The emission spectrum of IBr excited in the presence of Argon

Wavelengths and wavenumbers of the band heads in the region 3915–3540 Å are recorded as obtained from the measurements of the plates taken on a first order 21-feet grating spectrograph. Earlier workers recently reported 40 bands of this system covering the region 3900–3800 Å. All the bands of this system obtained in the present experiments are analysed as involving the³ Π (1) state for lower state. The constants for the lower state are such that they represent well the ΔG (v+1/2) values obtained in the present experiments fromv=0 tov=26 as well as those obtained by Brown fromv=9 tov=43. The vibrational constants of the two states involved are:

$$\begin{gathered} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } } \\ {137 \cdot 8 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } x_e ^{\prime \prime } } \\ {0 \cdot 571 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e ^{\prime \prime } y_e ^{\prime \prime } } \\ { - 0 \cdot 1156 cm.^{ - 1} } \\ \end{array} \begin{array}{*{20}c} {\omega _e z_e ^{\prime \prime } } \\ {3 \cdot 09 \times 10^{ - 3} cm.^{ - 1} } \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\omega _e ^{\prime \prime } t_e ^{\prime \prime } } \\ { - 2 \cdot 5 \times 10^{ - 5} cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime } \\ {90 \cdot 1 cm.^{ - 1} ,} \\ \end{array} \begin{array}{*{20}c} {\omega _e ^\prime x_e ^\prime } \\ {0 \cdot 15 cm.^{ - 1} } \\ \end{array} \hfill \\ \end{gathered} $$     

Existence and Concentration of Solutions For Sublinear Schrödinger-Poisson Equations

We concern the sublinear Schrödinger-Poisson equations \(\left\{ \begin{gathered}- \Delta u + \lambda V\left( x \right)u + \phi u = f\left( {x,u} \right)in{\mathbb{R}^3} \hfill \\- \Delta \phi = {u^2}in{\mathbb{R}^3} \hfill \\ \end{gathered} \right.\) where λ > 0 is a parameter, V ∈ C(R³,[0,+∞)), f ∈ C(R₃×R,R) and V^-1(0) has nonempty interior. We establish the existence of solution and explore the concentration of solutions on the set V^-1(0) as λ → ∞ as well. Our results improve and extend some related works.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Boundary value problems for complete quasi-parabolic differential equations of odd order with variable domains of operators

We prove the well-posed solvability in the strong sense of the boundary value Problems

$$\begin{gathered} ( - 1)\frac{{_m d^{2m + 1} u}}{{dt^{2m + 1} }} + \sum\limits_{k = 0}^{m - 1} {\frac{{d^{k + 1} }}{{dt^{k + 1} }}} A_{2k + 1} (t)\frac{{d^k u}}{{dt^k }} + \sum\limits_{k = 1}^m {\frac{{d^k }}{{dt^k }}} A_{2k} (t)\frac{{d^k u}}{{dt^k }} + \lambda _m A_0 (t)u = f, \hfill \\ t \in ]0,t[,\lambda _m \geqslant 1, \hfill \\ {{d^i u} \mathord{\left/ {\vphantom {{d^i u} {dt^i }}} \right. \kern-\nulldelimiterspace} {dt^i }}|_{t = 0} = {{d^j u} \mathord{\left/ {\vphantom {{d^j u} {dt^j }}} \right. \kern-\nulldelimiterspace} {dt^j }}|_{t = T} = 0,i = 0,...,m,j = 0,...,m - 1,m = 0,1,..., \hfill \\ \end{gathered} $$

where the unbounded operators A _s (t), s > 0, in a Hilbert space H have domains D(A _s (t)) depending on t, are subordinate to the powers A ^1?(s?1)/2m (t) of some self-adjoint operators A(t) ≥ 0 in H, are [(s+1)/2] times differentiable with respect to t, and satisfy some inequalities. In the space H, the maximally accretive operators A ₀(t) and the symmetric operators A _s (t), s > 0, are approximated by smooth maximally dissipative operators B(t) in such a way that

$$\begin{gathered} \mathop {lim}\limits_{\varepsilon \to 0} Re(A_0 (t)B_\varepsilon ^{ - 1} (t)(B_\varepsilon ^{ - 1} (t))^ * u,u)_H = Re(A_0 (t)u,u)_H \geqslant c(A(t)u,u)_H \hfill \\ \forall u \in D(A_0 (t)),c > 0, \hfill \\ \end{gathered} $$

, where the smoothing operators are defined by

$$B_\varepsilon ^{ - 1} (t) = (I - \varepsilon B(t))^{ - 1} ,(B_\varepsilon ^{ - 1} (t)) * = (I - \varepsilon B^ * (t))^{ - 1} ,\varepsilon > 0.$$

.

     

О некоторых свойства х частичных сумм рядо в Фурье—Уолша—Пэли

In this paper we consider the behaviour of partial sums of Fourier—Walsh—Paley series on the group62-01. We prove the following theorems: Theorem 1. Let {n _k } _k ^=1/∞ be some increasing convex sequence of natural numbers such that $$\mathop {\lim sup}\limits_m m^{ - 1/2} \log n_m< \infty $$ . Then for anyf∈L ^∞(G) $$\left( {\frac{1}{m}\sum\limits_{j = 1}^m {|Sn_j (f;0)|^2 } } \right)^{1/2} \leqq C \cdot \left\| f \right\|_\infty $$ . Theorem 2. Let {n _k } _k ^=1/∞ be a lacunary sequence of natural numbers,n _k+1/n _k≧q>1. Then for anyfεL ^∞(G) $$\sum\limits_{j = 1}^m {|Sn_j (f;0)| \leqq C_q \cdot m^{1/2} \cdot \log n_m \cdot \left\| f \right\|_\infty } $$ . Theorems. Let µ _k =2 ^k +2 ^k-2+2 ^k-4+...+2^α ₀,α ₀=0,1. Then $$\begin{gathered} \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in L^\infty (G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = 0(m)^2 \} .} \hfill \\ \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = \{ \{ a_k \} _{k = 1}^\infty ;\sum\limits_{k = 1}^m {a_k^2 = o(m)^2 \} = } \hfill \\ = \{ \{ S_{\mu _k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} \hfill \\ \end{gathered} $$ . Theorem 4. {{S ₂ k(f: 0)} _k ^=1/∞ ,f∈L ^∞(G)}=m. $$\{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G)\} = c. \{ \{ S_{2_k } (f:0\} _{k = 1}^\infty ;f \in C(G),f(0) = 0\} = c_0 $$ .     

A note on some results of Hua in short intervals

In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $

where U = N ^1/2?η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2k ^?1, k ? 2.

     

Divided differences for functions of two variables for irregularly spaced arguments

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inx _i ^j y _i ^k . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$     

The number of strictly increasing mappings of fences

Let X and Y be fences of size n and m, respectively and n, m be either both even or both odd integers (i.e., |m-n| is an even integer). Let \(r = \left\lfloor {{{(n - 1)} \mathord{\left/ {\vphantom {{(n - 1)} 2}} \right. \kern-0em} 2}} \right\rfloor\) . If 1<n<-m then there are \(a_{n,m} = (m + 1)2^{n - 2} - 2(n - 1)(\begin{array}{*{20}c} {n - 2} \\ r \\ \end{array} )\) of strictly increasing mappings of X to Y. If 1<-m<-n<-2m and s=1/2(n?m) then there are a _n,m+b _n,m+c _n of such mappings, where $$\begin{gathered} b_{n,m} = 8\sum\limits_{i = 0}^{s - 2} {\left( {\begin{array}{*{20}c} {m + 2i + 1} \\ l \\ \end{array} } \right)4^{s - 2 - 1} } \hfill \\ {\text{ }}c_n = \left\{ \begin{gathered} \left( {\begin{array}{*{20}c} {n - 1} \\ {s - 1} \\ \end{array} } \right){\text{ if both }}n,m{\text{ are even;}} \hfill \\ {\text{ 0 if both }}n,m{\text{ are odd}}{\text{.}} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $$      

Electron paramagnetic resonance of mn2+ in single crystals of Cesium Sulphate

Electron paramagnetic resonance (EPR) of divalent manganese ion has been studied at 9.5 KMc/sec. These studies reveal that Mn²⁺ ion substitutes at a β-Cs⁺ site and gets associated with a vacancy at a neighbouring β-Cs⁺ site in the aft-plane. Thez-axis of this ββ complex makes an angle of 25° with the crystallographicb-axis. The spectra observed have been described by the spin-Hamiltonian for Mn²⁺ in rhombic crystalline field. The temperature dependence of the parameters D and E has been studied in the range 293°–77° K. The spectra for the observed complex along its three-principal axes have been analysed using second order perturbation equations. The spin-Hamiltonian parameters obtained from the spectra taken at room temperature are: $$\begin{gathered} g_z = 2 \cdot 000 \pm 0 \cdot 003, g_x = 2 \cdot 015 \pm 0 \cdot 003, g_y = 2 \cdot 000 \pm 0 \cdot 003; \hfill \\ A_z = - 93 \pm 1, A_x = - 91 \pm 1, and A_y = - 91 \pm 1 G; \hfill \\ D = - 941 \pm 3 and E = - 14 \pm 4 G; \hfill \\ b^\circ _4 = \left( {0 \cdot 0} \right), b_4 ^2 = \left( {13} \right), and b_4 ^4 = - 77 \pm 5 G \hfill \\ \end{gathered} $$      

The D2Σ+ − X2Σ+ system of A10

Eleven bands of A10 belonging to the system (D² Σ ⁺-X² Σ ⁺) in the ultra-violet region have been analysed for their rotational structure. These are the 0-2, 1-3, 2-4, 0-3, 1-4, 3-0, 4-1, 3-1, 4-0, 5-1 and 6-1 bands lying at 2611.8Å, 2620.7Å, 2629.4Å, 2677.4Å, 2685.7Å, 2347.7Å, 2358.3Å, 2402.2Å, 2305.8Å, 2316.7Å and 2277.3Å respectively. The spin-splitting for the D² Σ ⁺ state has been determined from the doubling of the rotational lines observed for the bands 0-2, 1-3, 2-4 and 1-4. The rotational and vibrational constants (in cm^?1) evaluated for the D² Σ ⁺ state are: $$\begin{gathered} T_a = 40267 \cdot 6 \hfill \\ G(V) = 817 \cdot 47 (v + 1/2) - 4 \cdot 795 (v + 1/2)^2 - 0 \cdot 1107 (v + 1/2)^3 \hfill \\ B_v = 0 \cdot 56522 - 0 \cdot 0046 (v + 1/2) - 0 \cdot 00005 (v + 1/2)^2 \hfill \\ \gamma = 0 \cdot 004 \pm 0 \cdot 002 \hfill \\ \sigma = - 0 \cdot 4 \pm 0 \cdot 1 \times 10^{ - 6} \hfill \\ \end{gathered} $$      

On some results of Hua in short intervals

In this note, we prove some results of Hua in short intervals. For example, each sufficiently large integer N satisfying some congruence conditions can be written as

$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k}}, \hfill \\ {\left| {{p_j} - \sqrt {N/5} } \right| \leqslant U,\left| {p - {{\left( {N/5} \right)}^{\tfrac{1}{k}}}} \right|\leqslant UN - \tfrac{1}{2} + \tfrac{1}{k},j = 1,2,3,4,} \hfill \\ \end{array} } \right. $

where \( U = N\tfrac{1}{2} - \eta + \varepsilon \) with \( \eta = \frac{2}{{\kappa \left( {K + 1} \right)\left( {{K^2} + 2} \right)}} \) and \( K = {2^{k - 1}},k\geqslant 3. \)

     

Interpolation by polyhedral functions

A polyhedral functionlp(Δ_n) (f). interpolating a function f, defined on a polygon Φ, is defined by a set of interpolating nodes Δ_n ?Φ and a partition P(Δ_n) of the polygon Φ into triangles with vertices at the points of Δ_n. In this article we will compute for convex moduli of continuity the quatities $$\begin{gathered} E (H_\Phi ^\omega ; P (\Delta _n )) = sup || f - l_{p(\Delta _n )} (f)||, \hfill \\ f \in H_\Phi ^\omega \hfill \\ \end{gathered} $$ and also give an asymptotic estimate of the quantities $$\begin{gathered} E_n (H_\Phi ^\omega ) = infinf E (H_\Phi ^\omega ; P (\Delta _n )). \hfill \\ \Delta _n P(\Delta _n ) \hfill \\ \end{gathered} $$      

A decomposition for someU-type statistics

Define , $S_{k,n} = \Sigma _{1 \leqslant i_1< \cdot \cdot \cdot< l_k \leqslant n} X_{i_1 } \cdot \cdot \cdot X_{i_k } ,n \geqslant k \geqslant {\text{1}}$ where {X, X _n ,n≥1} are i.i.d. random variables withEX=0,EX ²=1 and letH _k (·) denote the Hermite polynomial of degreek. By establishing an LIL for products of correlated sums of i.i.d. random variables, the a.s. decomposition $$\begin{gathered} k!S_{k,n} = n^{k/2} H_k (S_{1n} /n^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ) - \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)S_{1.n}^{k - 2} \sum\limits_{i = 1}^n {(X_i^2 - 1)} \hfill \\ + O(n^{(k - 1)/2} (\log \log n)^{(k - 3/2} ) \hfill \\ \end{gathered} $$ valid whenEX ⁴<∞, elicits an LIL forη _k,n =k!S _k,n ?n ^k/2 H _k (S _1n /n ^1/2) under a reduced normalization. Moreover, whenE|X| ^p <∞ for somep in [2, 4], a Marcinkiewicz-Zygmund type strong law forη _k,n is obtained, likewise under a reduced normalization.     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

Hyperbolas over two-dimensional Fibonacci quasilattices

For the number n _s (α, β; X) of points (x ₁ , x ₂) in the two-dimensional Fibonacci quasilattices \( \mathcal{F}_m^2 \) of level m?=?0, 1, 2,… lying on the hyperbola x ₁ ² ? ??αx ₂ ² ?=?β and such that 0?≤?x ₁? ≤?X, x ₂? ≥?0, the asymptotic formula

$ {n_s}\left( {\alpha, \beta; X} \right)\sim {c_s}\left( {\alpha, \beta } \right)\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty $

is established, and the coefficient c _s (α, β) is calculated exactly. Using this, we obtain the following result. Let F _m be the Fibonacci numbers, A _i ∈ \( \mathbb{N} \), i?=?1, 2, and let \( \overleftarrow {{A_i}} \) be the shift of A _i in the Fibonacci numeral system. Then the number n _s (X) of all solutions (A ₁ , A ₂) of the Diophantine system

$ \left\{ {\begin{array}{*{20}{c}} {A_1^2 + \overleftarrow {A_1^2} - 2{A_2}{{\overleftarrow A }_2} + \overleftarrow {A_2^2} = {F_{2s}},} \\ {\overleftarrow {A_1^2} - 2{A_1}{{\overleftarrow A }_1} + A_2^2 - 2{A_2}{{\overleftarrow A }_2} + 2\overleftarrow {A_2^2} = {F_{2s - 1}},} \\ \end{array} } \right. $

0?≤?A ₁? ≤?X, A ₂? ≥?0, satisfies the asymptotic formula

$ {n_s}(X)\sim \frac{{{c_s}}}{{{\text{ar}}\cosh \left( {{{1} \left/ {\tau } \right.}} \right)}}\ln X\,\,\,\,{\text{as}}\,\,\,\,X \to \infty . $

Here τ?=?(?1?+?√5)/2 is the golden ratio, and c _s ?=?1/2 or 1 for s?=?0 or s?≥?1, respectively.

     

Enumerating rooted simple planar maps

The main purpose of this paper is to find the number of combinatorially distinct rooted simpleplanar maps,i.e.,maps having no loops and no multi-edges,with the edge number given.We haveobtained the following results.1.The number of rooted boundary loopless planar [m,2]-maps.i.e.,maps in which there areno loops on the boundaries of the outer faces,and the edge number is m,the number of edges on theouter face boundaries is 2,is(?)for m≥1.G_0~N=0.2.The number of rooted loopless planar [m,2]-maps is(?)3.The number of rooted simple planar maps with m edges H_m~s satisfies the following recursiveformula:(?)where H_m~(NL) is the number of rooted loopless planar maps with m edges given in [2].4.In addition,γ(i,m),i≥1,are determined by(?)for m≥i.γ(i,j)=0,when i>j.     

Fractal relaxed Dirichlet problems

Given aself similar fractal K ? ? ⁿ of Hausdorff dimension α>n?2, andc ₁>0, we give an easy and explicit construction, using the self similarity properties ofK, of a sequence of closed sets? _h such that for every bounded open setΩ?? ⁿ and for everyf ∈ L²(Ω) the solutions to $$\left\{ \begin{gathered} - \Delta u_h = f in \Omega \backslash \varepsilon _h \hfill \\ u_h = 0 on \partial (\Omega \backslash \varepsilon _h ) \hfill \\ \end{gathered} \right.$$ converge to the solution of the relaxed Dirichlet boundary value problem $$\left\{ \begin{gathered} - \Delta u + uc_1 \mathcal{H}_{\left| K \right.}^\alpha = f in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ (H _∣ ^α denotes the restriction of the α-dimensional Hausdorff measure toK). The condition α>n?2 is strict.     

Nontrivial solution of a quasilinear elliptic equation with critical growth in ℝn

Suppose Δ_n u = div (¦ ?u ¦^n-2?u) denotes then-Laplacian. We prove the existence of a nontrivial solution for the problem $$\left\{ \begin{gathered} - \Delta _n u + \left| u \right|^{n - 2} u = \int {(x,u)u^{n - 2} in \mathbb{R}^n } \hfill \\ u \in W^{1,n} (\mathbb{R}^n ) \hfill \\ \end{gathered} \right.$$ wheref(x, t) =o(t) ast → 0 and ¦f(x, t)¦ ≤C exp(α_n¦t¦^n/(n-1)) for some constantC > 0 and for allx∈?;t∈? with α_n =nω _n ^1/(n-1) , ω_n = surface measure ofS ^n-1.     

Infinitesimal form boundedness and Trudinger’s subordination for the Schrödinger operator

We give explicit analytic criteria for two problems associated with the Schrödinger operator H=-Δ+Q on L²(? ⁿ ) where Q∈D’(? ⁿ ) is an arbitrary real- or complex-valued potential.

First, we obtain necessary and sufficient conditions on Q so that the quadratic form \(\langle{Q}\cdot,\ \cdot\rangle\) has zero relative bound with respect to the Laplacian. For Q∈L¹_loc(? ⁿ ), this property can be expressed in the form of the integral inequality:

$\left\vert\int_{\mathbb{R}^n} |u(x)|^2 Q(x) dx \right\vert\leq\epsilon\| \nabla u \|^2_{L^2(\mathbb{R}^n)} + C(\epsilon) \|u \|^2_{L^2(\mathbb{R}^n)}, \quad\forall u \in C^{\infty}_0(\mathbb{R}^n),$

for an arbitrarily small ε>0 and some C(ε)>0. One of the major steps here is the reduction to a similar inequality with nonnegative function \(|\nabla(1-\Delta)^{-1} Q|^2 + |(1-\Delta)^{-1} Q|\) in place of Q. This provides a complete solution to the infinitesimal form boundedness problem for the Schrödinger operator, and leads to new broad classes of admissible distributional potentials Q, which extend the usual L ^p and Kato classes, as well as those based on the well-known conditions of Fefferman–Phong and Chang–Wilson–Wolff.

Secondly, we characterize Trudinger’s subordination property where C(ε) in the above inequality is subject to the condition C(ε)≤cε^-β(β>0) as ε→+0. Such quadratic form inequalities can be understood entirely in the framework of Morrey–Campanato spaces, using mean oscillations of \(\nabla(1-\Delta)^{-1}Q\) and \((1-\Delta)^{-1}Q\) on balls or cubes. A version of this condition where ε∈(0,+∞) is equivalent to the multiplicative inequality:

$\left\vert\int_{\mathbb{R}^n} |u(x)|^2Q(x)dx\right\vert\leq{C}\|\nabla{u}\|^{2p}_{L^2(\mathbb{R}^n)}\|u\|^{2(1-p)}_{L^2(\mathbb{R}^n)},\quad\forall{u}\in{C}^\infty_0(\mathbb{R}^n),$

with \(p=\frac\beta{1 + \beta}\in(0,1)\). We show that this inequality holds if and only if \(\nabla\Delta^{-1} Q \in{BMO}(\mathbb{R}^n)\) if \(p=\frac{1}{2}\). For \(0 < p < \frac{1}{2}\), it is valid whenever \(\nabla\Delta^{-1}Q\) is Hölder-continuous of order 1-2p, or respectively lies in the Morrey space \(\mathcal{L}^{2,\lambda}\) with λ=n+2-4p if \(\frac{1}{2} < p < 1\). As a consequence, we characterize completely the class of those Q which satisfy an analogous multiplicative inequality of Nash’s type, with \(\|u\|_{L^1(\mathbb{R}^n)}\) in placeof \(\|u\|_{L^2(\mathbb{R}^n)}\).

These results are intimately connected with spectral theory and dynamics of the Schrödinger operator, and elliptic PDE theory.     

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ? ₊ ^N with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ?? 3, ?? > 0, 0 < T ?? ??, and u ₀ ?? C ₀(? ₊ ^N ). We prove the existence of a threshold number ?? _N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ? ₊ ^N .