Schrödinger operators exhibiting parameter-dependent spectral transitions

Schrödinger operators exhibiting parameter-dependent spectral transitions Pavel Exner Doppler Institute for Mathematical Physics and Applied Mathematics Prague in collaboration with Diana Barseghyan, Andrii Khrabustovskyi, Vladimir Lotoreichik, and Miloš Tater A talk at the The fourth Najman Conference on Spectral Problems for Operators and Matrices Opatija, September 22, 2015 P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-1 -

Talk outline The simplest example: Smilansky model P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-2 -

Talk outline The simplest example: Smilansky model Spectral properties, the subcritical and supercritical case Numerical solution P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-2 -

Talk outline The simplest example: Smilansky model Spectral properties, the subcritical and supercritical case Numerical solution A regular version of Smilansky model Choice of the potential Spectral properties, the subcritical and supercritical case Intervals and multiple channels Another model: transition from a purely discrete to the real line Potential for which Weyl quantization fails Spectral properties, the subcritical and supercritical case The critical case P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-2 -

Talk outline The simplest example: Smilansky model Spectral properties, the subcritical and supercritical case Numerical solution A regular version of Smilansky model Choice of the potential Spectral properties, the subcritical and supercritical case Intervals and multiple channels Another model: transition from a purely discrete to the real line Potential for which Weyl quantization fails Spectral properties, the subcritical and supercritical case The critical case Summary & open questions P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-2 -

Smilansky model The model was originally proposed in [Smilansky 04] to describe a one-dimensional system interacting with a caricature heat bath represented by a harmonic oscillator. Mathematical properties of the model were analyzed in [Solomyak 04], [Evans-Solomyak 05], [Naboko-Solomyak 06]. More recently, time evolution in such a (slightly modified) model was analyzed [Guarneri 11] In PDE terms, the model is described through a 2D Schrödinger operator H Sm = 2 x 2 + 1 ) ( 2 2 y 2 + y 2 + λyδ(x) on L 2 (R) with various modifications to be mentioned later. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-3 -

Smilansky model The model was originally proposed in [Smilansky 04] to describe a one-dimensional system interacting with a caricature heat bath represented by a harmonic oscillator. Mathematical properties of the model were analyzed in [Solomyak 04], [Evans-Solomyak 05], [Naboko-Solomyak 06]. More recently, time evolution in such a (slightly modified) model was analyzed [Guarneri 11] In PDE terms, the model is described through a 2D Schrödinger operator H Sm = 2 x 2 + 1 ) ( 2 2 y 2 + y 2 + λyδ(x) on L 2 (R) with various modifications to be mentioned later. Due to a particular choice of the coupling the model exhibited a spectral transition with respect to the coupling parameter λ. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-3 -

A summary of results about the model Spectral transition: if λ > 2 the particle can escape to infinity along the singular channel in the y direction. In spectral terms, it corresponds to switch from a positive to a below unbounded spectrum at λ = 2. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-4 -

A summary of results about the model Spectral transition: if λ > 2 the particle can escape to infinity along the singular channel in the y direction. In spectral terms, it corresponds to switch from a positive to a below unbounded spectrum at λ = 2. At the heuristic level, the mechanism is easy to understand: we have an effective variable decoupling far from the x-axis and the oscillator potential competes there with the δ interaction eigenvalue 1 4 λ2 y 2. Eigenvalue absence: for any λ 0 there are no eigenvalues 1 2. If λ > 2, the point spectrum of H Sm is empty. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-4 -

A summary of results about the model Spectral transition: if λ > 2 the particle can escape to infinity along the singular channel in the y direction. In spectral terms, it corresponds to switch from a positive to a below unbounded spectrum at λ = 2. At the heuristic level, the mechanism is easy to understand: we have an effective variable decoupling far from the x-axis and the oscillator potential competes there with the δ interaction eigenvalue 1 4 λ2 y 2. Eigenvalue absence: for any λ 0 there are no eigenvalues 1 2. If λ > 2, the point spectrum of H Sm is empty. Existence of eigenvalues: for 0 < λ < 2 we have H Sm 0. The point spectrum is nonempty and finite, and N( 1 2, H Sm) 1 4 2(µ(λ) 1) holds as λ 2, where µ(λ) := 2/λ. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-4 -

Further results Absolute continuity: in the supercritical case λ > 2 we have σ ac (H Sm ) = R P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-5 -

Further results Absolute continuity: in the supercritical case λ > 2 we have σ ac (H Sm ) = R Extension of the result to a two channel case with different oscillator frequencies [Evans-Solomyak 05] Extension to multiple channels on a system periodic in x [Guarneri 11]. In this paper the time evolution generated by H Sm is investigated and proposed as a model of wavepacket collapse. The above results have been obtained by a combination of different methods: a reduction to an infinite system of ODE s, facts from Jacobi matrices theory, variational estimates, etc. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-5 -

Further results Absolute continuity: in the supercritical case λ > 2 we have σ ac (H Sm ) = R Extension of the result to a two channel case with different oscillator frequencies [Evans-Solomyak 05] Extension to multiple channels on a system periodic in x [Guarneri 11]. In this paper the time evolution generated by H Sm is investigated and proposed as a model of wavepacket collapse. The above results have been obtained by a combination of different methods: a reduction to an infinite system of ODE s, facts from Jacobi matrices theory, variational estimates, etc. Before proceeding further, let show how the spectrum can be treated numerically in the subcritical case. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-5 -

Numerical search for eigenvalues In the halfplanes ±x > 0 the wave functions can be expanded using the transverse base spanned by the functions ψ n (y) = 1 2 n n! π e y 2 /2 H n (y) corresponding to the oscillator eigenvalues n + 1 2, n = 0, 1, 2,.... Furthermore, one can make use of the mirror symmetry w.r.t. x = 0 and divide H λ into the trivial odd part H ( ) λ and the even part H (+) λ which is equivalent to the operator on L 2 (R (0, )) with the same symbol determined by the boundary condition f x (0+, y) = 1 αyf (0+, y). 2 P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-6 -

Numerical solution, continued We substitute the Ansatz with κ n := n + 1 2 ɛ. f (x, y) = c n e κnx ψ n (y) n=0 P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-7 -

Numerical solution, continued We substitute the Ansatz with κ n := n + 1 2 ɛ. f (x, y) = c n e κnx ψ n (y) n=0 This yields for solution with the energy ɛ the equation B λ c = 0, where c is the coefficient vector and B λ is the operator in l 2 with (B λ ) m,n = κ n δ m,n + 1 2 λ(ψ m, yψ n ). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-7 -

Numerical solution, continued We substitute the Ansatz with κ n := n + 1 2 ɛ. f (x, y) = c n e κnx ψ n (y) n=0 This yields for solution with the energy ɛ the equation B λ c = 0, where c is the coefficient vector and B λ is the operator in l 2 with (B λ ) m,n = κ n δ m,n + 1 2 λ(ψ m, yψ n ). Note that the matrix is in fact tridiagonal because (ψ m, yψ n ) = 1 2 ( n + 1 δm,n+1 + n δ m,n 1 ). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-7 -

Smilansky model eigenvalues 0.4 E n 0.2 0 0 0.5 1 λ In most part of the subcritical region there is a single eigenvalue, the second one appears only at λ 1.387559. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-8 -

Smilansky model eigenvalues 0.4 E n 0.2 0 0 0.5 1 λ In most part of the subcritical region there is a single eigenvalue, the second one appears only at λ 1.387559. The next thresholds are 1.405798, 1.410138, 1.41181626, 1.41263669,... P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-8 -

Smilansky model eigenvalues Close to the critical value, however, many eigenvalues appear which gradually fill the interval (0, 1 2 ) as the critical value is approached P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-9 -

Their number is as predicted The dots mean the eigenvalue numbers, the red curve is the above mentioned asymptotics due to Solomyak P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-10 -

Smilansky model ground state The numerical solution also indicates other properties, for instance, that the first eigenvalue behaves as ɛ 1 = 1 2 cλ4 + o(λ 4 ) as λ 0, with c 0.0156. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-11 -

In fact, we have c = 0.015625 Indeed, the relation B λ c = 0 can be written explicitly as µλ c0 λ + λ 2 2 cλ 1 = 0, kλ 2 2 cλ k 1 + k + 1λ k + µ λ ck λ + 2 ck+1 λ 2 = 0, k 1, where µ λ := 1 2 E 1(λ) and c λ = {c0 λ, cλ 1,... } is the corresponding normalized eigenvector of B λ. Using the above relations and simple estimates, we get ck λ 2 3 4 λ2 and c0 λ = 1 + O(λ 2 ) k=1 as λ 0+; hence we have in particular c λ 1 = λ 2 2 + O(λ2 ). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-12 -

In fact, we have c = 0.015625 The first of the above relation then gives as λ 0+, in other words µ λ = λ4 64 + O(λ5 ) E 1 (λ) = 1 2 λ4 64 + O(λ5 ). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-13 -

In fact, we have c = 0.015625 The first of the above relation then gives as λ 0+, in other words µ λ = λ4 64 + O(λ5 ) E 1 (λ) = 1 2 λ4 64 + O(λ5 ). And the mentioned coefficient 0.015625 is nothing else than 1 64.. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-13 -

Smilansky model eigenfunctions 0 1 =0.0678737 5 0 2 =0.1498950 5 0 0 0-5 -5-5 -10 y -15-10 -10-15 -15-20 -20-20 -25-25 4 2 0 5 x 10 0 4 =0.3143358 5 15-25 2 0-2 -4 5 x 10 15 0 5 =0.3960236 5 4 2 0-2 0 0-5 -5-5 -10 y -15-10 -10-15 -15-20 -20-20 -25-25 5 x 10 5 x 10 15 0 6 =0.4758990 5 0 2 0-2-4 0 3 =0.2321621 5-25 15 6 4 2 0-2 5 x 10 15 5 0-5 5 x 10 15 The first six eigenfunctions of HSm for λ = 1.4128241, in other words, λ = 2 0.0086105. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-14 -

A regular version of Smilansky model Our next aim is to show that one can observe a similar effect for Schrödinger operators with regular potentials. Now, however, the coupling cannot be linear in y and the profile of the channel has to change with y. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-15 -

A regular version of Smilansky model Our next aim is to show that one can observe a similar effect for Schrödinger operators with regular potentials. Now, however, the coupling cannot be linear in y and the profile of the channel has to change with y. Recall that the effect comes from competition between the oscillator potential with the principal eigenvalue of the transverse part of the operator equal to 1 4 λ2 y 2. We replace the δ by a family of shrinking potentials whose mean matches the δ coupling constant, U(x, y) dx y. This can be achieved, e.g., by choosing U(x, y) = λy 2 V (xy) for a fixed function V. This motivates us to investigate the following operator on L 2 (R 2 ), H = 2 x 2 2 y 2 + ω2 y 2 λy 2 V (xy)χ { x a} (x), where ω, a are positive constants, χ { y a} is the indicator function of the interval ( a, a), and the potential V with supp V [ a, a] is a nonnegative function with bounded first derivative. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-15 -

A regular version of Smilansky model, continued By Faris-Lavine theorem the operator is e.s.a. on C0 same is true for its generalization, (R2 ) and the H = 2 x 2 2 y 2 + ω2 y 2 N λ j y 2 V j (xy)χ { x bj a j }(x) with a finite number of channels, where functions V j are positive with bounded first derivative, with the supports contained in (b j a j, b j + a j ) and such that supp V j supp V k = holds for j k. Remark We note that the properties discussed below depend on the asymptotic behavior of the potential channels and would not change if the potential is modified in the vicinity of the x-axis, for instance, by replacing the above cut-off functions with χ y a (y) and χ y aj (y), respectively. j=1 P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-16 -

Subcritical case To state the result we employ a 1D comparison operator L = L V, L = d2 dx 2 + ω2 λv (x) on L 2 (R) with the domain H 2 (R). What matters is the sign of its spectral threshold; since V is supposed to be nonnegative, the latter is a monotonous function of λ and there is a λ crit > 0 at which the sign changes. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-17 -

Subcritical case To state the result we employ a 1D comparison operator L = L V, L = d2 dx 2 + ω2 λv (x) on L 2 (R) with the domain H 2 (R). What matters is the sign of its spectral threshold; since V is supposed to be nonnegative, the latter is a monotonous function of λ and there is a λ crit > 0 at which the sign changes. Theorem (Barseghyan-E 14) Under the stated assumption, the spectrum of the operator H is bounded from below provided the operator L is positive. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-17 -

Proof outline It is sufficient to prove the claim for λ = 1. We employ Neumann bracketing, similarly as for the previous model. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-18 -

Proof outline It is sufficient to prove the claim for λ = 1. We employ Neumann bracketing, similarly as for the previous model. Let h n and h n be respectively the restrictions of operator H to the strips G n = R {y : ln n < y ln(n + 1)}, n = 1, 2,..., and G n, their mirror images w.r.t. y, with Neumann boundary conditions; then H (h n h n ) ; n=1 We find a uniform lower bound σ(h n ) and σ( h n ) as n. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-18 -

Proof outline It is sufficient to prove the claim for λ = 1. We employ Neumann bracketing, similarly as for the previous model. Let h n and h n be respectively the restrictions of operator H to the strips G n = R {y : ln n < y ln(n + 1)}, n = 1, 2,..., and G n, their mirror images w.r.t. y, with Neumann boundary conditions; then H (h n h n ) ; n=1 We find a uniform lower bound σ(h n ) and σ( h n ) as n. Using the assumptions about V we find ( ) ( ) 1 ln n V (xy) V (x ln n) = O, y 2 ln 2 n = O n ln n n for any (x, y) G n, and analogous relations for G n. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-18 -

Proof outline continued This yields for for any (x, y) G n ( ) ln n y 2 V (xy) ln 2 n V (x ln n) = O n P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-19 -

Proof outline continued This yields ( ) ln n y 2 V (xy) ln 2 n V (x ln n) = O n for for any (x, y) G n which allows us to check that ( ) ln n inf σ(h n ) inf σ(l n ) + O, n where l n := 2 x 2 2 y 2 + ω 2 ln 2 n ln 2 n V (x ln n) on L 2 (G n ). The analogous relation holds for l n on L 2 ( G n ). It is important that all these operators have separated variables. Since the minimal eigenvalue of Neumann Laplacian d2 on the strips dy 2 ln n < y ln(n + 1), n = 1, 2,..., is zero, we have inf σ(l n ) = inf σ ( l n (1) ), where the last operator on L 2 (R) acts as l (1) n = d2 dx 2 + ω2 ln 2 n ln 2 n V (x ln n) P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-19 -

Proof outline concluded Note that the cut-off function χ { x a} plays no role in the asymptotic estimate as it affects a finite number of terms only. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-20 -

Proof outline concluded Note that the cut-off function χ { x a} plays no role in the asymptotic estimate as it affects a finite number of terms only. t By the change of variable x = ln n the last operator is unitarily equivalent to ln 2 n L which is non-negative as long as L is non-negative. In the same way one proves that l n is non-negative ; this concludes the proof. In the same way one can treat systems restricted in the x direction: Corollary Let H be our operator on ( c, c) R for some c a with Dirichlet (Neumann, periodic) boundary conditions in the variable x. The spectrum of H is bounded from below if L 0 holds, where L is the comparison operator on L 2 ( c, c) with Dirichlet (respectively, Neumann or periodic) boundary conditions. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-20 -

Supercritical case Once the transverse channel principal eigenvalue dominates over the harmonic oscillator contribution, the spectral behavior changes: P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-21 -

Supercritical case Once the transverse channel principal eigenvalue dominates over the harmonic oscillator contribution, the spectral behavior changes: Theorem (Barseghyan-E 14) Under our hypotheses, σ(h) = R holds if inf σ(l) < 0. Proof relies on construction of an appropriate Weyl sequence: we have to find {ψ k } k=1 D(H) such that ψ k = 1 which contains no convergent subsequence, and at same time Hψ k µψ k 0 as k. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-21 -

Supercritical case Once the transverse channel principal eigenvalue dominates over the harmonic oscillator contribution, the spectral behavior changes: Theorem (Barseghyan-E 14) Under our hypotheses, σ(h) = R holds if inf σ(l) < 0. Proof relies on construction of an appropriate Weyl sequence: we have to find {ψ k } k=1 D(H) such that ψ k = 1 which contains no convergent subsequence, and at same time Hψ k µψ k 0 as k. The construction is rather technical and we sketch just the main steps. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-21 -

Proof outline The claim is invariant under scaling transformations, hence we may suppose inf σ(l) = 1. The spectral threshold is a simple isolated eigenvalue; we denote the corresponding normalized eigenfunction by h. We want to show first that 0 σ ess (H). In fact, it would be enough for the proof to show that zero belongs to σ(h) but we get the stronger claim at no extra expense. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-22 -

Proof outline The claim is invariant under scaling transformations, hence we may suppose inf σ(l) = 1. The spectral threshold is a simple isolated eigenvalue; we denote the corresponding normalized eigenfunction by h. We want to show first that 0 σ ess (H). In fact, it would be enough for the proof to show that zero belongs to σ(h) but we get the stronger claim at no extra expense. We fix an ε > 0 and choose a natural k = k(ε) with which we associate a function χ k C0 2 (1, k) satisfying the following conditions k 1 1 z χ2 k (z) dz = 1 and k 1 z(χ k (z))2 dz < ε. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-22 -

Proof outline continued Such functions exist: as an example consider χ k (z) = 8 ln3 z ln 3 k χ {1 z 2 ln k 2 ln z k} (z) + ln k χ { k+1 z k 1} (z) +g k (z)χ { k<z< k+1} (z) + q k (z)χ {k 1<z k} (z), where g k and q k are interpolating functions chosen in such a way that χ k C0 2 (1, k), and define ( k ) 1/2 1 χ k (z) = z χ2 k (z) dz χ k (z). Given such functions χ k, put 1 ( ) ψ k (x, y) := h(xy) e iy 2 y /2 χ k + f (xy) n k y 2 e iy 2 /2 χ k ( y where f (t) := i 2 t2 h(t), t R, and n k N is a positive integer, which we choose using the following auxiliary result. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-23 - n k ),

Proof outline continued Lemma Let ψ k, k = 1, 2,..., as defined above; then for any given k one can achieve that ψ k L 2 (R 2 ) 1 2 holds by choosing n k large enough. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-24 -

Proof outline continued Lemma Let ψ k, k = 1, 2,..., as defined above; then for any given k one can achieve that ψ k L 2 (R 2 ) 1 2 holds by choosing n k large enough. We need one more auxiliary result: Lemma Let ψ k, k = 1, 2,..., be again functions defined above; then the inequality Hψ k 2 L 2 (R 2 ) < cε with a fixed constant c holds for k = k(ε). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-24 -

Proof outline continued Lemma Let ψ k, k = 1, 2,..., as defined above; then for any given k one can achieve that ψ k L 2 (R 2 ) 1 2 holds by choosing n k large enough. We need one more auxiliary result: Lemma Let ψ k, k = 1, 2,..., be again functions defined above; then the inequality Hψ k 2 L 2 (R 2 ) < cε with a fixed constant c holds for k = k(ε). Proofs are in both cases straightforward but rather tedious. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-24 -

Proof outline concluded Using the lemmata, we are able to complete the proof. We fix a sequence {ε j } j=1 such that ε j 0 holds as j and to any j we construct a function ψ k(εj ) in such a way that n k(εj ) > k(ε j 1 )n k(εj 1 ). The norms of Hψ k(εj ) are bounded from above with 9ε j on the right-hand side, and since the supports of ψ k(εj ), j = 1, 2,..., do not intersect each other by construction, their sequence converges weakly to zero. This yields the sought Weyl sequence for zero energy; for any nonzero real number µ we use the same procedure replacing the above ψ k with ( ) y ψ k (x, y) = h(xy) e iɛµ(y) χ k + f (xy) ( ) y n k y 2 e iɛµ(y) χ k, n k where ɛ µ (y) := y µ t 2 + µ dt, and furthermore, the functions f, χ k are defined in the same way as above. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-25 -

Restricted motion In the supercritical case, too, the result extends to systems restricted in the x direction: Theorem Let H be the our operator on L 2 ( c, c) L 2 (R) for some c > 0 with Dirichlet condition at x = ±c and denote by L the corresponding Dirichlet operator on L 2 ( c, c). If the spectral threshold of L is negative, the spectrum of H covers the whole real axis. Observing the domains of the quadratic form associated with such operators we get Corollary The claim of the above theorem remains valid if the Dirichlet boundary conditions at x = ±c are replaced by any other self-adjoint boundary conditions. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-26 -

The multichannel case The above results are interesting not only per se or to deal with the Guarneri-type periodic modification of the model. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-27 -

The multichannel case The above results are interesting not only per se or to deal with the Guarneri-type periodic modification of the model. Using a simple bracketing argument we can show how the spectral-regime transition looks like in the multichannel case: Theorem (Barseghyan-E 14) Let H be our operator with the potentials satisfying the stated assumptions, namely the functions V j are positive with bounded first derivative and supp V j supp V k = holds for j k. Denote by L j the comparison operator on L 2 (R) with the potential V j and set t V := min j inf σ(l j ). Then H is bounded from below if and only if t V 0 and in the opposite case its spectrum covers the whole real axis. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-27 -

Another model class Consider next a related family of systems in which the transition is even more dramatic passing from purely discrete spectrum in the subcritical case to the whole real line in the supercritical one. Recall that there are situations where Weyl s law fails and the spectrum is discrete even if the classically allowed phase-space volume is infinite. A classical example due to [Simon 83] is a 2D Schrödinger operator with the potential V (x, y) = x 2 y 2 or more generally, V (x, y) = xy p with p 1. Similar behavior one can observe for Dirichlet Laplacians in regions with hyperbolic cusps see [Geisinger-Weidl 11] for recent results and a survey. Moreover, using the dimensional-reduction technique of Laptev and Weidl one can prove spectral estimates for such operators. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-28 -

Another model class Consider next a related family of systems in which the transition is even more dramatic passing from purely discrete spectrum in the subcritical case to the whole real line in the supercritical one. Recall that there are situations where Weyl s law fails and the spectrum is discrete even if the classically allowed phase-space volume is infinite. A classical example due to [Simon 83] is a 2D Schrödinger operator with the potential V (x, y) = x 2 y 2 or more generally, V (x, y) = xy p with p 1. Similar behavior one can observe for Dirichlet Laplacians in regions with hyperbolic cusps see [Geisinger-Weidl 11] for recent results and a survey. Moreover, using the dimensional-reduction technique of Laptev and Weidl one can prove spectral estimates for such operators. A common feature of these models is that the particle motion is confined into channels narrowing towards infinity. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-28 -

Adding potentials unbounded from below This may remain true even for Schrödinger operators with unbounded from below in which a classical particle can escape to infinity with an increasing velocity. The situation changes, however, if the attraction is strong enough As an illustration, let us analyze the following class of operators: ( L p (λ) : L p (λ)ψ = ψ + xy p λ(x 2 + y 2 ) p/(p+2)) ψ, p 1 on L 2 (R 2 ), where (x, y) are the standard Cartesian coordinates in R 2 and the parameter λ in the second term of the potential is non-negative; unless the value of λ is important we write it simply as L p. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-29 -

Adding potentials unbounded from below This may remain true even for Schrödinger operators with unbounded from below in which a classical particle can escape to infinity with an increasing velocity. The situation changes, however, if the attraction is strong enough As an illustration, let us analyze the following class of operators: ( L p (λ) : L p (λ)ψ = ψ + xy p λ(x 2 + y 2 ) p/(p+2)) ψ, p 1 on L 2 (R 2 ), where (x, y) are the standard Cartesian coordinates in R 2 and the parameter λ in the second term of the potential is non-negative; unless the value of λ is important we write it simply as L p. 2p Note that p+2 < 2 so the operator is e.s.a. on C 0 (R2 ) by Faris-Lavine theorem again; the symbol L p or L p (λ) will always mean its closure. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-29 -

The subcritical case The spectral properties of L p (λ) depend crucially on the value of λ and there is a transition between different regimes as λ changes. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-30 -

The subcritical case The spectral properties of L p (λ) depend crucially on the value of λ and there is a transition between different regimes as λ changes. Let us start with the subcritical case which occurs for small values of λ. To characterize the smallness quantitatively we need an auxiliary operator which will be an (an)harmonic oscillator Hamiltonian on line, H p : H p u = u + t p u on L 2 (R) with the standard domain. Let γ p be the minimal eigenvalue of this operator; in view of the potential symmetry we have γ p = inf σ(h p ), where H p : H p u = u + t p u on L 2 (R + ) with Neumann condition at t = 0. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-30 -

The subcritical case continued The eigenvalue γ p = inf σ(h p ) equals one for p = 2; for p it becomes γ = 1 4 π2 ; it smoothly interpolates between the two values. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-31 -

The subcritical case continued The eigenvalue γ p = inf σ(h p ) equals one for p = 2; for p it becomes γ = 1 4 π2 ; it smoothly interpolates between the two values. Since x p 1 χ [0,1] (x) we have γ p ɛ 0 0.546, where ɛ 0 is the ground-state energy of the rectangular potential well of depth one. In fact, a numerical solution gives true minimum γ p 0.998995 attained at p 1.788; in the semilogarithmic scale the plot is as follows: 2.5 2 γ p 1.5 1 10 0 10 1 10 2 p P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-31 -

The subcritical case continued The spectrum is naturally bounded from below and discrete if λ = 0; our aim is to show that this remains to be the case provided λ is small enough. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-32 -

The subcritical case continued The spectrum is naturally bounded from below and discrete if λ = 0; our aim is to show that this remains to be the case provided λ is small enough. Theorem (E-Barseghyan 12) For any λ [0, λ crit ], where λ crit := γ p, the operator L p (λ) is bounded from below for p 1; if λ < γ p its spectrum is purely discrete. Idea of the proof: Let λ < γ p. By minimax we need to estimate L p from below by a s-a operator with a purely discrete spectrum. To construct it we employ bracketing imposing additional Neumann conditions at concentric circles of radii n = 1, 2,.... P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-32 -

The subcritical case continued The spectrum is naturally bounded from below and discrete if λ = 0; our aim is to show that this remains to be the case provided λ is small enough. Theorem (E-Barseghyan 12) For any λ [0, λ crit ], where λ crit := γ p, the operator L p (λ) is bounded from below for p 1; if λ < γ p its spectrum is purely discrete. Idea of the proof: Let λ < γ p. By minimax we need to estimate L p from below by a s-a operator with a purely discrete spectrum. To construct it we employ bracketing imposing additional Neumann conditions at concentric circles of radii n = 1, 2,.... In the estimating operators the variables decouple asymptotically and the spectral behavior is determined by the angular part of the operators. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-32 -

Subcritical behavior the proof Specifically, in polar coordinates we get direct sum of operators acting as L (1) n,pψ = 1 ( r ψ ) 1 2 ( ) ψ r 2p r r r n 2 ϕ 2 + 2 p sin 2ϕ p λr 2p/(p+2) ψ on the annuli G n := {(r, ϕ) : n 1 r < n, 0 ϕ < 2π}, n = 1, 2,... with Neumann conditions imposed on G n. Obviously σ(l (1) n,p) is purely discrete for each n = 1, 2,..., hence it is sufficient to check that inf σ(l (1) n,p) holds as n. We estimate L (1) n,p from below by an operator with separating variables, note that the radial part does not contribute and use the symmetry of the problem; for ε (0, 1) the question is then to analyze ( L (2) n,p : L (2) n n,pu = u 2p+2 + 2 p sin p 2x λ ) 1 ε n(4p+4)/(p+2) u on L 2 (0, π/4) with Neumann conditions, u (0) = u (π/4) = 0. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-33 -

Subcritical behavior proof continued We have n 2 inf σ(l (1) n,p) inf σ(l (2) n 1,p ) if n is large enough, specifically for n > ( 1 (1 ε) (p+2)/(4p+4)) 1, hence it is sufficient to investigate the spectral threshold µ n,p of L (2) n,p as n. The trigonometric potential can be estimated by a powerlike one with the similar behavior around the minimum introducing, e.g. ( ( ) p L (3) n,p := d2 2 dx 2 + n2p+2 x p χ (0,δ(ε)] (x) + χ [δ(ε),π/4)(x)) λ ε n (4p+4)/(p+2) π for small enough δ(ε) with Neumann boundary conditions at x = 0, 1 4 π, where we have denoted λ ε := λ(1 ε) p 1. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-34 -

Subcritical behavior proof continued We have n 2 inf σ(l (1) n,p) inf σ(l (2) n 1,p ) if n is large enough, specifically for n > ( 1 (1 ε) (p+2)/(4p+4)) 1, hence it is sufficient to investigate the spectral threshold µ n,p of L (2) n,p as n. The trigonometric potential can be estimated by a powerlike one with the similar behavior around the minimum introducing, e.g. ( ( ) p L (3) n,p := d2 2 dx 2 + n2p+2 x p χ (0,δ(ε)] (x) + χ [δ(ε),π/4)(x)) λ ε n (4p+4)/(p+2) π for small enough δ(ε) with Neumann boundary conditions at x = 0, 1 4 π, where we have denoted λ ε := λ(1 ε) p 1. We have L (2) n,p (1 ε) p L (3) n,p. To estimate the rhs by comparing the indicated potential contributions it is useful to pass to the rescaled variable x = t n (2p+2)/(p+2). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-34 -

Subcritical behavior proof concluded In this way we find that µ n,p := inf σ(l (3) n,p) satisfies µ n,p n 2 as n P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-35 -

Subcritical behavior proof concluded In this way we find that µ n,p := inf σ(l (3) n,p) satisfies µ n,p n 2 as n Through the chain of inequalities we come to conclusion that inf σ(l (1) n,p) holds as n which proves discreteness of the spectrum for λ < γ p. If λ = γ p the sequence of spectral thresholds no longer diverges but it remains bounded from below and the same is by minimax principle true for the operator L p (λ). P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-35 -

Subcritical behavior proof concluded In this way we find that µ n,p := inf σ(l (3) n,p) satisfies µ n,p n 2 as n Through the chain of inequalities we come to conclusion that inf σ(l (1) n,p) holds as n which proves discreteness of the spectrum for λ < γ p. If λ = γ p the sequence of spectral thresholds no longer diverges but it remains bounded from below and the same is by minimax principle true for the operator L p (λ). Remark It is natural to conjecture that σ(l p (γ p )) R +. There may be a negative discrete spectrum in the critical case; we return to this question a little later. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-35 -

The supercritical case Theorem (E-Barseghyan 12) The spectrum of L p (λ), p 1, is unbounded below from if λ > λ crit. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-36 -

The supercritical case Theorem (E-Barseghyan 12) The spectrum of L p (λ), p 1, is unbounded below from if λ > λ crit. Idea of the proof: Similar as above with a few differences: P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-36 -

The supercritical case Theorem (E-Barseghyan 12) The spectrum of L p (λ), p 1, is unbounded below from if λ > λ crit. Idea of the proof: Similar as above with a few differences: now we seek an upper bound to L p (λ) by a below unbounded operator, hence we impose Dirichlet conditions on concentric circles the estimating operators have now a nonzero contribution from the radial part, however, it is bounded by π 2 independently of n the negative λ-dependent term now outweights the anharmonic oscillator part so that inf σ(l (1,D) n,p ) holds as n P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-36 -

The supercritical case Theorem (E-Barseghyan 12) The spectrum of L p (λ), p 1, is unbounded below from if λ > λ crit. Idea of the proof: Similar as above with a few differences: now we seek an upper bound to L p (λ) by a below unbounded operator, hence we impose Dirichlet conditions on concentric circles the estimating operators have now a nonzero contribution from the radial part, however, it is bounded by π 2 independently of n the negative λ-dependent term now outweights the anharmonic oscillator part so that inf σ(l (1,D) n,p ) holds as n Using suitable Weyl sequences similar to those the previous model, however, we are able to get a stronger result: Theorem (Barseghyan-E-Khrabustovskyi-Tater 15) σ(l p (λ)) = R holds for any λ > γ p and p > 1. P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-36 -

Spectral estimates: bounds to eigenvalue sums Let us return to the subcritical case and define the following quantity: α := 1 ( 1 + ) 2 5 5.236 > γ 1 p 2 P. Exner: Spectral transitions... 4th Najman Conference September 22, 2015-37 -