higher-order Krein extensions, denoted by A K,,2m (a, b, q), of the closed, strictly positive, higher-order di erential operator in L 2 ( ) (the minimal operator) A min,,2m (a, b, q) = 2m (a, b, q), dom(a min,,2m (a, b, q)) =W 2m,2 0 ( ), corresponding to the higher-order di erential expression 2m (a, b, q) = Pn j,k=1 ( i@ j b j (x))a j,k (x)( i@ k b k (x)) + q(x) m, m 2 N. Moreover, we need an extension of A min,,2m (a, b, q) to all of R n, denoted by the operator e A 2m (a, b, q) in L 2 (R n ), ea 2m (a, b, q) = 2m (a, b, q), dom e A 2m (a, b, q) = W 2m,2 (R n ). Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

30 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Extensions Hypothesis. (i) Let;6= R n be open and bounded, n 2 N. (ii) Letm 2 N. Assumethat b =(b 1, b 2,...,b n ) 2 W (2m 1),1 (R n ) n, bj real-valued, 1 apple j apple n, 0 apple q 2 W (2m 2),1 (R n ), a := {a j,k } 1applej,kapplen 2 C (2m 1) R n, R n2, and assume that there exists " a > 0, such that (uniform ellipticity...) P n j,k=1 a j,k(x)y j y k " a y 2, x 2 R n, y =(y 1,...,y n ) 2 R n. In addition, assume that the n n matrix-valued function a equals the identity I n outside a ball B n (0; R) containing, thatis,thereexistsr > 0suchthat a(x) =I n, x R, B n (0; R). Given these assumptions, 0 apple e A 2m (a, b, q) is self-adjoint in L 2 (R n ). We need one more spectral theory assumption on e A 2m (a, b, q) so that the Fourier transform as the eigenfunction transform of on dom( )=W 2,2 (R n )can be replaced by the eigenfunction transform, i.e.,adistorted Fourier transfom, associated with e A 2m (a, b, q) on dom e A 2m (a, b, q) = W 2m,2 (R n ): Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

31 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Extensions Hypothesis. In addition to the previous hypotheses on a, b, q and, assume the following: (i) Suppose there exists : R n R n! C, such that the operator (Ff )( ) :=(2 ) n/2 R R n f (x) (x, ) d n x, 2 R n, originally defined on functions f 2 L 2 (R n ) with compact support, can be extended to the unitary operator in L 2 (R n ), such that f 2 W 2,2 (R n ; d n x) if and only if 2 (Ff )( ) 2 L 2 (R n ; d n ), and e A 2 (a, b, q) = F 1 M 2F, where M 2 represents the maximally defined operator of multiplication by 2 in L 2 (R n ; d n ). (ii) In addition, suppose that sup 2R n k (, )k L 2 ( ) < 1. Note. Condition (ii) is tricky, but appropriate limiting absorption theorems can be shown to imply it in many cases of interest. Condition (i) impliestheabsence of eigenvalues and any singular continuous spectrum of A e 2 (a, b, q). Itis simplest implemented by choosing (a I n ), b, q all of compact support. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

32 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Extensions Theorem ([AGLMS16]). Under these hypotheses, and with A K,,2m (a, b, q) and A F,,2m (a, b, q) the Krein and Friedrichs extensions of A,2m (a, b, q), the following estimates holds: N( ; A K,,2m (a, b, q)) apple v n (2 ) n 1+ 2m n/(2m) sup k (, )k 2 L 2m + n 2 ( ) 2R n N(, A F,,2m (a, b, q)) apple v n (2 ) n 1+ 2m n n/(2m) sup 2R n k (, )k 2 L 2 ( ) n/(2m), >0, m 2 N. n/(2m), Here v n := n/2 / ((n + 2)/2) denotes the (Euclidean) volume of the unit ball in R n and (, ) represent the suitably normalized generalized eigenfunctions of ea 2 (a, b, q) satisfying ea 2 (a, b, q) (, ) = 2 (, ) in the distributional sense. In particular, k (, )k 2 L 2 ( ) = if a = I n, b = q = 0. Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33

33 Bounds on Eigenvalue Counting Functions Eigenvalue Counting Functions (contd.): Extensions Much of this is new in the Krein extension context, no matter what assumptions on the but there are some cases which overlap with earlier work by M. Sh. Birman M. Solomyak and G. Rozenblum. The case of the Friedrichs extension in the special case of the Laplacian (i.e., if a = I n, b = q = 0), actually, the more general case of the fractional Lalacian, is due to A. Laptev 97. Thank you! Fritz Gesztesy (Baylor) Krein von Neumann extensions September 29, / 33